Unemployment and Endogenous Reallocation over the Business Cycle

Unemployment and Endogenous Reallocation over the Business Cycle November 2021 Unemployment and Endogenous Reallocation over the Business Cycle *† October 2021 Abstract This paper studies the extent to which the cyclicality of occupational mobility shapes that of aggregate unemployment and its duration distribution. Using the SIPP, we document the relation between workers’ occupational mobility and unemployment duration over the long run and business cycle. To interpret this evidence, we develop a multi-sector business cycle model with heterogenous agents. The model is quantitatively consistent with several important features of the US labor market: procyclical gross and countercyclical net occupational mobility, the large volatility of unemployment and the cyclical properties of the unemployment duration distribution, among others. Our analysis shows that occupational mobility due to workers’ changing career prospects interacts with aggregate conditions to drive fluctuations of aggregate unemployment and its duration distribution. Keywords: Unemployment, Business Cycle, Rest, Search, Occupational Mobility. JEL: E24, E30, J62, J63, J64. 1 Introduction Occupational mobility is an important part of an unemployed worker’s job finding process. On average 44% of workers who went through a spell of unemployment in the US changed “major occupational groups” at re-employment.1 These occupation movers also take longer to find a job and contribute to the cyclical changes in long-term unemployment. For every extra month it takes an occupation stayer to find a job during a downturn, movers take 40% longer. This suggests that the willingness and ability of individuals to move across different sectors of the economy can have important consequences for aggregate labor market fluctuations. This paper builds on this evidence and studies the implications of unemployed workers occupational mobility for the cyclical behaviour of the unemployment duration distribution and the aggregate unemployment rate. We propose and quantitatively assess a multi-sector business cycle model in which the unemployed face search frictions in, and reallocation frictions across, heterogeneous occupations. The economy we consider further exhibits idiosyncratic worker-occupation productivity shocks, orthogonal to occupation-wide productivities, to capture the evolving career prospects of a worker within an occupation. Workers accumulate occupation-specific human capital through learning-by-doing, but face skill loss during unemployment. Even with this rich level of heterogeneity, workers’ job separations and reallocation decisions can be characterised by simple reservation productivity cutoffs that respond to aggregate and occupational-wide productivities. A key success of the framework is that it can simultaneously generate the observed cyclical fluctuations of aggregate unemployment and its duration distribution as well as a strongly downwardsloping Beveridge curve. Underlying these fluctuations, the cyclical responses of the model’s aggregate job separation and job finding rates are in line with the data (see Shimer, 2005, Hall and Milgrom, 2008, and Hagedorn and Manovskii, 2008). In addition, the model generates the observed procyclicality of gross occupational mobility among the unemployed and the stronger countercyclicality of unemployment duration among occupational movers. It also generates the observed increase in net reallocation of workers across occupations during recessions (see Dvorkin, 2014, Pilossoph, 2014, Wiczer, 2015 and Chodorow-Reich and Wieland, 2020). Our approach provides a novel insight. We find that it is the interaction between worker’s evolving career prospects within an occupation and aggregate conditions, and not occupation-wide productivity differences, that drive cyclical unemployment. The main mechanism is as follows. The estimation implies that within each occupation the job separation cutoffs consistently lie above the reallocation cutoffs. With uncertain returns and costly reallocation, those unemployed with idiosyncratic productivities between the cutoffs prefer the option of waiting and remaining attached to their pre-separation occupations instead of reallocating. During recessions the area between these cutoffs widens endogenously and workers spend a longer period of their jobless spells waiting even though there are no jobs posted for them. This drives up (long-term) unemployment more for occupational movers than stay1 Major occupational groups are broad categories that can be thought of as representing one-digit occupations. For example, managers, sales, mechanic and repairers, construction/extraction, office/admin support, elementary trades, etc. The above proportion is obtained after correcting for measurement error. 1 ers during recessions, and helps create strong cyclical amplification in the aforementioned aggregate labor market variables. The importance of idiosyncratic career productivity shocks in the model’s mechanism reflects the prominence of excess mobility, i.e. moves that cancel each other out at the occupation level, in driving the unemployed’s occupational mobility patterns in the data. We use the observed high propensity to change occupations and its increase with unemployment duration to uncover the stochastic process of the idiosyncratic career shocks. The estimated process then shapes workers’ incentive to wait, generates procyclical excess and gross mobility inline with the data, and determines the cyclical performance of the model. A prominent literature of multi-sector models in the spirit of Lucas and Prescott (1974) “islands” framework typically emphasise countercyclical net reallocation of unemployed workers across sectors (or “islands”) as the main underlying force behind unemployment fluctuations (see Lilien, 1982, Rogerson, 1987). Countercyclical unemployment can arise when more workers engage in time consuming switches from sectors that have been affected harder in a recession to those sectors which offer relatively higher job finding prospects. To incorporate these insights we use an imperfect directed search approach to model search across occupations over the business cycle (see also Cheremukhin et al. 2020 and Wu, 2020). Nevertheless, as gross occupational mobility flows are an order of magnitude greater than net flows, adding this dimension does not change the importance of workers’ evolving career prospects over occupation-wide productivities in explaining labor market fluctuations or the procyclical nature of occupational mobility. This occurs because the option value of waiting in the pre-separation occupation remains important within (cyclically) declining and expanding occupations. Although net mobility has a small role in explaining aggregate unemployment fluctuations, it has a clear cyclical pattern. During recessions a higher proportion of workers lose their jobs in routine manual occupations and do not come back to these jobs; while a higher proportion of workers find jobs in non-routine manual occupations at re-employment. We show that these patterns substantially contribute to the long-run decline of the employment share of routine occupations and long-run increase in the employment share of non-routine occupations (see also Cortes et al., 2020). Hence, there is no contradiction between changing career prospects playing a very important role in shaping cyclical unemployment, and worker flows through unemployment contributing meaningfully to the changing sizes of occupations particularly during recessions. The empirical study of occupational (or industry) mobility focused exclusively on workers who went through unemployment has received relatively little attention. This is in contrast to the larger amount of research investigating occupational mobility among pooled samples of employer movers and stayers (see Jovanovic and Moffitt, 1990, Kambourov and Manovskii, 2008, and Moscarini and Thomsson, 2007, among others).2 There is no reason, a priori, to conclude that the mobility patterns uncovered by these studies apply to the unemployed. Therefore, we use data from the Survey of Income and Programme Participation (SIPP) between 1983-2014 to document relevant patterns link2 A few recent exceptions are Şahin et al. (2014), Fujita and Moscarini (2017), Carrillo-Tudela et al. (2016), Faberman and Kudlyack (2019) and Huckfeldt (2021). 2 ing individuals’ occupational mobility with their unemployment duration outcomes. We also use the Panel Survey for Income Dynamics (PSID) and the Current Population Survey (CPS) to corroborate our results. As the levels of gross and excess occupational mobility are crucial for our analysis, a major concern is the extent to which coding errors creates spurious mobility and inflate our statistics (see Kambourov and Manovskii, 2008, and Moscarini and Thomsson, 2007). We show that one cannot use existing correction estimates based on samples pooling all workers when attempting to correct the occupational mobility of the unemployed. Instead we develop a novel classification error model that allows us to estimate the extent of coding error at the level of each occupation. We calibrate our model using simulation method of moments and find that the nature of unemployment changes over the cycle. Rest/wait unemployment becomes relative more prominent in recession and search unemployment in expansions.3 Alvarez and Shimer (2011) also study the relative importance of rest and search unemployment using a multi-sector model, but in an aggregate steady state. Their analysis implies that the individual transitions between work, rest and search are not determined.4 In contrast, the estimated dynamics of workers’ career prospects in our framework determines the transitions between employment and the different types of unemployment. This allows us to analyse the relationship between unemployment duration, occupational mobility and job finding probabilities, both in the long-run and over the cycle. The large and persistent rise in unemployment observed during and in the aftermath of the Great Recession generated a renewed interest in multi-sector business cycle models as useful frameworks to investigate cyclical unemployment. Like in our paper, Pilossoph (2014) finds a muted effect of net reallocation across sectors on aggregate unemployment. Chodorow-Reich and Wieland (2020) build on this work and link net reallocation of workers across industries/locations to increases in total unemployment. They find this link only for the recession-to-recovery phase of the cycle, arguing for a crucial role of wage rigidity. In these papers, gross mobility is constant or countercyclical, which is at odds with the data.5 Their focus is also not on the cyclical unemployment duration patterns nor cyclical patterns in the relationship between individuals’ unemployment duration and their occupational mobility, features that are central to our paper. Closer to our analysis is Wiczer (2015), who studies the role of occupations on long-term unemployment over the cycle in a multi-sector model. In contrast, unemployed workers in our framework take into account the potential recovery of their occupational productivities when making job separations and occupational mobility decision. It is this feature that takes us a long way in replicating the overall volatility of cyclical unemployment, while remaining consistent with the cyclical behaviour of the short and long-term unemployed. 3 The concept of rest/wait unemployment was introduced by Jovanovic (1987), using a business cycle multi-sector model. However, he did not link it to occupational mobility, the unemployment duration distribution, or investigated its quantitative properties. See also Hamilton (1988) and Gouge and King (1997). 4 With perfectly competitive labor markets workers in their model are considered search unemployed when in transit between sectors and are indifferent between work and rest or between work, rest and search. 5 To the best of our knowledge Dvorkin (2014) is the only one who attempts to reproduce the procyclicality of gross mobility together with the countercyclicality of net mobility. However, his calibrated model generates nearly acyclical series and hence is not able to reproduce the observe strong cyclicality of these series (see his Table 9). 3 As we are interested on how occupational mobility affects unemployment fluctuations, worker heterogeneity in our model is naturally time variant. There is also a growing literature that incorporates time-invariant worker heterogeneity to the Mortensen and Pissarides (1994) model to generate enough cyclical volatility in aggregate unemployment (see Bils et al., 2012, Chassambouli, 2013 and Murtin and Robin, 2018). We share with these papers that some unemployed workers do not provide incentive for vacancies to enter the labor market during periods in which their productivities lie below the job separation cutoff. These models, however, do not capture that during recessions the stronger lengthening of unemployment spells among the larger group of occupation movers significantly contributes to the increase in long-term unemployment. The addition of the reallocation cutoff enable us to explain the cyclical behaviour of short and long-term unemployment of occupational movers and stayers. In addition to workers’ evolving career prospects within an occupation, occupational human capital plays an important in our analysis. Like in Kambourov and Manovskii (2009a) and Alvarez and Shimer (2012), it generates an additional waiting motive that implies older, more experienced workers tend to switch occupations less than younger less experienced ones. Different to these papers, however, differences in human capital implies that during recessions the composition of unemployment and separations moves towards the (on-average) more productive group of prime-aged workers (see Mueller, 2017). The rest of the paper proceeds as follows. Section 2 presents the empirical evidence that motivates our paper. Section 3 describes and characterises the model and its main implications. In Sections 4 and 5 we quantitatively assess this model and show the importance of changing career prospect in explaining cyclical unemployment outcomes. Section 6 concludes. All proofs, detailed data, quantitative analysis and robustness exercises are relegated to several online appendices. 2 Occupational Mobility of the Unemployed Our main statistical analysis is based on the sequence of 1984-2008 SIPP panels, covering the 19832014 period. The sample restricts attention to those workers who were observed transiting from employment to unemployment and back within a given panel (EU E flows), and excludes those in self-employment, in the armed forces, or in the agricultural occupations.6 In our baseline analysis we consider workers who have been unemployed throughout their non-employment spells, but show that our main results also hold when using mixed unemployment/out-of-labor-force spells. To minimize the effects of censoring that arise due to the SIPP structure, we consider unemployment spells for which re-employment occurs as from month 16 since the start of the corresponding SIPP panel and impose that workers at the moment of re-employment have at least 14 months of continuous labor market history within their panel. In Supplementary Appendix B.7 we provide further details and 6 The self-employed are not included in our analysis as they might face a very different frictional environment and choices than those in dependent employment. Indeed, we find that 50% of those who transited from self-employment to unemployment in the SIPP went to back self-employment. This suggests that self-employment begets self-employment, a feature we do not capture in our model. On the other hand, 96% of those who transited from dependent employment into unemployment returned to dependent employment and are captured in our model. 4 analyse the implications of these restrictions. We found that among all unemployment to employment transitions, only about 5% transited into self-employment. Furthermore, 50% of those who transited from self-employment to unemployment went to back self-employment. This suggests that self-employment begets self-employment. An individual is considered unemployed if he/she has not been working for at least a month after leaving employment and reported “no job - looking for work or on layoff”. Since we want to focus on workers who have become unattached from their previous employers, we consider those who report to be “with a job - on layoff”, as employed.7 After dropping all observations with imputed occupations, we compare each workers’ reported occupations before and after the non-employment spell. To capture meaningful career changes we use the 21 “major” occupational groups of the 2000 Census Occupational Classification (2000 SOC) as well as their aggregation into the task-based occupational categories proposed by Autor and Dorn (2013) and Cortes et al. (2020). In the SIPP, however, the occupation information of a worker newly hired from unemployment is collected under independent interviewing, which is known to generate occupational coding errors.8 Without correcting for miscoding we could potentially be inflating the importance of occupational mobility among the unemployed. We address this issue by developing a classification error model, which we briefly present in the next subsection. After adjusting our data for misclassification error we use the relationship between occupational mobility and unemployment duration to investigate the degree of “attachment” workers have to their pre-separation occupations and how it evolves with their spell duration. We also investigate how this attachment differs across demographic groups, occupational categories and across unemployment spells, how it depends on excess and net mobility (defined below) and the business cycle. Supplementary Appendix B present a more detailed analysis of our long-run and business cycle findings, as well as extensive robustness exercises and provides details on the data construction and measurement. 2.1 Correcting for Coding Errors in Occupation Mobility In order to correctly measure the level and cyclicality of excess and net occupational mobility we propose an approach that allows us (i) to correct for the potentially large heterogeneity in (the propensity of) coding errors in the flows between particular occupations, and thereby capture more accurately coding errors for those occupations that weigh more among the unemployed; (ii) to correct for the effect of miscoding on net mobility; (iii) to correct sequential occupational mobility observations across 7 Fujita and Moscarini (2017) find that the unemployed (as typically defined by the BLS) consist of two groups that behave very differently: “temporary laid-off workers” and “permanent separators”. The latter group are those who lost their job with no indication of recall. Similarly, Hornstein (2013) and Ahn and Hamilton (2018) consider two groups among the unemployed in terms of fixed characteristics: those with “high job finding rates” and those with “low job finding rates”. Excluding from our unemployment measure those workers who are “with a job - on layoff” and those who find employment within a month means that our unemployment sample is close to Fujita and Moscarini’s “permanent separators” sample and to Hornstein’s and Ahn and Hamilton’s “low job finding rate” workers. In Supplementary Appendix B.4.4, we further discuss this issue. 8 This implies that the occupational question is asked without reference to the answers giving by the respondent in previous interviews. A professional coder then assigns an occupational code based on the respondent’s answer, also without reference to occupational codes previously assigned or previous answers given by the same respondent. 5 two unemployment spells, where a single coding mistake can create two spurious moves; and (iv) to easily incorporate it in our quantitative analysis. Suppose that coding errors are made according to a garbling matrix Γ of size OxO, where O denotes the number of occupational categories. The element γij is the probability that the true occuP pation i = 1, 2, ..., O is coded as occupation j = 1, 2, ..., O, such that O j=1 γij = 1. Let M denote the matrix that contains workers’ true occupational flows, where element mij is the flow of workers from occupation i to occupation j. Under independent interviewing such a matrix appears as MI = Γ′ MΓ, where the pre- and post-multiplication by Γ takes into account that the observed occupations of origin and destination would be subject to coding error. Knowledge of Γ (and of its invertibility) allows us to de-garble M as Γ−1′ MI Γ−1 .9 Online Appendix A and Supplementary Appendix A describes formally this correction methodology. There we prove that Γ can be identified and estimated from our data by making three assumptions. (A1) Independent classification errors: conditional on the true occupation, the realization of an occupational code does not depend on workers’ labor market histories, demographic characteristics or the time it occurred in our sample. (A2) “Detailed balance” in miscoding: coding mistakes are symmetric in that the number of workers whose true occupation i gets mistakenly coded as j is the same as the number of workers whose true occupation j gets mistakenly coded as i. (A3) Strict diagonal dominance: It is more likely to correctly code occupation i than to miscoded it. In Supplementary Appendix A we also use SIPP, PSID and CPS data to evaluate the plausibility of these assumptions. We then implement our method using the change from independent to dependent interviewing that occurred between the 1985 and 1986 SIPP panels. Applying the Γ-correction to the occupational flows of workers who go through unemployment results in an average miscoding of about 10% each time information is collected when using “major” occupational categories of the 2000 SOC. This implies that at re-employment true occupational stayers have on average about a 20% chance of appearing as occupational movers. Further, we do find that different occupations have very different propensities to be assigned a wrong code and, given a true occupation, some coding mistakes are much more likely than others. This matters for our measures of net mobility, where we find a sizeable relative increase in net mobility after correction. Supplementary Appendix A also presents an alternative correction based on the PSID retrospective occupation - industry supplementary data files (see also Kambourov and Manovskii, 2008) to evaluate the robustness of our Γ-correction. We show that the level and cyclicality of the Γ-corrected occupational mobility rate at re-employment are in line with the ones derived from the PSID. In Supplementary Appendix B we use the SIPP to provide further robustness based on two alternative measures of occupational mobility: (i) simultaneously mobility of major occupational and major industrial groups at re-employment and (ii) self-reported duration of occupational tenure obtained from the topical modules. The first measure is considered less sensitive to miscoding as it typically requires 9 This formulation builds on Poterba and Summers (1986) and Abowd and Zellner (1985), who focus on miscoding of labor force status. They are able to directly observe miscoding from CPS re-interviews, where discrepancies in labor force status are explicitly reconciled by the Census, under the assumption that re-interviews uncovers the true worker’s status. In contrast, our challenge is that we do not observe the garbling matrix of occupations directly from the data. 6 Figure 1: Extent of occupational mobility by unemployment duration (b) Age groups - Major occupations .1 .1 Proportion Moving Occupation Upon Exit U .3 .4 .6 .2 .5 Proportion Moving Occupation Upon Exit U .2 .3 .4 .5 .6 .7 .7 (a) Overall 1 2 3 4 5 6 7 8 9 Time in Unemployment (months) G-corrected Major Occupation Groups (2000 Clsf) 4-cat. NR/Rout Cogn/Man 10 11 12 1 2 3 4 5 6 7 8 Time in Unemployment G-corrected Young (20-30yo) Prime-aged (35-55yo) Uncorrected 95% CI Smoothed 9 10 11 12 Code-error Uncorrected Young (raw data) Prime-aged (raw data) Notes: Each mobility-duration profile shows for a given unemployment duration x, the proportion of workers who changed occupations at reemployment among all workers who had unemployment spells which lasted at least x months. errors to be made simultaneously along two dimensions. The second captures the worker’s own perception of occupational mobility and is not based on occupational coding. We find a very consistent picture across all methods. In what follows, all statistics are corrected for miscoding, unless otherwise stated. 2.2 Gross Occupational Mobility and Unemployment Duration We now document the degree of attachment workers have to their pre-separation occupation as their unemployment duration increases. In Figure 1 we pool the SIPP panels to generate mobility-duration profiles. They show, for a given unemployment duration x, the proportion of workers who changed occupations at re-employment among all workers who had unemployment spells which lasted at least x months. Figure 1a shows that 44.4% of workers who had at least one month in unemployment changed occupation at re-employment, while 53.7% of workers who had at least 9 months in unemployment changed occupation at re-employment. This evidence thus shows that gross occupational mobility at re-employment is high and increases moderately with unemployment duration. The moderate increase implies that a large proportion of long-term unemployed, over 45%, still return to their previous occupation at re-employment.10 The figure shows that a similar pattern arises when considering mobility across four task-based occupational categories: non-routine cognitive (NRC), routine cognitive (RC), non-routine manual (NRM) and routine manual (RM) occupations. In Supplementary Appendix B.1 we show this pattern also holds when using non-employment spell that include at least one month of unemployment, simultaneous industry/occupation mobility or self-reported duration of occupational 10 Kambourov and Manovskii (2008) compare two measures of year-to-year occupational mobility of pooled employer movers and stayers using the PSID, one that includes and one that excludes the unemployed. They find that the inclusion of unemployed workers raises the year-to-year occupational mobility rate by 2.5 percentage points, using a two-digit aggregation. In Supplementary Appendices A and B.5 we relate in more detail our analysis to theirs. Moscarini and Thomsson (2007) find high occupational mobility among employer-to-employer movers in the CPS, using a sample of workers who changed employers directly or with an intervening spell of non-employment of at most one month. 7 Tra n sp o nS mi Ad cc C o Le m Ed gal pO/Ma uc ati cc th Oc He on c Arc alth hit Pra ct/ ct En . gin ee rin g Co ns tru ctio n up rt O po rt ain t./I ns tal l ir/M pa Re Fo od pre p/s erv Art He s, En alt trtm hc are nt, su Spo pp rts ort Pro du ctio nO cc Se rv. ng ildi Bu CLife/ P Pe omm rs. /hSys/S Ca ocSoc re er Sc Ma Se v ie n na rvi ge ce ce me s Pro nt Bu tectiv s& e Fin Serv Op . s. Sa les -.4 Occupational Mobility Rate .4 .6 .8 -.2 0 .2 1 Figure 2: Gross and Net Occupational Mobility per Occupation 0 20 40 60 Occupational composition of unemployment spells (in pp) 80 100 Mean Gross Occ. Mobility Gross Mob. p.Occ: NRC NRM RC RM Mean Net Occ. Mobility Net Mob. p.Occ: NRC NRM RC RM Mean Net Occ Mob, excl. Mgt Notes: Gross mobility: The height of each light-colored bar is given by Ei U E−i /Ei U E, where Ei U E−i denotes the number of EU E spells of individuals who lost their jobs in occupation i and found re-employment in occupations other than i; and Ei U E denotes all EU E spells of those who lost their job in occupation i, including those who were re-employed in i. The width of each bar corresponds to Ei U E/EU E. Occupations are then sorted in decreasing order by workers’ gross mobility. Net mobility: The height of each dark-colored bar corresponds (E−i U Ei − Ei U E−i )/Ei U E. A positive value refers to net inflows, while a negative value refers to net outflows. The area of each of these bars gives the occupation-specific net flows as a proportion of all EU E transitions. Those occupations within the same task-based category are displayed in the same color, where N RC = non-routine cognitive, RC = routine cognitive, N RM = non-routine manual, RM = routine manual. The solid line correspond to the average gross occupational mobility rate. The dashed lines correspond to the average net mobility rate with and without managerial occupations. All data is corrected for miscoding using the method outlined in Section 2.1. tenure. Demographics In Supplementary Appendix B.1 we also show that the high level of occupational mobility and the moderate loss of attachment with duration is shared across men and women, education and race groups. However, we find that the level of gross occupational mobility decreases substantially with age, from 52.5% when young, (20-30yo), to 39.7% when prime-aged, (35-55yo). Figure 1b shows that the mobility-duration profile of prime-aged workers is below of that of young workers typically by about 9-13 percentage points but has a very similar slope. Thus, prime-aged workers display a higher level of attachment to their occupation but lose it in a similar gradual way with duration as young workers. Mobility by occupation Figure 2 shows that most occupations share high mobility rates. The gross mobility of an occupation i (height of each light-colored bar) is defined as the percentage of unemployed workers previously employed in i finding employment in a different occupation. This finds that occupations with gross occupational mobility rates above 40% cover more than 80% of all EU E spells in our data. Apart from small and specialized occupations (as engineers, architects, and doctors), construction is the only large occupation with a meaningfully lower occupational mobility, which is still close to 25%. In Supplementary Appendix B.1 we show that the moderate increase of occupational mobility with unemployment duration is also shared across (origin) occupations. Further, we cannot reject the equality of the slopes (and semi-elasticities) across all occupation-specific mobility-duration profiles. The slope of the aggregate duration profile does not arise because some occupations with relatively 8 high unemployment durations have particularly high occupational outflows – rather, it appears that the unemployed across all occupations lose their attachment gradually. 2.3 Excess and Net Mobility To assess the importance of occupational moves that result in certain occupations experiencing net inflows (outflows) through unemployment, we divide gross occupational mobility into net and excess mobility. Denote by Ei U E−i the number of unemployment spells that involve a move from occupation i to any of the other occupations. The dark bars in Figure 2 depict the net flows per occupation, defined as (E−i U Ei − Ei U E−i )/Ei U E, where the numerator denotes the difference between gross inflows and outflows for occupation i and the denominator captures all unemployment spells that originate from occupation i. It is evident that net flows are an order of magnitude smaller than gross outflows across almost all occupations, the main exception being managerial occupations. The avP P erage net mobility rate, 0.5 i |E−i U Ei − Ei U E−i |/EU E (where EU E = i Ei U E) equals only 4.2%. This means that 4.2% of workers’ EU E spells make up the contribution of the unemployed to the changing size of occupations.11 Although small relative to unemployment flows, we observe a clear pattern: net outflows from the routine manual occupations (RM) and net inflows into the non-routine manual (NRM) occupations. Excess mobility is the most important component of occupational mobility, across all occupations except for management. Aggregating across occupations, the average excess mobility rate P min{E−i U Ei , Ei U E−i }/EU E implies that 40.2% of all EU E spells represent excess mobility, about 90% of all gross mobility. Supplementary Appendix B.2 extends this analysis and shows these results are robust to alternative occupational classifications and considering non-employment spells instead of only unemployment spells.12 Excess and net mobility-duration profile Figure 3 shows that the moderate increase of gross mobility with unemployment duration is associated with an increase in excess mobility. For each duration x = 1, 2, 3, ..., 12 we re-compute the average net and excess mobility rates defined above but using only those EU E spells that have unemployment episodes of at least x months. We then decompose the mobility-duration profile into three categories: excess mobility, net mobility among nonmanagement occupations (by dropping all the management flows), and the difference between these two, which we label “residual net mobility”. This finds that, while for the long-term unemployed occupational moves are more common, those moves still overwhelmingly cancel out at an occupational level. Therefore, although there is a small absolute increase in net mobility with duration, Figure 3 does not support the conjecture that long-term unemployment is associated with a subset of occupations that workers are particularly eager to leave for a different and disjunct set of occupations that 11 The pre-multiplication by 0.5 reflects that each net outflow in some occupation is simultaneously also counted as a net inflow in other occupations. Note that coding error matters for the level of net mobility, where in the raw data the average net mobility rate is below 3%. Miscoding will mistakenly convert some true mobility flows into occupational stays, while miscoding for stayers is completely symmetric with respect to origin and destination occupations, and therefore should not give rise to spurious net mobility. 12 For pooled samples of employer stayers and movers, Jovanovic and Moffitt (1990) and Kambourov and Manovskii (2008) have also highlighted the importance of excess relative to net mobility across industries or occupations. 9 Figure 3: Gross, Net and Excess Occupational Mobility by unemployment duration .5 (b) Task-based categories (NRC/RC/NRM/RM) Excess Reallocation (smoothed) Residual Net Mobility, incl. Mgt Net mobility excl. Mgt (smoothed) Overall Excess Realloc. (All spells) Excess Reallocation (smoothed) Residual Net Mobility, incl. Mgt Net mobility excl. Mgt (smoothed) Overall Excess Realloc. (All spells) 0 0 .1 Proportion Moving Occupation Upon Exit U .3 .2 .4 Proportion Moving Occupation Upon Exit U .1 .6 .2 .3 .4 .5 .7 (a) Major Occupational Groups (2000 SOC) 1 2 3 4 5 6 7 8 9 10 Length of (Incomplete) Unemployment Spell 11 12 1 2 3 4 5 6 7 8 9 10 Length of (Incomplete) Unemployment Spell 11 12 P Notes: At each duration x = 1, 2, 3, ..., 12 we compute the average net mobility rate 0.5 |E−i U Ei − Ei U E−i |/EU E using EU E spells that have completed unemployment episodes of at least x months. The net mobility rate among non-management occupations drops all management flows P from this calculation. The average excess mobility rate is computed as min{E−i U Ei , Ei U E−i }/EU E, once again only using the EU E spells that have completed unemployment episodes of at least x months. The horizontal dashed line across these graphs reflect the average excess mobility rate of 40.2% among those workers who had at least one month in unemployment. offer better conditions. 2.4 Repeat Mobility The SIPP allows us to investigate the evolution of a worker’s attachment to occupations across multiple unemployment spells. These “repeat mobility” statistics tell us whether typically workers who changed (did not change) occupations after an unemployment spell, will change occupation subsequently after a following unemployment spell.13 Here we can also use the Γ-correction to counteract coding error in three-occupation histories (surrounding two unemployment spells).14 We find that from all those stayers who became unemployed once again, 64.9% of these workers remain in the same occupation after concluding their second unemployment spell. This percentage is higher for prime-aged workers, 69.3%, and lower for young workers, 57.1%. However, the loss of occupational attachment itself also persists. Among workers who re-enter unemployment after changing occupations in the preceding unemployment spell, we find that 55.8% of these workers move again. This percentage is lower for prime-aged workers, 50.8%, and higher for the young, 63.8%. Supplementary Appendix B.5 shows a similar pattern in the PSID. 13 Our repeat mobility statistics are measured within the SIPP 3.5 to 5 years windows and are based on 610 of observations of individuals with multiple spells across all panels when considering only pure unemployment spells and 1,306 when considering non-employment spells that include months of unemployment. For further details see Supplementary Appendix B.7. Note that workers with two consecutive unemployment spells within this window are not necessarily representative of all unemployed workers, nor of behavior in unemployment spells that are further apart. Nevertheless, these statistics are valuable and will inform our modelling choices and quantitative analysis, where we construct our simulated measures in the same way as we do in the SIPP. 14 With O the total number of occupations, let the matrix Mr (with elements mrijk ) be the O × O × O matrix of true repeat flows. Then, this matrix relates to the observed repeat flow matrix Mr,obs through vec(Mr )′ = vec(Mr,obs )′ (Γ ⊗ Γ ⊗ Γ)−1 , where vec(M) is the vectorization of matrix M, and ⊗ denotes the Kronecker product. Since Γ is invertible, Γ ⊗ Γ ⊗ Γ is also invertible. 10 2.5 Occupational Mobility of the Unemployed over the Cycle Unemployed workers’ attachment to their previous occupations changes over the business cycle. In expansions unemployed workers change occupations more frequently than in recessions. Panel A of Table 1 investigates the cyclicality of occupational mobility by regressing the (log) gross occupational mobility rate on the (log) unemployment rate. Columns (i) and (ii) relate the HP-filtered quarterly series of the Γ-corrected and uncorrected occupational mobility rates obtained from the SIPP to HPfiltered series of the unemployment rate, all with a filtering parameter of 1600. Because there are proportionally more stayers and hence more spurious mobility in recessions, the corrected series yields a somewhat stronger cyclicality than the uncorrected one. Column (iii) presents the regression results based on (uncorrected) occupational mobility data from the CPS for the period 1979-2019. We use the CPS as in this case the quarterly mobility series does not suffer from gaps as does the SIPP (see also Supplementary Appendix B.5.15 We observe that the uncorrected SIPP and CPS series have a very similar degree of procyclicality, suggesting that data gaps do not meaningfully affect our conclusion. Table 1: Occupational mobility and unemployment duration over the business cycle HP-filtered Qtrly Occ. Mobility (i) SIPP Γ-corrected HP U Controls HP U Dur. coef Controls -0.170*** (0.060) - (ii) SIPP uncorrected (iii) CPS uncorrected Unfiltered Occ Mobility (iv) SIPP Γ-corrected (v) SIPP uncorrected (vi) CPS uncorrected (vii) SIPP uncorrected Panel A: Mobility regression, not controlling for non-employment duration -0.100*** -0.106*** -0.154** -0.114** -0.087*** -0.129*** (0.030) (0.039) (0.062) (0.049) (0.032) (0.043) D,T, S.O. Panel B: Mobility regression, controlling for non-employment duration -0.199*** -0.150*** -0.116*** -0.174*** (0.063) (0.050) (0.035) (0.044) 0.0161*** 0.0133*** 0.0102*** 0.0142*** (0.002) (0.002) (0.001) (0.002) D,T, S.O. Notes: */**/*** denotes significance at 10%, 5%, 1% level. SIPP sample is restricted to quarters where the data allows the full spectrum of durations between 1-12 months to be measured. Standard errors clustered on quarters. Dur Coef. is the coefficient on completed durations. Underlying the regression sample are spells with completed durations between 1 and 14 months, not involving agricultural occupations; for further restrictions, see Supplementary Appendix B.7. Regressions (i)-(iii) and (iv)A on quarterly data; (iv)B on (Γ-corrected) quarter x duration data; (v)-(vii) on individuallevel panel data. CPS data described and cyclicality further analyzed in Supplementary Appendix B.5, (vi) on period 1984-2014 for comparability with SIPP. Controls: D=demographic controls (gender, race, education, and a quartic in age); T=time controls (linear time trend, and a dummy for the classification in which data was originally reported); S.O.= source occupation. Columns (iv)-(vii) presents the results of regressions relating unfiltered occupational mobility series to the HP-filtered unemployment rate as further robustness. Again, both SIPP and CPS data sets give a broadly similar procyclicality. The last column adds further individual-level controls and shows that these do not meaningfully change our results. This indicates that the procyclicality of occupational mobility is not the result of a compositional shift towards occupations or demographics characteristics that are associated with higher mobility when the economy is in an expansion. In 15 These occupational mobility series have data missing due to non-overlapping SIPP panels combined with our sampling restrictions (to avoid censoring issues), as described in Supplementary Appendix B.7. To deal with these gaps, we use TRAMO-SEATS (Gomez and Maravall, 1996) for interpolation, HP-filter the series and then discard all quarters that were interpolated. 11 Figure 4: Cyclicality of occupational mobility, 1985-2014 (b) Net Occupational Mobility - Task-based categories 1 2 3 4 5 6 7 8 Time in Unemployment times of high U 95% CI 9 10 11 12 -.02 .4 -.01 .45 0 .01 .02 Occupational Mobility .6 .55 .5 .03 .65 (a) Occ. Mobility - Duration Profile Shift NRC times of low U 95% CI RC Times of High U NRM RM Times of Low U Notes: Left panel:The circular markets depict the raw data and the solid curves represent the smoothed mobility-duration profile. The thickness of the profiles indicates the amount of spells surviving at a given duration. Right panel: The net mobility rate for each task-based category is computed excluding Managers and constructed as (E−i U Ei − Ei U E−i )/EU E, separately for periods of high and low unemployment. The version including Managers can be found in the Supplementary Appendix B. Supplementary Appendix B.3 we provide an extensive set of robustness exercises based on the SIPP all showing the procyclicality of gross occupational mobility. Supplementary Appendix B.5 further shows a procyclical occupational mobility rate among the unemployed when using the PSID for the period 1968 to 1997. Cyclicality of the mobility-duration profile Figure 4a depicts the cyclical shift of the mobilityduration profile. It plots the profile separately for those spells that ended in times of high unemployment and for spells that ended in times of low unemployment. Times of high (low) unemployment are defined as periods in which the de-trended (log) unemployment rate was within the bottom (top) third of the de-trended (log) unemployment distribution. The thickness of the profiles indicates the amount of spells surviving at a given duration, showing the faster reduction of spells with duration in expansions. Occupational mobility at any unemployment duration is lower in recessions, corroborating the procyclicality of gross occupational mobility documented in Table 1. Both in times of high and low unemployment, an increase in unemployment duration is associated with a moderate loss of attachment to workers’ previous occupation. Panel B of Table 1 shows that a roughly similar vertical shift of the mobility-duration profile over the cycle is found across SIPP and CPS and this is robust to including demographics and (origin) occupations controls. The cyclicality of net occupational mobility Figure 4b shows the cyclical behavior of the net mobility rates for each of the four task-based categories. We compute the net mobility rate as (E−i U Ei − Ei U E−i )/EU E, separately for periods of high and low unemployment.16 Differently from Section 2.3 we normalise net flows in each task-based category by the total number of EU E spells observed in periods of either high or low unemployment. This allows us to control for the fact that the number of unemployment episodes changes over the cycle. 16 In this case we define times of high (low) unemployment as periods in which the de-trended (log) unemployment rate was within the top (bottom) third (half) of the de-trended (log) unemployment distribution. We chose this partition as it minimises small sample bias. In Supplementary Appendix B.3 we show that the same patterns hold when defining periods of low and high unemployment in many different ways. 12 It is clear from the graph that across all task-based categories net mobility increases in periods of high unemployment relative to periods of low unemployment. RM occupations increase their net outflows in downturns relative to expansions, while N RM occupations increase their net inflows in downturns relative to expansions.17 The countercyclicality of net mobility therefore implies that the cyclicality of excess mobility is the main driver behind the procyclical behaviour of gross occupational mobility among unemployed workers.18 Comparing unemployment spells between movers and stayers The above patterns imply that occupational movers have longer spells than stayers, on average by 0.5 month.19 In recession, this difference grows to 1.11 months. This increase does not result from cyclically different demographics of unemployed movers or because they are more likely to be in long-duration occupations in recessions (see Supplementary Appendix B.4 for a formal regression analysis). Although the occupational mobility of the unemployed decreases in a recession, the lengthening of unemployment spells among movers is proportionally stronger. Occupational movers thus contribute to the increase in aggregate unemployment, and especially strongly so, to the increase in long-term unemployment. In Section 5 we further discuss these empirical findings put them in the context of our theoretical framework. 3 Theoretical Framework We now develop a theory of occupational mobility of the unemployed to explain the above empirical results and link them to the cyclical behaviour of long and short term unemployment as well as the aggregate unemployment rate. 3.1 Environment Time is discrete t = 0, 1, 2, . . . A mass of infinitely-lived, risk-neutral workers is distributed over a finite number of occupations o = 1, . . . , O. At any time t, workers within a given occupation can be either employed or unemployed and differ in two components: an idiosyncratic productivity, zt , and human capital, xt . We interpret the z-productivity as a “career match” which captures in a reduced form the changing career prospects workers have in their occupations (see Neal, 1999). These zproductivities follow a common and bounded first-order stationary Markov process, with transition law F (zt+1 |zt ).20 Their realizations affect a worker both in employment and in unemployment and will drive excess occupational mobility in our model. To capture the different levels of attachment to occupations found across age groups, workers’ accumulate occupational human capital through a learning-by-doing process while employed, and are subject to human capital depreciation while 17 In Supplementary Appendix B.3 we show that the exclusion of the managerial occupations from our calculation implies that RC occupations are now experiencing net outflows instead of net inflows as suggested by Figure 2. 18 Kambourov and Manovskii (2008) using PSID data also find countercyclical net mobility and procyclical gross mobility among a pooled sample of employer stayers and movers. 19 This difference is economically significant: it represents nearly half of the differences between the average unemployment spell in periods of high versus low unemployment. 20 The assumption that the z-productivity process is common across workers and occupations is motivated by our evidence showing that the change of occupational mobility with unemployment duration does not seem to differ across occupations or demographic groups. 13 unemployed. Conditional on the worker’s employment status, his human capital xt is assumed to evolve stochastically following a Markov chain with values xt ∈ {x1 , ..., xH }, x1 > 0 and xH < ∞ . Each occupation is subject to occupation-wide productivity shocks. Let po,t denote the productivity of occupation o at time t and pt = {po,t }O o=1 the vector that contains all the occupations’ productivities at time t. Differences across po,t will drive net mobility. Business cycle fluctuations occur due to changes in aggregate productivity, At . We allow the occupation-wide productivity process to depend on aggregate productivity. Both po,t and At follow bounded first-order stationary Markov processes. There is a mass of infinitely-lived risk-neutral firms distributed across occupations. All firms are identical and operate under a constant return to scale technology, using labor as the only input. Each firm consists of only one job that can be either vacant or filled. The output of a worker with current productivity zt and human capital xt employed in a firm in occupation o is given by the production function y(At , po,t , zt , xt ). The production function is strictly increasing and continuous in all of its arguments and differentiable in the first three. All agents discount the future at rate β. Workers retire stochastically, receiving a fixed utility flow normalized to zero. They are replaced by new entrants, inexperienced workers with x1 that are allocated across occupations following an exogenous distribution ψ. We rescale β to incorporate this retirement risk. Match break-up can occur with an exogenous (and constant) probability δ, but can also occur if the worker and the firm decide to do so, and after a retirement shock. Once the match is broken, the firm decides to reopen the vacancy and, unless retired, the worker stays unemployed until the end of the period. We assume that any unemployed worker receives b each period. Wages will be determined below. To study business cycle behavior in a tractable way, we focus on Block Recursive Equilibria (BRE). In this type of equilibrium the value functions and decisions of workers and firms only depend on ω = {z, x, o, A, p} and not on the joint productivity distribution of unemployed and employed workers over all occupations. An occupation can be segmented into many labor markets, one for each pair (z, x) such that workers in different markets do not congest each other in the matching process. Each of these labor market has the Diamond-Mortensen-Pissarides (DMP) structure. Each has a constant returns to scale matching function which governs the meetings of unemployed workers and vacancies within a market. We assume that all these markets have the same random matching technology. Each market exhibits free entry of firms, where posting a vacancy costs k per period. Once an unemployed worker’s z or x changes, the relevant labor market for this worker changes accordingly.21 Searching across occupations Instead of searching for jobs in their own occupation, unemployed workers can decide to search for jobs in different occupations. This comes at a per-period cost c and entails re-drawing their z-productivity. Workers rationally expect their initial career match in 21 In Supplementary Appendix C we show that a competitive search model in the spirit of Moen (1997) and Menzio and Shi (2010, 2011) endogenously generates this sub-market structure, such that unemployed workers with current productivities (z, x) optimally participate only in the (z, x) market. Here we proceed by assuming this sub-market structure from the start in order to reduce unnecessary complexity in the analysis. The allocations and equilibrium outcomes are the same under both approaches (see Carrillo-Tudela and Visschers, 2013). 14 any occupation to be a draw from F (z), which we take to be the ergodic distribution associated with the Markov process F (zt+1 |zt ). The i.i.d. nature of the re-draws allows us to capture that some occupational movers end up changing occupations again after a subsequent jobless spell, as suggested by the repeat mobility patterns documented earlier. The differences in occupation-wide labor market conditions po imply that workers are not indifferent from which occupation the draw of z comes from. To capture that in the data excess mobility is much larger than net mobility and hence that workers not always specialise their search in the occupation with the highest po , we model the choice of occupation following an imperfectly directed search approach in the spirit of Fallick (1993). During a period, workers have a unit of search effort to investigate their employment prospects in the remaining occupations. They can only receive at most one new draw of z per period without recall. A worker must then chose how much effort to allocate to each one of these occupations in order to maximise the probability to receiving a z. Let sõ denote the P search effort devoted to occupation õ such that õ∈O− sõ = 1, where O− denotes the set of remaining occupations. Each sõ maps into a probability of receiving the new z from occupation õ. Conditional on switching from o, this probability is denoted by α(sõ ; o), where α(.; o) is a continuous, weakly P increasing and weakly concave function of s with α(0; o) = 0 and õ∈O− α(sõ ; o) ≤ 1 for all o ∈ O. P Hence, 1 − õ∈O− α(sõ ; o) is the probability that a worker does not receive a new z during the period. If no z is received, the above process is repeated the following period. If a z is received, the worker must sit out one period unemployed in the new occupation õ before deciding whether to sample another z from a different occupation.22 If the worker decides to sample once again the above process is repeated. However, if the worker decides to accept the z, he starts with human capital x1 in the new occupation. The worker’s z and x then evolve as described above.23 3.2 Agents’ Decisions The timing of the events is summarised as follows. At the beginning of the period the new values of A, p, z and x are realised. After these realisations, the period is subdivided into four stages: separation, reallocation, search and matching, and production. To keep notation complexity to a minimum, we leave implicit the time subscripts, denoting the following period with a prime. 22 Note that this implies that the worker is forced to move to the new occupation even if the z turns out to be low enough. To further simplify we also assume that after the worker is in the new occupation, he can sample z-productivities from previous occupations. This way we avoid carrying around the histories of occupations ever visited by a worker in the state space. 23 Our data suggests that c and the loss of x when changing occupation should be incorporated in our model as mobility costs. This is because we find (i) a substantial proportion of stayers among young workers, which are typically associated with low levels of human capital, and (ii) substantial occupational staying among those who moved occupations but subsequently have become unemployed again. Since this occurs within the duration of a SIPP panel, these workers’ occupational tenure is low, yet they also display significant occupational attachment. 15 Worker’s Problem Consider an unemployed worker currently characterised by (z, x, o). The value function of this worker at the beginning of the production stage is given by  n h io U ′ ′ ′ ′ E ′ ′ U ′ W (ω) = b+βEω′ max ρ(ω )R(ω )+(1−ρ(ω )) λ(θ(ω ))W (ω )+(1−λ(θ(ω )))W (ω ) , ′ ρ(ω ) (1) where θ(ω) denotes the ratio between vacancies and unemployed workers currently in labor market (z, x) of occupation o, with λ(.) the associated job finding probability. The value of unemployment consists of the flow benefit of unemployment b, plus the discounted expected value of being unemployed at the beginning of next period’s reallocation stage, where ρ(ω) takes the value of one when the worker decides to search across occupations and zero otherwise. The term R(ω) denotes the expected net value of searching across occupations and is given by Z z  X X W U (z̃, x1 , õ, A, p)dF (z̃) + (1 − α(sõ (ω)))Ŵ U (ω) − c , (2) R(ω) = max α(sõ (ω)) S(ω) õ∈O− z õ∈O− where Ŵ U (ω) = b + βEω′ R(ω ′ ), S denote a vector of sõ for all õ ∈ O− and the maximization is P subject to sõ ∈ [0, 1] and õ∈O− sõ = 1. The first term denotes the expected value of drawing a new z and losing any accumulated human capital, while the second term denotes the value of not obtaining a z and waiting until the following period to search across occupations once again. The formulation of Ŵ U (ω) is helpful as it implies that R(ω) and {sõ } become independent of z. It is through R(ω) that expected labor market conditions in other occupations affect the value of unemployment, and indirectly the value of employment, in the worker’s current occupation. The worker’s decision to reallocate is captured by the choice between the expected net gains from drawing a new z̃ in another occupation and the expected payoff of remaining in the current occupation. The latter is given by the expression within the inner squared brackets in equation (1). Now consider an employed worker currently characterised by (z, x, o). The expected value of employment at the beginning of the production stage, given wage w(ω), is h i ′ E ′ ′ U ′ W E (ω) = w(ω) + βEω′ max {(1 − d(ω ))W (ω ) + d(ω )W (ω )} . (3) ′ d(ω ) The second term describes the worker’s option to quit into unemployment in next period’s separation stage. The job separation decision is summarised in d(ω ′ ), such that it take the value of δ when W E (ω ′ ) ≥ W U (ω ′ ) and the value of one otherwise. Firm’s Problem Consider a firm posting a vacancy in sub-market (z, x) in occupation o at the start of the search and matching stage. The expected value of a vacancy solves the entry equation V (ω) = −k + q(θ(ω))J(ω), (4) where q(.) denotes firms’ probability of finding an unemployed worker and J(ω) denotes the expected value of a filled job. Free entry implies that V (ω) = 0 for all those sub-markets that yield a θ(ω) > 0, and V (ω) ≤ 0 for all those sub-markets that yield a θ(ω) ≤ 0. In the former case, the entry condition simplifies (4) to k = q(θ(ω))J(ω). Now consider a firm employing a worker currently characterized by the pair (z, x, o) at wage w(ω). The expected lifetime discounted profit of this firm at the beginning of the production stage 16 can be described recursively as J(ω) = y(A, po , z, x) − w(ω) + βEω′ h n oi ′ ′ ′ ′ max (1 − σ(ω ))J(ω ) + σ(ω )V (ω ) , ′ σ(ω ) (5) where σ(ω ′ ) takes the value of δ when J(ω ′ ) ≥ V (ω ′ ) and the value of one otherwise. Wages We assume that wages are determined by Nash Bargaining. Consider a firm-worker match currently characterised by (z, x, o) such that it generates a positive surplus. Nash Bargaining implies that the wage, w(ω), solves     (1 − ζ) W E (ω) − W U (ω) = ζ J(ω) − V (ω) , (6) where ζ ∈ [0, 1] denotes the worker’s exogenous bargaining power. This guarantees that separation decisions are jointly efficient, d(ω) = σ(ω). In what follows we impose a Cobb-Douglas matching function and the Hosios condition, such that 1 − ζ = η, where η denotes the elasticity of the job finding probability with respect to labor market tightness within sub-market (z, x). This will guarantee that firms post the efficient number of vacancies within sub-markets. It will also guarantee efficiency of our decentralized economy. 3.3 Equilibrium and Characterization In a BRE outcomes can be derived in two steps. In the first step, decision rules are solved independently of the joint productivity distribution of unemployed and employed workers over all occupations, using (1)-(5). Once those decision rules are determined, we fully describe the dynamics of the workers’ distribution, using the workers’ flow equations. To prove existence and uniqueness we build on the proofs of Menzio and Shi (2010, 2011) but incorporate the value of reallocation across occupations and show it preserves the block recursive structure. The formal definition of the BRE is relegated to Supplementary Appendix C, where we also present the derivation of the flow equations and the proofs of all the results of this section. Existence Let M (ω) ≡ W E (ω) + J(ω) denote the joint value of the match. To prove existence and uniqueness of the BRE we define an operator T that is shown to map M (ω), W U (ω) and R(ω) from the appropriate functional space into itself, with a fixed point that implies a BRE. Given this result and the Banach’s Fixed Point Theorem, this fixed point exists and is unique. To prove efficiency we show that the unique solution to the planner’s problem in the general state space (which includes the distribution of workers across occupations) coincides with the solution to the decentralized economy problem in the space ω. The key step is to ensure that a worker’s value of searching across occupations coincides with the planner’s value of making the worker search across occupations. Proposition 1. Given F (z ′ |z) < F (z ′ |z̃) for all z, z ′ when z > z̃: (i) a BRE exists and it is the unique equilibrium; and (ii) the BRE is constrained efficient. Characterization The decision to separate from a job and the decision to search across occupations can be characterised by z-productivity cutoffs, which are themselves functions of A, p, o and x. The job separation cutoff function, z s (.) solves M (ω) − W U (ω) = 0 such that the match surplus becomes 17 Figure 5: Relative positions of the reservation productivities (a) z r > z s z Employed workers, Search Unemployment (b) z s > z r z z r (A; PO , x, o) Employed workers, search unemployment z s(A; PO , x, o) Employed workers, Unemployed workers reallocate Rest unemployment z r (A; P0, x, o) All workers reallocate z s(A; PO , x, o) A All workers reallocate A zero. As in Mortensen and Pissarides (1994) it characterises endogenous separations. However, in our setup z refers to the worker’s idiosyncratic productivity in an occupation, rather than to a matchspecific productivity with a firm. This difference implies that when the worker becomes unemployed, his z is not lost or is reset when re-entering employment in the same occupation. Instead, the worker’s z continuous evolving during the unemployment spell. It is only when the worker searches across occupations that he can reset his z. The reallocation cutoff function, z r (.) solves R(ω) = W U (ω) and determines when an unemployed worker decides to search across occupations. The latter occurs if and only if z < z r (.). The relative position and the slopes of z r (.) and z s (.) are crucial determinants of the long-run and cyclical outcomes in our model. To show this, we first discuss the implications of their relative position and then those of their slopes. Figure 5a illustrates the case in which z r > z s for all A, holding constant p, o and x. Here having a job makes a crucial difference on whether a worker stays or leaves his occupation. When an employed worker has a current z ∈ [z s , z r ), the match surplus is enough to keep him attached to his occupation. For an unemployed worker with a current z in the same interval, however, the probability of finding a job is sufficiently small to make searching across occupations the more attractive option, even though this worker could generate a positive match surplus if he were to become employed in his pre-separation occupation. For values of z < z s , all workers search across occupations. For values of z ≥ z r , firms post vacancies and workers remain in their occupations, flowing between unemployment and employment over time as in the canonical DMP model. Figure 5b instead shows the case in which z s > z r for all A. Here workers who endogenously separate into unemployment, at least initially, do not search across occupations, while firms do not create vacancies in sub-markets associated with values of z < z s . These two cutoffs create an area of inaction, in which workers become rest unemployed during the time their z lies in [z r , z s ): they face a very low – in the model (starkly) zero – contemporaneous job finding probability, but still choose to remain attached to their occupations. The stochastic nature of the z process, however, implies that these workers can face a positive expected job finding probability for the following period. Only after the worker’s z has declined further, such that z < z r , the worker searches for a new z across 18 occupations. For values of z ≥ z s , the associated sub-markets function as in the DMP model. An unemployed worker is then considered search unemployed during the time in which his z ≥ z s , as in the associated labor markets firms are currently posting vacancies. A worker whose current z < z r is considered reallocation unemployed only during the time in which he is trying to find another occupation that offers him a z > z r . Once he finds such an occupation, he continues his unemployment spell potentially with periods in search and rest unemployment, depending on the relative position of z s and z r and the initial draw and evolution of his z in such an occupation. The stochastic nature of the z process implies that search, rest and reallocation unemployment are not fixed characteristics, but transient states during an unemployment spell. Therefore, to be consistent with the analysis of Section 2, an occupational mover is a worker who left his old occupation, went through a spell of unemployment (which could encompass all three types of unemployment) and found a job in a different occupation. A key decision for an unemployed worker is whether to remain in his occupation, waiting for his z to improve, or to search across occupations, drawing a new z. Periods of rest unemployment arise when the option value of waiting in unemployment is sufficiently large. However, search frictions imply that there is also an option value associated with waiting in employment in an existing job match. In the face of irreversible match destruction, workers remain employed at lower output levels relative to the frictionless case because of potential future improvements in their z-productivities. This drives the separation cutoff function down. The tension lies in that these two waiting motives work against each other. Which one dominates depends on parameter values. Using a simplified version of the model without aggregate or occupation-specific shocks, we show that the difference z s − z r increases when c, b or x increase (see Supplementary Appendix C.1). Although it is intuitive that a higher c or x reduces z r by making occupational mobility more costly, they also reduce z s by increasing the match surplus and making employed workers less likely to separate. We show that, overall, the first effect dominates. A rise in b decreases z r by lowering the effective cost of waiting, while decreasing the match surplus by increasing W U (.) and hence increasing z s , pushing towards rest unemployment. We also show that a higher degree of persistence in the z process decreases z s − z r as it decreases the option value of waiting. Figure 5 shows the case of countercyclical job separation decisions (∂z s (.)/∂A < 0) and procyclical occupational mobility decisions (∂z r (.)/∂A > 0), as suggested by the data. The relative position of z s and z r is an important determinant of the cyclicality of occupational mobility decisions. Using a simplified version of the model without occupation-specific shocks, we show that when z s > z r we obtain procyclical occupational mobility decisions without the need of complementarities in the production function (see Supplementary Appendix C.1). This arises as with search frictions wages and job finding probabilities increase with A, and complement each other to increase the expected value of occupational mobility (relative more than in the frictionless case). In addition, the presence of rest unemployment reduces the opportunity cost of mobility, making the latter less responsive to A. This occurs as any change in A does not immediately affect the utility flow of the rest unemployed. 19 The relative position of z s and z r also affects the cyclicality of job separation decisions. When z s is sufficiently above z r , job separation decisions mainly reflect whether or not an employed worker should wait unemployed in his current occupation for potential improvement of his z. Occupational mobility is only one possible future outcome and hence it is discounted. This implies that a sufficiently large z s − z r > 0 moderates the feedback of procyclical occupational mobility decisions on the cyclicality of job separation decisions. As the position and slope of the z s and z r cutoffs can only be fully determined through quantitative analysis, we now turn to estimate the model and investigate its resulting cyclical properties. 4 Quantitative Analysis 4.1 Calibration Strategy We set the model’s period to a week and the discount factor β = (1 − d)/(1 + r) is such that the exit probability, d, is chosen to match an average working life of 40 years and r such that β matches an annual real interest rate of 4%. To keep the population constant every worker that leaves the economy is replaced by a new unemployed worker. We target occupational mobility statistics based on the 2000 SOC and aggregate the simulated data to ‘major’ occupational groups and task-based categories (nonroutine cognitive N RC, routine cognitive RN , non-routine manual N RM and routine manual RM ) as done in Section 2. Our classification error model then allows us to easily correct for aggregate and occupation-specific levels of miscoding by imposing the Γ-correction matrix on simulated worker occupational flows at the required level of aggregation. Aggregate and occupation productivities The production function is assumed multiplicative and given by yo = Apo xz for all o ∈ O, chosen to keep close to a ‘Mincerian’ formulation. The logarithm of aggregate productivity, ln At , follows an AR(1) process with persistence and dispersion parameters ρA and σA . For a given occupation o, the logarithm of the occupation-wide productivity is given by ln po,t = ln po + ϵo ln At , where po denotes this occupation’s constant productivity level and ϵo its loading with respect to changes in aggregate productivity. This formulation implies that different occupations can have different sensitivities to the aggregate shock and hence different relative attractiveness to workers over the business cycle.24 We consider occupation-wide productivity differences at the level of task-based categories, O = {N RC, RC, N RM, RM }. All major occupations within a task-based category o ∈ O then share the same po,t . This approach not only simplifies the computational burden by reducing the state space of the calibrated model, but is also consistent with the evidence presented in Figure 2 showing that within the majority of task-based categories all major occupations’ net flows exhibit the same sign. To further simplify we normalize both the employment weighted average of po and of ϵo across o ∈ O to one. 24 The evidence presented in Supplementary Appendix C.3 suggests that our approach is consistent with the observed cyclical behaviour of net occupational flows, where the majority of occupations exhibit a very similar cyclical pattern across several recession/expansion periods. 20 Worker heterogeneity within occupations The logarithm of the worker’s idiosyncratic productivity, ln zt , is also modelled as an AR(1) process with persistence and dispersion parameters ρz and σz . We include a normalization parameter z norm that moves the entire distribution of z-productivities such that measured economy-wide productivity averages one. Occupational human capital is parametrized by a three-level process h = 1, 2, 3, where x1 = 1. Employed workers stochastically increase their human capital one level after five years on average. With probability γd the human capital of an unemployed worker depreciates one level until it reaches x1 . To allow for differences in the separation rates across young and prime-age workers that are not due to the interaction between z and x, we differentiate the probability of an exogenous job separation between low (x1 ) and high human capital (x2 , x3 ) workers: δL and δH . The matching function within each sub-market (z, x) in any occupation is given by m(θ) = θη . Search across occupations The probability that a worker in a major occupation within task-based category o receives the new z from a different major occupation in task-based õ is parametrized as (1−ν) α(sõ ; o) = αo,õ sνõ for all o, õ pairs in O = {N RC, RC, N RM, RM } and sõ ∈ [0, 1].25 The parameter ν ∈ [0, 1] governs the responsiveness of the direction of search across occupations that is related P to differences in po . The parameter αo,õ is a scaling factor such that õ∈O αo,õ = 1. It captures the extent to which an unemployed worker in a major occupation within task-based category o has access P to job opportunities in another major occupation in task-based category õ. Since õ∈O α(sõ ; o) ≤ 1, this formulation implies that if a worker in o wants to obtain a new z with probability one, he will choose sõ = αo,õ for all õ ∈ O. If a worker wants to take into account current occupation-wide productivity differences, he will choose sõ ̸= αo,õ for at least some õ. The cost of doing so is the P possibility of not receiving a new z at all (i.e. õ∈O α(sõ ; o) < 1) and paying c again the following period. The parameter ν determines the extent of this cost, with higher values of ν leading to lower probabilities of not receiving a new z. The formulation of α(sõ ; o) is convenient for it implies that the optimal value of sõ can be solved explicitly, R (1−ν) 1 s∗õ (ω) =P e 1−ν log[αo,õ õ∈O− 1 ( z z W U (z̃,x1 ,õ,A,p)dF (z̃)−Ŵ U (ω))] (1−ν) e 1−ν log[αo,õ R ( zz W U (z̃,x1 ,õ,A,p)dF (z̃)−Ŵ U (ω))] P with õ∈O− s∗õ (ω) = 1 and takes a similar form as the choice probabilities obtained from a multinomial logit model.26 Note that parameters αo,õ appear directly inside the closed form and can freely shape bilateral flows between occupations.27 This leaves parameter ν free to capture the responsive25 The identity of the major occupation within task-based õ from which the new z comes from is randomly drawn following a uniform distribution. 26 To derive this result note that for each sõ;o equation (2) yields the first order condition s∗õ (ω) = h να(1−ν) R i1/(1−ν) P z o,õ U U W (z̃, x , õ, A, p)dF (z̃) − Ŵ (ω) , where λ is the multiplier of the constraint õ∈O− s∗õ (ω) = 1. 1 λ z 1 1 Substituting out s∗õ (ω) in the constraint and using the change of variable X 1/(1−ν) = e 1/(1−ν) log(X) leads to the above expression. See Carrillo-Tudela et. al, (2021). 27 Many multi-sector models use the random utility model to drive excess mobility, where additive taste shocks are distributed i.i.d Type 1 Extreme Value (see Chodorow-Reich and Wieland, 2020, Wiczer, 2015, Dvorkin, 2014 and Pilossoph, 2014, among others). In the most tractable of such settings, underlying gross flows are constant at all times (e.g. Chodorow-Reich and Wieland, 2020). More generally, when the reallocation involves maxo∈O {Uo (.) + ϵo }, where 21 ness to cyclically changing occupation-wide productivities, which in turn allows us to capture net mobility flows over the cycle. It also leaves free the persistent career match z-process to drive excess mobility in a way that is consistent with the patterns documented in Section 2. Given that our data analysis covers three decades, we need to distinguish the observed long-run changes in the employment-size distribution across task-based categories from their cyclical changes. For this we first externally calibrate the initial size distribution in the simulations to match the one observed in the SIPP in 1984. This results in setting the employment proportions for N RC, RC, N RM , RM to 0.224, 0.292, 0.226 and 0.258, respectively, at the start of the simulation. In addition to the occupational mobility decisions of the unemployed, we allow this size distribution to change over time due to the mobility decisions of new entrants. Let ψo denote the exogenous probability that P a new entrant to the economy is allocated to task-based category o such that o∈O ψo = 1. This worker is then randomly allocated to a major occupation within the drawn task-based category at the point of entry, and is allowed to search across occupations to obtain first employment somewhere else. Simulation method of moments In the above parametrization [c, ρz , σz , z norm ] governs occupational mobility due to idiosyncratic reasons (excess mobility); [x2 , x3 , γd , δL , δH ] govern differences in occupational human capital; [po , ϵo , αo,õ , ν, ψo ] for all o, õ ∈ {N RC, RC, N RM, RM } govern occupational mobility due to occupation-wide productivity differences (net mobility); and the remainder parameters [k, b, η, ρA , σA ] are shared with standard DMP calibrations. All these parameters are estimated by minimising the sum of squared distances between a set of model simulated moments and their data counterparts. For consistent measurement we generate ‘pseudo-SIPP panels’ within one hundred time-windows each of 30 year length and follow the same procedures and definitions to construct the moments in data and in model simulations. Figure 6 and Table 2 show the set of moments used to recover these parameters as well as the fit of the model. The calibrated model provides a very good fit to the data across all the targeted dimensions. The mobility-duration profiles and survival functions primarily inform the excess mobility and the human capital parameters. Employer separations patterns inform the parameters shared with DMP calibrations, except for the persistence and standard deviation of the aggregate productivity process, ρA and σA , which are informed by the corresponding parameters of the series of output per worker (outpw) obtained from the BLS, ρoutpw and σoutpw , and measured quarterly for the period 1983-2014.28 The net mobility patterns across task-based categories inform the occupation-specific productivities, occupation distribution for new entrants and the imperfect direct search technology. Uo (.) is the value of being in occupation o and ϵo is the taste shock, this imposes a symmetry. All mobile workers who are considering occupations in set O have the same distribution over the destinations in O, independently of where they originated. Here we want to explicitly break this symmetry to be consistent with the bilateral flows of the transition matrix, a feature we can do without giving up on a convenient closed form. Our formulation also decouples the cyclical responsiveness from the cross-sectional flows, again without giving up on the closed form. In contrast, in the additive taste shock setting hitting cross-sectional patterns constrains the mobility response to cyclical shifts in Uo (.): both dimensions rely on how differences in Uo (.) translate into differences in the cdf of ϵo (or a transformation of the latter). 28 We cannot set ρA and σA directly because the composition of the economy changes with the cycle due to workers’ endogenous separation and reallocation decisions. We measure output in the model and data on a quarterly basis (aggregating the underlying weekly process in the model). For the data, we HP-filtered the series of (log) output per worker for the period 1970 to 2016. Then, we use the persistence and the variance parameters of this series calculated over the period 1983-2014, which is the period that the SIPP and the BLS series overlap. 22 The latter adds a number of extra parameters to the estimation, particularly the scale parameters αo,õ . As mentioned above these allow us to capture very well the relevant difference observed across occupations. We now present the arguments that justify the choice of moments, keeping in mind that all parameters need to be estimated jointly. 4.2 Gross occupational mobility and unemployment duration A worker’s attachment to his pre-separation occupation during an unemployment spell depends on the properties of the z process, the human capital process and the reallocation cost c. The aggregate and age-group mobility-duration profiles depicted in Figures 6a and 6b (see also Section 2) play an important role in informing these parameters. The aggregate mobility-duration profile contains information about c and ρz . As shown in Lemma 1 (see Supplementary Appendix C.1) changes in the overall level of mobility lead to opposite changes in c. The slope of the profile informs ρz primarily through the time it takes unemployed workers to start searching across occupations.29 A lower ρz (keeping constant F (z)) increases the relative number of unemployed workers deciding to search across occupations at shorter durations, decreasing the slope of the model’s mobility-duration profile. Lemma 1, however, also implies that a lower ρz reduces overall mobility (ceteris paribus), creating a tension between c and ρz such that an increase in ρz must go together with an increase in c to fit the observed mobility-duration profile as depicted in Figures 6a. To help identify σz we match instead the mobility-duration profiles of young and prime-aged workers. For given values of x, a larger value of σz leads to a smaller importance of human capital differences relative to z differences in workers’ output. This brings the simulated occupational mobility patterns across age groups closer together, creating a negative relationship between σz and the difference between the mobility-duration profiles of young and prime-aged workers. Figure 6b shows that the model is able to resolve this tension very well. Online Appendix C.1 shows that in addition the model remains fully consistent with the much larger contribution of excess mobility relative to net mobility in accounting for the mobility-duration profile at all durations, as depicted in Figure 3b. To inform the human capital parameters x2 and x3 we target the overall level of occupational mobility among young and prime-aged workers (see Lemma 2, Supplementary Appendix C.1) as well as the observed five and ten-year returns to occupational experience. As it is difficult to accurately estimate the later with the SIPP due to the relative short nature of its panels, we use the OLS estimates for 1-digit occupations reported in Kambourov and Manovskii (2009b) from the PSID and estimate the same OLS regression in simulated data.30 29 Other factors that might allow the calibrated model to generate the observed mobility-duration profile do not appear important in our estimation. In particular, the large extent of occupational mobility at short unemployment durations implies that the time it takes a typical worker to decide to search in a given occupation is small and hence does not drive the observed unemployment duration differences between occupational movers and stayers. Further, since the structure of the model implies that exogenous separated workers and occupational movers have very similar realised zdistributions, composition effects in post-reallocation outcomes do not play an important role. Finally, the changes in the mean-reversion of the z-productivity process brought about by changes in ρz seem to only play a minor role in shaping the mobility-duration profile. 30 We use the OLS estimates because occupation selection occurs both in the model and in the data, where selection 23 Figure 6: Targeted Moments. Data and Model Comparison (b) Mob-Dur (c) Cyclical .7 .55 .6 .45 .5 .55 .5 .25 Corr. Mob. Data Corr. Mob. Model Corr. Mob. Model 11 4 6 8 10 12 14 16 Months in Unemployment 18 20 1 2 3 4 5 6 7 8 9 Months in Unemployment (f) Survival 10 11 12 Profile Movers-Stayers Occ. Stayers Model Data .05 .1 .2 .3 .4 .5 .7 1 Survival (Log Scale) Occ. Movers Model Data .05 .1 .2 .3 .4 .5 .7 1 Survival (Log Scale) Prime-Aged Target - Data Target - Model 0 0 0 2 12 Profile Young-Prime Young Unemployed Target - Data Target - Model .05 .1 .2 .3 .4 .5 .7 1 Survival (Log Scale) 10 .02 Survival in Unemployment .2 .4 .6 .8 Target - Model Corr. Mob. Data Mobility - Model 4 5 6 7 8 9 Months in Unemployment (e) Survival LHS axis - Log Target - Data Target - Model Target - Model Mobility - Data Mobility - Model .02 3 Target - Model Mobility - Data .15 2 Target - Model 1 1 Target - Data Survival in Unemployment .2 .4 .6 .8 .25 12 Profile (All) 1 RHS axis - Linear Target - Data Target - Model 11 Times of Low U Target - Data .02 (d) Survival 10 Times of High U Target - Data .2 .3 .35 .4 .45 .4 .35 .3 4 5 6 7 8 9 Months in Unemployment .2 3 Prime-Aged Target - Data 1 2 Young Unemployed Survival in Unemployment .2 .4 .6 .8 1 Target - Model Mobility - Model Corr. Mob - Model Mob-Dur Profile .6 .65 .7 .65 .6 .55 .5 .45 .4 .35 Target - Data Mobility - Data Corr. Mob - Data .3 .25 Young-Prime .65 Profile (All) .75 (a) Mobility-Duration 2 4 6 8 10 12 14 16 Months in Unemployment 18 20 1 2 3 4 5 6 7 8 9 10 11 12 Months in Unemployment Calibrations with or without occupational human capital depreciation yield very similar long-run moments (see Online Appendix C.2). This occurs because the gradual loss of occupational attachment with unemployment duration underlying the observed mobility-duration profile can be generated by human capital depreciation or the z process. To differentiate these two forces we instead use the cyclical shift of the mobility-duration profile. During recessions longer unemployment spells imply that expected human capital depreciation is higher, making employed workers more attached to their jobs and unemployed workers less attached to their occupations. At the same time low aggregate productivity interacted with z typically makes employed workers less attached to their jobs and unemployed workers more attached to their occupations. To inform this tension and recover γd we fit the mobility-duration profile in recessions and expansions as depicted in Figure 6c (see also Figure 4a). We target the unemployment survival function depicted in Figure 6d to additionally inform us about the z and x processes. The extent of duration dependence is linked to the properties of the z process (and the importance of search frictions) through its effect on the extent of true duration arises as measured returns are a result of two opposing forces: human capital acquisition and z-productivity mean reversion. 24 Table 2: Targeted Moments. Data and Model Comparison Panel A: Economy-wide moments Moment Model Data Agg. output per worker mean Agg. output per worker persistence, ρoutpw Agg. output per worker st. dev., σoutpw Mean unemployment Task-based gross occ. mobility rate Repeat mobility: occ. stay after stay Occ. mobility young/prime-aged Occ. mobility-duration profiles: 0.999 1.000 0.764 0.753 0.009 0.009 0.036 0.036 0.280 0.288 0.600 0.649 1.167 1.163 Fig 7a,b,c Moment Model Data Rel. separation rate young/prime-aged Rel. separation rate recent hire/all Prob (unemp. within 3 yr for empl.) Empirical elasticity matching function 5-year OLS return to occ. tenure 10-year OLS return to occ. tenure Average u. duration movers/stayers U. survival profiles 1.999 2.044 5.180 4.945 0.151 0.124 0.526 0.500 0.143 0.154 0.219 0.232 1.181 1.140 Fig 7d,e Panel B: Occupation-Specific Moments, Long-run Proportion empl. size o2014 NRC RC NRM RM Net mobility Mean Transition Matrix Model Data Model Data Model Data NRC RC NRM RM NRC RC NRM RM 0.337 0.246 0.260 0.157 0.329 0.258 0.260 0.154 0.008 0.006 -0.027 0.011 0.006 0.001 -0.021 0.015 0.763 0.129 0.034 0.037 0.164 0.681 0.065 0.069 0.055 0.144 0.760 0.247 0.018 0.047 0.141 0.647 0.722 0.078 0.020 0.013 0.167 0.681 0.115 0.066 0.084 0.168 0.710 0.188 0.028 0.066 0.155 0.733 Panel C: Occupation-Specific Moments, Cyclical Recessions NRC RC NRM RM Net mobility Expansions Rec-Exp ∆exp−rec (inflow o/all flows) εU Do ,u /εU Davg ,u Model Data Model Data Model Data Model Data Model Data -0.012 -0.009 0.034 -0.017 -0.011 -0.005 0.033 -0.017 -0.002 -0.005 0.017 -0.006 -0.003 -0.001 0.011 -0.008 -0.010 -0.004 0.017 -0.011 -0.008 -0.004 0.022 -0.009 -0.003 0.006 -0.066 0.027 -0.010 0.003 -0.054 0.061 0.996 1.054 0.874 1.081 1.096 1.027 0.761 1.122 dependence and dynamic selection in our model, where the latter is driven by worker heterogeneity in x and z at the moment of separation. We use the cumulative survival rates at intervals of 4 months to reduce the seam bias found in the SIPP. The model also reproduces well the associated hazard functions (see Figures 1 and 2, Online Appendix C.1). Both in the model and the data duration dependence is different across (ex post) occupational stayers and movers and across age groups, where duration dependence is stronger among occupational stayers relative to movers and among young relative to prime-aged workers. Young occupational stayers have especially high job finding at low durations, which decrease faster as duration increases. In addition, the model replicates the (untargeted) incomplete unemployment duration distribution among all workers and separately by age groups, in particular the empirical amount of long-term unemployment that occurs in the face of high occupational mobility (see Table 1, Online Appendix C.1). Finally, we target the ratio between the average unemployment durations of occupational movers and stayers. The very good fit of all the above moments shows that the z and x processes capture very well the main forces behind the aggregate and age specific mobility-duration profiles and unemployment survival functions and duration distributions. The elasticity of the matching function, η, at the submarket (z, x) level is obtained by estimating through OLS a log-linear relation between the aggregate job 25 finding rate (the proportion of all unemployed workers in the economy who have a job next month) and aggregate labor market tightness (aggregate vacancies over aggregate unemployed) across quarters, in simulated data. The estimated elasticity η̂ is targeted to be 0.5 (Petrongolo and Pissarides, 2001) and allows us to indirectly infer η. 4.3 Employer separations A worker’s attachment to employment depends on the size of search frictions. A higher value of k leads to stronger search frictions through its effect on firm entry and labor market tightness. This pushes down the z s cutoff relative to z r , reducing the extent of endogenous separations.31 Therefore to inform k (and the relative position of z s and z r ) we use as targets the proportion of separations observed within a year of workers leaving unemployment relative to the overall yearly separation rate (“Rel. separation rate recent hire/all”) and the concentration of unemployment spells over a SIPP panel among the subset of workers who start employed at the beginning of the panel (“Prob (unemp. within 3yr for empl.)”). The probability that an occupational stayer becomes an occupational mover in the next unemployment spell (“Repeat mobility”) also informs endogenous separations and how these relate to occupational mobility.32 Given the job-finding moments, the overall separation rate follows from targeting the average unemployment rate. As we focus on those who held a job previously, we use the most direct counterpart and construct the unemployment rate only for those who were employed before and satisfied our definition of unemployment (see Section 2). Note that this unemployment rate (3.6%) is lower than the BLS unemployment rate, but we find it responsible for more than 0.75 for every one percentage point change in the BLS unemployment rate (see Online Appendix C.1 and Supplementary Appendix B.7 for details), consistent with the results of Hornstein (2013), Fujita and Moscarini (2017) and Ahn and Hamilton (2018). The ratio of separation rates between young and prime-aged workers (“Rel. separation rate young/prime-aged”) as well as their survival functions in Figure 6e inform δL , δH and b. A relatively higher δL shifts the realised z-distribution of newly separated inexperienced workers away from their z s cutoff towards higher z levels and hence affect their extent of duration dependence in unemployment, especially at shorter durations. The extent of separations for young and prime-aged workers also informs us about b through the positions of the z s cutoffs of low and high human capital workers Intuitively, note that with z s < z r and a persistent z process (as in the calibration) workers who endogenously separate will immediately change occupation (see Figure 5). Since these workers will be above their z r cutoffs in the new occupation, they face a lower risk of further endogenous separations damping down this margin. However, with z s > z r workers who endogenously separate and managed to become re-employed in the same occupation will remain close to z s , facing once again a high job separation probability. Among those who changed occupations, there will still be a mass of workers close to their z s cutoffs who face a high risk of future job separation. This leads to a larger amount of endogenous separations for both stayers and movers when z s > z r . As shown below, in the calibrated model z s > z r and the hazard rate of job separations among new hires out of unemployment is greater for occupational stayers, 0.035, than for occupational movers, 0.027, as suggested by the previous arguments. This is qualitatively consistent with SIPP data, where we find a hazard rate among new hires of 0.026 for stayers and 0.024 for movers. 32 The model also remains consistent with the (untargeted) probability that a worker who changed occupation after an unemployment spell, changed occupation after a subsequent unemployment spell. This probability is 0.54 in the model and 0.56 in the data (see Section 2.4). 31 26 relative to the average of these workers’ productivities. 4.4 Net occupational mobility Variation over the business cycle can naturally inform the cyclical sensitivity of occupation-wide productivities. In particular, to recover ϵo we target the levels of net mobility each task-based category exhibits in recessions and expansions (“Net mobility o, Recessions and Net mobility o, Expansions”) as well as their implied difference (“Net mobility o, Rec-Exp”). We also regress (for each o) the completed (log) unemployment durations of those workers whose pre-separation task-based category was o on the (log) unemployment rate and a time trend, and target the ratio between the estimated unemployment duration elasticity and the average elasticity across task-based categories, εU Do ,u /εU Davg ,u (see Online Appendix C.1 for details). The advantage of this approach is that it allows us to leave untargeted the cyclicality of aggregate unemployment, which we separately evaluate in Section 5. To inform the values of po we target the average net mobility level of each o (“Net mobility o, Mean”). We also use cyclical variation to inform the degree of directness in workers’ search across occupations. To recover ν we exploit the observed differences in the cyclicality of inflows across task-based categories. As ν increases, workers should be more sensitive (ceteris paribus) to cyclical differences in po when choosing occupations, making the inflows to occupations with the higher po respond stronger. To capture how cyclically sensitive are the inflows we compute, separately for expansions and recessions, the ratio of inflows into task-based category o over the sum of all flows. For each o we target the difference between the expansion and recession ratios, ∆exp−rec (inflow o/all flows). To recover αo,õ we target the observed task-based occupation transition matrix. To recover the set of ψo we use the employment-size distribution of task-based categories observed in 2014, the end of our sample period, “Prop (empl. size o2014 )”. We also target the average gross mobility rate across task-based categories (“Task-based gross occ. mobility rate”) so that the model remains consistent with gross mobility at this level of aggregation. Table 3: Calibrated Parameters ρA σA b k η Occ. human capital process 0.9985 x2 0.0020 x3 0.830 γd 124.83 δL 0.239 δH Occupational mobility 1.171 c 1.458 ρz 0.0032 σz 0.0035 z norm 0.0002 ν Agg. prod. and search frictions Occupation-specific po 7.603 ϵo 0.9983 ψo 0.0072 αo,N RC 0.354 αo,RC 0.04 αo,N RM αo,RM Non-routine Cognitive Routine Cognitive Non-routine Manual Routine Manual 1.019 0.988 1.000 0.988 1.082 1.120 0.532 1.283 0.620 0.145 0.087 0.147 0.436 0.407 0.000 0.000 0.560 0.383 0.093 0.140 0.004 0.210 0.384 0.767 0.000 0.000 0.524 0.094 27 4.5 Estimated parameters Table 3 reports the resulting parameter values implied by the calibration. The estimated value of b represents about 80% of total average output, y, not too far off from Hall and Milgrom’s (2008) estimate, though we use different information. Vacancy cost k translates to a cost of about 30% of weekly output to fill a job. The elasticity of the matching function in each submarket (z, x) within an occupation is estimated to be η = 0.24, about half of ηb = 0.5 when aggregating across all submarkets across occupations.33 The actual returns to occupational experience x2 and x3 are higher than the OLS returns, because occupational entrants select better z-productivities that typically mean-revert over time, dampening the average evolution of composite xz-productivity. The parameter γd implies that a year in unemployment costs an experienced worker in expectation about 5% of his productivity. The estimated values of δL and δH imply that exogenous separations are much more prevalent for low rather than high human capital workers, leading to a larger importance of endogenous separations among the latter, as implied by the prime-aged survival and mobility-duration profiles. The estimated value of c and the sampling process imply that upon starting a job in a new occupation, a worker has paid on average a reallocation cost of 15.18 weeks (or about 3.5 months) of output. This suggests that reallocation frictions are important and add to the significant lose in occupational human capital when changing occupation.34 The process driving workers’ idiosyncratic productivities within an occupation has a broadly similar persistence (at a weekly basis) as the aggregate shock process driving the business cycle. However, its larger variance implies there is much more dispersion across workers’ z-productivities than there is across values of A. We also find that workers’ idiosyncratic productivities are much more dispersed than occupation-wide productivities. For example, the max-min ratio of po is 1.13 (1.09) at the highest (lowest) value of A, where the RM task-based category is the most responsive to aggregate shocks and N RM the least. In contrast, the max-min ratio among z-productivities is 2.20. To gauge whether the dispersion across z-productivities is reasonable we calculate the implied amount of frictional wage dispersion using Hornstein et al. (2012) M m ratio. These authors find an M m between 1.46 and 1.90 33 The difference between η and ηb is mainly due to the effect of aggregation across submarkets that exhibit rest unemployment. Workers in episodes of rest unemployed entail no vacancies, have zero job finding rates, do not congest matching in other submarket, but are included in the aggregate number of unemployed. Hence they are included in the denominator of the aggregate labor market tightness and the aggregate job finding rate. It can be shown that this creates s s a wedge between η and η̂ = 0.5 that is governed by 0.5−η 1−η εθ̂,A = εu ,A , where εθ̂,A and εu ,A denote the cyclical elasticity of aggregate labor market tightness, θ̂, and the proportion of search unemployment over total unemployment, us , respectively. Since in the calibrated model both elasticities are positive, 0.5−η 1−η must also be strictly positive and hence η < ηb = 0.5. In addition, each submarket within an occupation has its own concave matching function and hence aggregating these concave functions across submarkets also imply that the calibrated value of η will further diverge from 0.5. 34 The average reallocation cost is computed as the product of c and the number of times workers sample a new occupation, which is 1.996 times. The value of c reported in Table 3 is consistent with the large proportion of unemployed workers who changed occupation. Given that in the data occupational changes are typically accompanied by changes in industries (based in our own calculations) and, to a lesser extent, by geographical location (see Papageorgiou, 2018), the estimated value of c could also be capturing the moving costs associated with these changes. Indeed, Alvarez and Shimer (2011) find also large reallocation costs across industries, while Kennan and Walker (2011) and Papageorgiou (2018) find large reallocation costs across geographical locations. 28 using the PSID, while the estimated z-dispersion yields 1.40. The estimated value of ν implies that the ability of workers to access job opportunities in other task-based categories plays an important role in shaping the direction of their search. The estimated values of αo,õ imply that on average workers in N RC have a low probability of drawing a new z from manual occupations and vice versa; while workers in N RM and RM occupations mostly draw a new z from these same two categories, although drawing from RC is not uncommon. In addition, the value of ν implies workers significantly adjust their direction of search as a response to cyclical occupation-wide productivity differences. This is evidenced by the ability of the model to reproduce the observed cyclical changes in the net mobility patterns presented in Section 2 and Table 2, where RM occupations have the strongest cyclical response of net outflows, increasing in recessions, as well as the strongest response in the inflow proportion, also larger in recessions. In contrast, N RM occupations are the ones which experience the largest increase in net inflows in recessions and the largest increase in inflows as destination category (see also Online Appendix C.1, Figure 4). Taken together, these estimates show a high degree of directness when workers search across task-based categories. Further, Table 6 in Section 5 shows an important role of occupational mobility through unemployment in changing the relative sizes of N RM and RM occupations. In contrast, the high value of ψN RC captures that the N RC category did not increase its size between 1984 and 2014 because of inflows through unemployment, but rather because of a significant proportion of labor market entrants taking up jobs there. 5 Cyclical Unemployment Outcomes We now turn to investigate the cyclical patterns of aggregate unemployment and its duration distribution generated by the model, noting that these were not targeted in our estimation procedure. Our aim is to evaluate the importance of excess and net occupational mobility in generating these patterns. We first present the implications of the full model as estimated above. We then discuss the implications of a re-estimated version of the model where we shut down the heterogeneity in occupation-wide productivities.35 . An alternative exercise would be to maintain productivity differences across occupations but not allow workers to chose in which occupations to search on. Given that the estimated dispersion of z-productivities is much larger than that of po productivities, this exercise would not generate meaningfully different results. A second alternative could be to re-estimate a version of the model where we shut down the z-productivity process, making workers decide whether to change occupations based only on po productivities differences. It is clear, however, that this version of the model will not be able to reproduce many of the occupational mobility patterns documented in Section 2. With a slight abuse of terminology, we label this version “excess mobility model” as unemployed workers’ occupational mobility decisions are based solely on the changing nature of their z-productivities and their interaction with A and x. In Online Appendix C.2 we present the estimation 35 In this version the observed net mobility patterns can be imposed exogenously to keep the model’s gross occupational mobility patterns consistent with the evidence presented in Section 2 and Supplementary Appendix B 29 results of the excess mobility model. Table 4: Logged and HP-filtered Business Cycle Statistics. Data (1983-2014) and Model u Data σ ρt−1 Full Model σ ρt−1 Exc. Mob. Model σ ρt−1 v Volatility and Persistence θ s f outpw occm u Correlations with u and outpw v θ s f outpw occm 0.14 0.11 0.25 0.10 0.09 0.98 0.99 0.99 0.94 0.91 0.01 0.93 0.03 0.91 u outpw 1.00 -0.92 0.56 -0.98 0.80 0.51 -0.39 -0.82 0.27 -0.47 1.00 -0.52 0.38 0.14 0.05 0.17 0.07 0.10 0.93 0.90 0.92 0.87 0.92 0.01 0.88 0.04 0.93 u outpw 1.00 -0.61 0.76 -0.96 0.79 0.96 -0.90 -0.88 0.93 -0.94 1.00 -0.82 0.83 0.14 0.05 0.18 0.07 0.10 0.95 0.89 0.94 0.88 0.93 0.01 0.94 0.04 0.90 u outpw 1.00 -0.63 0.77 -0.97 0.78 0.96 -0.87 -0.88 0.93 -0.94 1.00 -0.80 0.83 Note: The excess mobility model considers only occ. mobility decisions based on the z-productivity process. Each model’s aggregate time series arise from the distributions of employed and unemployed workers across all labor markets, combined with agents’ decisions. Times series are centered 5Q-MA series of quarterly data (both model and data), to smooth out the discreteness in the relatively flat cutoffs (relative to the grid) discussed further in the computational appendix. The cyclical components of the (log) of these time series are obtained by using an HP filter with parameter 1600. See Online Appendix C.1 for further details and results without the 5Q-MA smoothing. Aggregate unemployment Table 4 shows the cyclical properties of the aggregate unemployment, vacancy, job finding and separation and gross occupational mobility rates, computed from the data and the simulations.36 It shows that the full model is able to generate a countercyclical unemployment rate, together with a countercyclical job separation rate, procyclical job finding and gross occupational mobility rates. Table 4 also shows that the cyclical volatilities and persistence of the aggregate unemployment, job finding, separation and gross occupational mobility rates are very close to the data. Note that this aggregate behavior is not driven by a higher cyclicality of young workers’ unemployment rate. In Online Appendix C.1 we show that the responsiveness of the unemployment rate to aggregate output per worker is slightly stronger for prime-aged workers than for young workers, leading to a countercyclical ratio of unemployment rates between young and prime-aged workers. Therefore, in the model the pool of unemployment shifts towards high human-capital, prime-aged workers during recessions, a feature noted by Mueller (2017). This occurs mostly due to the larger increase in endogenous job separations among prime-aged relative to young workers. The model also generates a strongly negatively-sloped Beveridge curve as in the data. The latter stands in contrast with the canonical DMP model, where it is known that endogenous separations hamper this model from achieving a Beveridge curve consistent with the data. It also stands in contrasts with the predictions of many multi-sector models where unemployment fluctuations arise from the time-consuming reallocation of workers from the sector that experienced a negative shock to the one that experienced a positive shock. As argued by Abraham and Katz (1986), these models typically imply an upward sloping Beveridge curve as more vacancies are created in the latter sector (see Chodorow-Reich and Wieland, 2020, for a recent exception). We return to this point in Section 5.2, where we discuss the role of occupation-wide heterogeneity. Unemployment duration distribution Panel A in Table 5 evaluates the ability of the model to reproduce the shifts in the incomplete unemployment duration distribution with respect to changes in the unemployment rate. It shows that the shares of unemployed workers by duration exhibit a very 36 Both in the model and data the unemployment, job finding and separation rates are computed based on the same unemployment definition used in the previous sections, while the cyclical properties of the occupational mobility rate are computed using major occupational groups and after applying the Γ-correction matrix. In Online Appendix C.1 we provide the full set of correlations. 30 .2 .3 .4 .5 .6 .7 Note: The elasticities are constructed using the cyclical component of the series of the shares of unemployed workers by durations, the aggregate unemployment rate. .1 Panel A: Cyclicality of Duration Distribution Elasticity wrt u Semi-elasticity wrt u Full Excess Full Excess Model Model Data Model Model Data 1 − 2m -0.435 -0.447 -0.464 -0.168 -0.165 -0.169 1 − 4m -0.316 -0.329 -0.363 -0.178 -0.179 -0.186 5 − 8m 0.388 0.350 0.320 0.074 0.070 0.071 9 − 12m 1.083 1.033 0.864 0.061 0.060 0.072 > 13m 1.787 1.513 1.375 0.047 0.048 0.044 Panel B: Semi-Elasticity Duration wrt u by Occupational Mobility Unemp. HP-filtered Log u linearly detrended Duration Full Excess Full Excess by Mob. Model Model Data Model Model Data Movers 2.9 2.9 3.2 2.4 2.3 2.0 Stayers 1.5 1.4 2.5 1.2 1.2 1.6 Unemp. Duration 0 20 40 60 80 Percentile of Distribution Times of High U Times of Low U Corr. Mob. - Data Corr. Mob. Model Corr. Mob. Model gdcmdur_m 95% CI 95% CI 100 Figure 7: Cyclical Shift of Distribution Table 5: Cyclical duration distribution similar degree of responsiveness with cyclical unemployment as in the data. Crucially the elasticity measure shows that the model creates a strong response in the shares of unemployment at long durations. When using the semi-elasticity measure the model generates a nearly perfect fit. Thus, in our model as in the data cyclical changes in the aggregate unemployment rate are driven by particularly strong cyclical changes in long-term unemployment. An important force behind the increase in long-term unemployment during recessions is the larger increase in the unemployment duration of occupational movers relative to stayers. Panel B in Table 5 shows the cyclical responses of the average unemployment duration of movers and stayers using different measures. Along all of these measures the model’s average unemployment duration of occupational movers increases more than the average unemployment duration of stayers, an increase that is consistent with the data. Stayers’ durations respond somewhat less relative to the data, between 60% (relative to the log HP-filtered unemployment measure) and 80% (relative to the linearly detrended unemployment measure). Relative to the lack of amplification in conventional DMP models, this still constitutes a large response. As in the data, the lengthening of movers’ unemployment duration contributes meaningfully to the increase in long-term unemployment during recessions. Figure 7 shows how the untargeted shift in unemployment durations combines with the targeted shift of the mobility-duration profile. At any percentile of the unemployment duration distribution, the model generates a drop in occupational mobility in recessions. By comparing the observations’ x-coordinates, this figure also illustrates that the cyclical shift of the model’s duration distribution follows the data. Excess vs. net mobility A key insight from Tables 4 and 5 is that the aforementioned cyclical patterns are nearly identical to the ones generated by the excess mobility model. In Online Appendix C.2 we show that this model also fits very well the economy-wide targets described in Table 2 and the estimated values of [c, ρz , σz , z norm , x2 , x3 , γd , δL , δH , k, b, η, ρA , σA ] are nearly identical to the estimated in the full model. This comparison demonstrates that allowing workers to chose in which 31 occupations to search in due to occupation-wide productivity differences is not the reason why the model is able to replicate the cyclical patterns of aggregate unemployment and its duration distribution. Instead, the excess mobility calibration highlights the importance of the worker-occupation idiosyncratic productivity process and its interaction with aggregate productivity in generating these cyclical patterns. The excess mobility model and the full model calibrations are successful in these dimensions because they yield similar implications for search, rest and reallocation unemployment during workers’ unemployment spells. In Section 5.1 we first demonstrate this claim using the excess mobility model’s calibration. This also allows us to show in more detail the importance of having a persistent zproductivity process for the cyclical performance of the model. In Section 5.2 we show that the same forces occur within each task-based category in the full model, although modulated by differences in the level and cyclical responsiveness of po across occupations. 5.1 Main mechanism As argued in Section 3.3, the relative position and slopes of z s and z r are key determinants of the longrun and cyclical implications of our model. We now discuss these in the context of the calibrations. Relative position of z s and z r Figure 8a depicts the cutoff functions generated by the excess model calibration as a function A given x, where all occupations share the same cutoff functions. It shows that z s ≥ z r for nearly all A and h = 1, 2, 3. The exception being z s (A; x1 ) < z r (A; x1 ) for the highest values of A. This implies that periods of search, rest and reallocation unemployment can occur within the same unemployment spell as A and z evolve. Further z s and z r decrease with x such that, as predicted by our theory, workers with higher human capital are less likely to change occupations relative to those with lower human capital. As z s (., x3 ) < z s (., x1 ) the average level of separations is also lower for high human capital workers (noting that δL and δH also contribute to this difference). Once separated, high human capital workers spend on average a longer time in unemployment due to the larger distance between their z s and z r cutoffs. Given the values of x, our theory shows that c, ρz and σz are key determinants of the distance between z s and z r , and therefore of the presence of episodes of rest unemployment. To illustrate why values of these parameters that lead to z s ≥ z r allow the model to match the mobility-duration profile and survival functions, consider a set of workers with the same x who just endogenously separated. Given z s ≥ z r and a persistent z process (as in the calibration), these workers will be initially close to z s . A small positive shock would then suffice to move them above z s , while only large negative shocks would take them below z r . Hence at short durations these workers face relatively high job finding rates and, if re-employed, they will be most likely occupational stayers. Those who stayed unemployed for longer would have on average experienced further negative z shocks and would face a higher probability of crossing z r . However, the stochastic properties of the z-process imply there will still be many of these workers that end up crossing z s . As a result, the likelihood of an occupational move increases moderately with unemployment duration, while the job finding rate decreases with unemployment duration. 32 1 1 .8 Search u. Rest u. zs(A, x2) zr(A, x2) zs(A, x1) zr(A, x1) zs(A, x2) zr(A, x2) zs(A, x3) zs(A, x3) zr(A, x3) .96 .98 1 1.02 1.04 Aggregate Productivity Shock (a) Unemployment Distribution 1.06 0 .4 .4 zr(A, x3) .94 Unemployment by Type .4 .6 zs(A, x1) zr(A, x1) .2 z-productivity level .6 .8 z-productivity level .6 .8 Reall. u .94 .96 .98 1 1.02 1.04 Aggregate Productivity Shock (b) Employment Distribution 1.06 .95 1 Aggregate Productivity Shock (A) 1.05 (c) Unemployment Decomposition Figure 8: Cutoffs, Unemployment Distribution and Decomposition Slope of z s and z r Figures 8a,b show that ∂z s /∂A < 0 and ∂z r /∂A > 0 for each x. This property implies that during recessions there is an increased scope for episodes of rest unemployment; while in expansions there is an increased scope for episodes of search unemployment. Figure 8c illustrate this last feature by showing the proportion of workers facing an episode of search, rest or reallocation unemployment for a given value of A. Although both rest and search unemployment are countercyclical, search unemployment episodes are relatively more common when the economy moves from mild recessions up to strong expansions. It is only as recessions get stronger that rest unemployment episodes become more common. The slopes of the cutoffs reveal a cyclical area of inaction, [z r (A; x), z s (A; x)], for each x. These areas of inaction are key to understand the cyclical performance of unemployment and vacancies. The negative slope of the z s cutoffs together with the large mass of workers right above them (see Figure 8b) imply that a decrease in A leads to a large increase in the inflow of workers into rest unemployment episodes. The positive slope of the z r cutoffs implies that the same decrease in A also leads to a large decrease in the outflow from rest unemployment via reallocation. These forces significantly add to the density of unemployed workers already “trapped” within these areas (see Figure 8a). Given that no firm in an occupation expects to be able to make a profit by hiring these workers, vacancy creation falls as well. As conditions improve the areas of inaction narrow considerably such that rest unemployed workers are now much more likely to get a z shock that takes them below (or above) z r (z s ).37 As the surplus from hiring these workers becomes positive and higher occupational mobility flows help workers increase their z-productivities, vacancy creation goes up across all occupations. The strong cyclical responses of rest and search unemployment to the changes in the areas of inaction imply that aggregate unemployment, u, also becomes highly responsive to A. Episodes of reallocation unemployment, however, make a small contribution to the cyclicality of u because they only capture the time spent transiting between occupations, which is about 2 weeks on average, after which workers continue their jobless spell in episodes of rest or search before finding a job in a new 37 In recessions that involve a 5% reduction in A relative to the mean, workers still face an average probability of about 25% of transitioning out of rest unemployment within a month; and this probability sharply increases with aggregate productivity. 33 occupation. In Online Appendix C.2 we show that these patterns occur across low and high human capital levels, explaining why we obtain unemployment, job finding and separations rates across age groups with similar cyclical responses. The widening of the area of inaction during recessions also imply that long-term unemployed workers require a sequence of more and larger good z shocks before becoming search unemployed in their pre-separation occupation. They would also require a sequence of more and larger bad shocks before deciding to change occupations. In contrast, for those workers who have just endogenously separated z s is the cutoff that weighs most on their future outcomes. For these workers the distance to the nearest cutoff is therefore not as responsive to A as for the long-term unemployed. Hence, we observe that the outflow rate of long-term unemployed workers responds more to changes in aggregate conditions relative to the outflow rate of shorter-term unemployed workers. This mechanism then translates into a stronger increase in the share of long-term unemployed in recessions as shown in Table 5, stronger than the one predicted based on the decline of f alone. The same mechanism also implies that at low values of A the time spent in rest unemployment increases more for (ex-post) occupational movers than for occupational stayers. This rationalizes the stronger increase in average unemployment duration among occupational movers relative to stayers during recessions documented in Section 2.4. The role of human capital depreciation Human capital depreciation is important in determining these dynamics as it affects the cyclical changes in the areas of inaction. Workers with a z-productivity much lower than z s take into account that even with a sequence of positive z realizations they might experience depreciation and reallocate anyway, decreasing the option value of waiting in their occupations and flattening the z r cutoff. At separation a similar argument operates: increases in z s − z r during recessions imply that depreciation more often leads to a reallocation than otherwise, increasing the option value of staying employed and flattening the z s cutoff. In Online Appendix C.2 we demonstrate that this mechanism is important by re-estimating the model without human capital depreciation. Such a version of the model exhibits a stronger amplification of rest unemployment and, as a consequence, generates too large a volatility of the aggregate unemployment rate as well as too little occupational mobility during recessions. The role of occupational mobility The cyclical sensitivity of the areas of inaction is also tightly linked with the existence of the z r cutoff and the properties of the z-productivity process. To show this we re-estimate the model not allowing workers to change occupations. We use all the same moments outlined before except those pertaining to occupational mobility. In Online Appendix C.3 we show that the calibrated one-sector model with no occupational mobility can do well in fitting most of the targeted long-run moments, particularly the unemployment survival functions for all workers and by age groups. However, the aggregate unemployment, vacancy, job finding and separation rates now exhibit below half the cyclical volatility observed in their data counterparts, 0.04, 0.02, 0.03 and 0.03, respectively, and the correlation between unemployment and vacancies drops to -0.32. The cyclicality of the unemployment duration distribution is also far from the data, generating too little cyclical response across all durations, but particularly among the long-term unemployed. 34 The main reason why this version generates such a low cyclical response is that the new area of rest unemployment is defined by the set of z-productivities that lie in [z, z s (A; x)], where z denotes the lowest value of z. This implies that any cyclical changes in the size of this area now solely depend on the responsiveness of z s relative to the workers’ z distribution. Although ∂z s /∂A < 0 and hence separations are countercyclical, this model cannot resolve a key trade-off: in the absence of the z r cutoff the z process is less persistent and exhibits a much larger standard deviation, which creates enough heterogeneity in unemployment durations to allow it to match the empirical unemployment survival functions. However, the new estimated properties of the z process also increase the heterogeneity in z-productivities relative to the cyclical range of A. This dampens the model’s cyclical performance as it implies less responsive z s cutoffs, weakening the cyclical responses of job separations and the rate at which workers leave the area of rest unemployment. In Online Appendix C.3 we show that an alternative version of the one-sector calibration with a more persistent and less volatile z process can create a much larger cyclical amplification of the unemployment rate and a stronger Beveridge curve, but at the cost of missing many of the unemployment duration targets and generating too much long-term unemployment even in expansions. It then also misses the cyclicality of the unemployment duration distribution, generating too little response in long-term unemployment. Thus, the one-sector version of our model appears unable to reconcile the observe cyclical fluctuations in aggregate unemployment with those of its duration distribution. This trade-off disappears once unemployed workers are allowed to weigh the option of waiting for their conditions to improve in their occupation with that of reallocating, as the z r cutoffs create narrower and more cyclically sensitive areas of inactions for each x. 5.2 Occupation Heterogeneity and Cyclical Unemployment We now show that the same mechanisms described above hold within each task-based category but their strength varies across these occupational groups. Consequently, unemployed workers face different unemployment outcomes that depend also on the identity of the occupation. Both the long-run and cyclical dimensions of occupation-wide productivity differences are relevant. To understand the former, column 5 in Table 6 shows the contribution of unemployed occupational switchers in changing the observed sizes of the task-based categories in our calibration. This is compared to the contribution of the exogenous entry and exit process as captured by d and ψo (column 4 “Entrants”), such that for each task-based category the two values add up to the change in the employment stock. The calibration shows that N RM occupations increased in size due to more unemployed workers switching to these occupations than away from them. In contrast, RM and RC decrease in size as more unemployed workers move away from these occupations than to them. The last two columns of Table 6 show the contribution of mobility through unemployment separately by periods of high and low unemployment, where we categorise these periods by comparing the HP-filtered unemployment rate to its median. We observe that it is during recessions that mobility through unemployment particularly accelerates the changing size of N RM and RM occupations, representing about two-thirds and three-quarters of the total contribution of this channel, respectively. 35 Table 6: Role Unemployment in the Changing Size of Occupations Distributions Task-Based Occupational Categories Non-routine Cognitive Routine Cognitive Non-routine Manual Routine Manual Initial Model Decomposition of Distribution Change End Distribution Entrants Occ. Mob through Unemployment Distribution Data Model All Qtrs All Qtrs Qtrs u < umedian Qtrs u ≥ umedian 0.224 0.292 0.226 0.258 0.329 0.258 0.260 0.154 0.337 0.246 0.260 0.157 0.133 -0.019 -0.036 -0.067 -0.020 -0.027 0.070 -0.034 -0.011 -0.009 0.025 -0.008 - 0.009 - 0.018 0.045 - 0.026 Jaimovich and Siu (2020) already documented the importance of recessions in changing the size of routine occupations. Here we show that the net mobility patterns described in Section 2 together with the endogenous response in unemployment yield precisely such a pattern within our model. Figure 9 illustrates the mechanism behind this. Figure 9a shows the levels and cyclicalities of the estimated occupation-wide productivities for the range of A. Reflecting the estimated values of ϵo , it shows that RM and RC occupations are strongly negatively affected in recessions, but catch up with the average in expansions. In contrast, N RM occupations are the least attractive in expansions but become more attractive in recessions. N RC occupations are consistently above average over the cycle (more so in expansions). Figures 9b, 9c, 9e, 9f show that these different cyclical productivities result in different separation and reallocation cutoffs. Although their levels are not that different across task-based categories, in RM occupations the separation cutoffs decreases more steeply, while the reallocation cutoffs are nearly horizontal. In N RM occupations the separation cutoffs are nearly horizontal and the reallocation cutoffs are strongly upward-sloping. This implies that in recessions job separations are more prominent in RM than in N RM occupations. Despite the differences in slopes, all task-based categories exhibit cutoffs with the z s > z r property. Further, the distance between these cutoffs creates areas of inaction that increase in recessions and narrow in expansions as described earlier. Figure 9d shows that as a result rest unemployment episodes are more common than search unemployment episodes in recessions within each task-based category. As the economy recovers search unemployment episodes are the most common ones. The observed countercyclical net mobility patterns then occur for mainly two reasons: (i) a differential cyclical response in the outflows across task-based categories, such that some task-based categories shed more workers during recessions relative to the average; and (ii) a differential cyclical response in the inflows, such that those workers who have decided to change occupations choose their destination task-based category differently in recessions than in expansions. The widening of the area of inactions as A decreases implies that overall occupational mobility falls during recessions in all task-based categories. However, the differential responses in occupation-wide productivities across the cycle imply that the decrease in outflows is stronger in N RM occupations and weaker in RM occupations relative to the average, as observed in the data. At the same time, Table 2 shows that the model is also able to reproduce the shift in the inflow distribution towards RM and away from N RM occupations that occurs in recessions. 36 Figure 9: Heterogeneity across Occupation across the Cycle NRC RC NRM RM (b) NRC Cutoffs & U. Distr. (c) RC 1 Aggregate Productivity Component A U. Decomposition z-productivity level .8 .6 .96 .98 1 1.02 1.04 Aggregate Productivity Shock (e) NRM 1.06 Cutoffs & U. Distr. (f) RM .98 1 1.02 1.04 Aggregate Productivity Shock 1.06 Cutoffs & U. Distr. 1.05 .4 .6 z-productivity level .8 .6 .4 1 Aggregate Productivity Component A .96 RC RM 0 .95 .94 1 NRC NRM All Ratio of Rest/Search Unemployment .5 1 1.5 .94 z-productivity level .8 2 (d) Cyclical 1.05 1 .95 .4 .4 .9 .6 z-productivity level .8 Occupation Productivity (incl. A) .95 1 1.05 Average Cutoffs & U. Distr. 1 Occ. Productivty 1 1.1 (a) Cycl. .94 .96 .98 1 1.02 1.04 Aggregate Productivity Shock 1.06 .94 .96 .98 1 1.02 1.04 Aggregate Productivity Shock 1.06 Accompanying these countercyclical net mobility patterns, vacancy creation in every occupation is procyclical. As mentioned earlier, this feature stands in contrast with many multi-sector models in the spirit of Lucas and Prescott (1974) were vacancy creation increases in recessions, generating an upward sloping Beverdige curve. In our framework, in contrast, all occupation-wide productivities co-move with the common aggregate productivity shock and the loadings ϵo only create relative productivity differences across occupations. These relative difference are economically important, driving the net mobility patterns. 6 Conclusions In this paper we show that there is no tension between the cyclical behavior of individual unemployment outcomes, procyclical gross occupational mobility and countercyclical net mobility through unemployment. While individual outcomes are to a large extent driven by the interaction between worker-occupation idiosyncratic and aggregate shocks, net mobility is affected by occupation-wide productivity differences and unemployed workers’ differential responses to these. Further, given that net mobility increases in recessions, transitions through unemployment play a meaningful role in shaping the changing size of RM , RC and N RM occupations. 37 Along with the high mobility rate increasing with duration, many long-term unemployed still return to their previous occupation. The model interprets this as a sizeable option value of waiting for prospects to improve in one’s previous occupation. In recessions, this option value becomes more important and increases the unemployment durations of stayers and more so of movers, a pattern observed in the data. This implies that in the model the nature of unemployment changes over the cycle. In expansions (and mild recessions) the typical worker is not able to find jobs that are currently available to him due to standard search frictions and search unemployment becomes the main source of aggregate unemployment. As recessions get stronger the typical worker is not able to find jobs because there are no jobs posted for him. In this case, rest or wait unemployment becomes the main source of aggregate unemployment. These dynamics translate into large cyclical changes in aggregate unemployment and its duration distribution. The concept of rest unemployment is closely related to that of mismatch, stock-flow and rationing unemployment. Shimer (2007) defines mismatch unemployment as those workers who remain attached to a local labor market even though there are currently no jobs for them. In stock-flow matching, unemployed workers in the stock wait for new jobs to arrive, as existing vacancies do not offer suitable employment opportunities. In Michaillat (2012) rationing unemployment occurs because workers are currently unproductive and no jobs are posted for them. As conditions improve, they become productive and employable once again. A key difference with all these models, is that here workers in rest unemployment episodes always have the option of looking for jobs in alternative occupations. Crucially, the occupational mobility decision changes over the cycle, with a larger proportion of workers deciding not to use this option in recessions. Throughout, our analysis we have considered workers who are currently in a rest unemployment episodes as part of the labor force, still searching and expecting a positive job finding probability in the near future. Episodes of rest unemployment, however, could conceptually be extended to incorporate marginally attached workers. In terms of occupational mobility patterns, Supplementary Appendix B shows that our analysis is robust to introducing periods of non-participation within workers’ jobless spells. Online Appendix C.2 shows that considering non-participation periods in our targeted statistics does not alter the quantitative performance of our model. These exercises suggest that our results are robust to inclusion of the marginally attached. Although other models have been successful in replicating some of the cyclical unemployment patterns described here, Bils et al. (2011) argue that these models would typically have difficulty in jointly explaining the observed cyclicality of the aggregate unemployment rate and generating realistic dispersion in wage growth. This should not be an issue in our framework. As shown in Sections 4 and 5 our calibration generates the observed cyclicality of unemployment together with a realistic amount of wage dispersion as measured by Hornstein et al. (2011) M m ratio. In this paper we have emphasised labor market flows pertaining to the unemployed, but extending the analysis to include heterogeneity in firm-worker matches and on-the-job search would allows us to study the cyclical relationship between wages, occupational mobility, unemployment fluctuations. We leave this for future research. 38 Appendices • Online Appendix: Different calibrations, in detail; summary of the Occupational Code ErrorCorrection model, and Theory https://github.com/lpvscotland/website-files/raw/main/Online_Appendix_Oct_2021.pdf • Supplementary Appendix A: Occupational Coding Errors and Their Correction: Full study of Occupational Coding Errors for the Unemployed in the SIPP and PSID, and the Code Error Correction Method we estimate on the SIPP, including proofs, robustness, out-of-sample predictions, etc. https://github.com/lpvscotland/website-files/raw/main/Supp_Appendix_A_October_2021.pdf • Supplementary Appendix B: Occupational Mobility of the Unemployed in the Data: extensive robustness investigations in the SIPP, CPS, PSID, and the data construction appendix covering the data in the main text. https://github.com/lpvscotland/website-files/raw/main/Supp_appendix_B_Oct_2021.pdf • Supplementary Appendix C: Theory: Further Exposition of the Theory, Proofs etc. https://github.com/lpvscotland/website-files/raw/main/Supp_Appendix_C_October_2021.pdf References [1] Abraham, K. and L. 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