Survey design and use of data. We consider the period 1983 - 2013. To cover this period we use 13 panels in total: the 1984-1988, 1990-1993, 1996, 2001, 2004 and 2008 panels. For the 1984-1988 and 1990-1993 panels we have used the Full Panel files as the basic data sets, but appended the monthly weights obtained from the individual waves (sometimes referred to as core wave data). Until the 1993 panel we use the occupational information from the core waves. We do this for two reasons: (i) the full panel files do not always have an imputation flag for occupations; and (ii) between the 1990 and 1993 panels firm identities were retrospectively recoded, based on core wave firm identifiers. For our study it is important to be clear to which firm the occupation belongs. We exclude the 1989 SIPP panel because the US Census Bureau does not provide the Full Panel file for the 1989 data set and this panel was discontinued after only three waves (12 months). Since we want to be conservative regarding censoring, we opted for not using this data set. This is at a minor cost as the 1988 panel covers up to September 1989 and the 1990 panel collects data as from October 1989. For the 1996, 2001, 2004 and 2008 panels there is no longer a Full Panel file nor a need for one. One can simply append the individual wave information using the individual identifier “lgtkey” and merge in the person weights of those workers for whom we have information from the entire panel (or an entire year). In this case, the job identifier information is also clearly specified. Two important differences between the post and pre-1996 panels are worth noting. The pre-1996 panels have an overlapping structure and a smaller sample size. Starting with the 1996 panel the sample size of each panel doubled in size and the overlapping structure was dropped. We have constructed our pre-1996 indicators by obtaining the average value of the indicators obtained from each of the overlapping panels.

The SIPP's sample design implies that in all panels the first and last three months have less than four rotation groups and hence a smaller sample size. For this reason, in our time series analysis, we only consider months that have information for all four rotation groups. For statistics for which the distribution of unemployment duration matters we require that workers have at least 14 months of labor market history at the moment of re-employment in their corresponding SIPP panel. If necessary (and discussed in detail below), we impose further restrictions to deal with censored spells in order to generate a representative distribution of unemployment spells for at least up to one year. This restriction addresses that e.g. short completed unemployment spells typically have lower mobility, while in the first waves of a panel spells started and completed within the panel are necessarily of short duration. This is even more important when constructing the job finding rates, especially when we want to focus on the job finding rates in completed spells for which we know the occupational mobility outcomes. For the cumulative survival profile in unemployment, we consider all spells that at their start have at least 32 months of subsequent continuous presence in the sample and restrict these observations to be in the first 4 waves of the panel. For job finding rates in the time series, which are based on incomplete spells, we require that all workers whose job finding rate is measured at duration $x$ remain continuously in the sample for at least 19-$x$ more months.

The data also shows the presence of seams effects between waves, where transitions are more likely to occur at a seam (i.e. between waves, and therefore at 4,8, 12…months) than based on other characteristics, e.g. duration. When we consider time series and given the above restrictions, there is always one rotation at the seam in every month we consider which effectively smoothes out the clustering at the seam. In the case of the duration statistics for which the seam effect matters, we either consider observations in 4 months bins (e.g. survival at 4, 8, 12, 16 months of unemployment) or use the standard methods to reducing the seam bias by smoothing out the survival rates or considering “cumulative” mobility rates with duration (see e.g. for the mobility-duration profile reported in Section 2.2 of the main text).

We use the person weights per wave (“wpfinwgt”, and equivalent), but normalize these such that the average weight within a panel is equal to one. This is done because the size of panels is not constant, and we do not want to weigh panels with less observations more heavily as within a wave of a panel “wpfinwgt” adds up to population totals and thus is higher on average when sample size is smaller. We think of our normalization as a reasonably agnostic approach that keeps the relative weights within a panel intact, but also takes into account the number of available observations.

Sample selection and labor market status. For the 1984-2008 panels, we consider all workers between 18 and 65 years of age who are not in self-employment or in the armed forces nor in the agricultural occupations.[As agricultural occupations could be miscoded nonagricultural occupations and vice versa, in our code-error corrected measure, we take agricultural workers according to reported occupational codes into account, apply our correction method, and remove agricultural workers after correction from our sample and associated statistics.] We measure an individual's monthly labor force status in the SIPP using two sources of information. The first one relies on the labor force status reported at the second week of each month. The second relies on the monthly employment status recode. Using these two sources (and using the SIPP 2001 wording as an example) we consider a worker to be employed during a month if the individual reported in the second week of that month that he/she was “with job/business - working”, “with job/business - not on layoff, absent without pay” or “with job/business - on layoff, absent without pay”. The category “with a job” (Census 2008) is assigned if the person either (a) worked as paid employees (or worked in their own business or profession or on their own farm or worked without pay in a family business or farm) or (b) “were temporarily [emphasis added] absent from work either with or without pay.” Thus, the employment status recode category “with job/business - on layoff, absent without pay” appears to capture temporary layoffs. Note that as a result our definition of employment differs from the CPS definition of employment. As the SIPP documentation points out: “ “With a job” includes those who were temporarily absent from a job because of layoff and those waiting to begin a new job In 30 days; in the CPS these persons are not considered employed.”

We consider a worker to be employed if the individual reported in the monthly employment status recode variable that he/she was “with a job entire month, worked all weeks", but also when “with a job all month, absent from work without pay 1+ weeks, absences not due to layoff”, or “with a job all month, absent from work without pay 1+ weeks, absences due to layoff”. If workers have spent part of the month in employment and part of the month in unemployment, workers are nonemployed only if they are nonemployed in week 2 and have been nonemployed for at least four weeks in total. That is, those who have less than a month of nonemployment in week 2 are still counted as employed. If the worker is “no job/business - looking for work or on layoff” during one of the weeks in nonemployment (i.e. in the "no job/business") state, we consider the worker to be unemployed. We have chosen this classification, because we want entry into unemployment to capture the serious weakening of the link with the previous firm of employment, rather than to be a definite period of nonproduction after which the worker would return to the previous employer. The restriction of nonemployment for at least four weeks is meant to further limit the role of short-term absences from the same firm and temporary layoffs. This is motivated by the analysis of Fujita and Moscarini (2017), who document that many workers with very short unemployment spells return to their previous employer. We want to focus on those unemployed who at least consider employment in other firms and possibly other occupations.

[Sample: quarters after seam-correction SIPP]
[Sample: quarters with workers in sample >14months ]
Unemployment Rate SIPP (red, our measures) and BLS-CPS (blue, official)

With these restrictions, our measured unemployment rate is somewhat lower than the official unemployment rate. In Figure <ref>, we plot the unemployment rate in our SIPP sample, constructed according to our definition, next to the standard (CPS-based) unemployment rate from the BLS. The left panel displays the unemployment rate in the SIPP when excluding those quarters in which panels phase in or out. In this case all quarters considered are symmetric with respect to seams between waves: each month there is at least one rotation group which switches interview wave. The right panel considers only observations that have been in the sample for at least 14 months. This necessarily means bigger gaps between panels, but addresses issues of left-censoring when considering complete unemployment spells for which we can observe the occupational mobility. This is important when e.g. considering the distribution of unemployment durations of completed spells as at the beginning of the panel the only completed unemployment spells are necessarily those of short duration, which have a lower occupational mobility rate. Not taking this into account would lead to mistaken time series patterns of occupational mobility of the unemployed.

Both panels of Figure <ref> show that our restrictions lower the unemployment rate by about 1.5pp. This level effect is nearly uniform over time and over the cycle. That is, the time-series evolution of our measure of unemployment follows BLS unemployment closely, while the level effect is nearly unaffected by the business cycle. If anything, our measure responds slightly stronger to the business cycle (a 1pp rise in the official rate raises our unemployment measure by 1.04pp, this is small but statistically significant). The correlation between our unemployment measure and the BLS measure (for those months for which we have our SIPP measure of unemployment) is 98-99% for both unemployment series in Figure <ref>. This implies that the cyclical pattern in unemployment, to a very large extent, originates in the set of unemployed we consider in this paper (see also Hornstein, 2013, and Ahn and Hamilton, 2019).

We further check that these properties also hold when we restrict unemployment to those who start and end unemployment within the SIPP sample, taking care to address both the left- and right-censoring involved in this measure. The correlation of this unemployment rate (smoothed, because of the smaller set of observations) with the BLS unemployment rate is over 94% when considering nonemployment spells that include months of unemployment, and 93% when considering pure unemployment spells, where workers are unemployed in every month of nonemployment.

Assigning “source”/“destination” - occupations to unemployed workers. The SIPP collects information on a maximum of two jobs an individual might hold simultaneously. For each of these jobs we have information on, among other things, hours worked, total earnings, 3-digit occupation and 3-digit industry codes. We drop all observations with imputed occupations (and industries). If the individual held two jobs simultaneously, we consider the main job as the one in which the worker spent more hours. We break a possible tie in hours by using total earnings. The job with the highest total earnings will then be considered the main job. In most cases individuals report to work in one job at any given moment. In the vast majority of cases in which individuals report two jobs, the hours worked are sufficient to identify the main job. Once the main job is identified, the worker is assigned the corresponding two, three or four digit occupation.[For the 1990-1993 panels we correct the job identifier variable following the procedure suggested by Stinson (2003).]

Each unemployment spell that is started and finished inside the panel can be assigned a “source”-occupation (main occupation right before the start of the unemployment spell), and a “destination”-occupation (main occupation right after becoming employed again). If the occupation code is missing just before the unemployment spell (e.g. due to imputation) and an occupation code is reported in a previous wave, while employment is continuous from the time that the occupation was reported until the start of the unemployment spell under consideration, we carry the latter occupation forward as source occupation. A worker is an occupation mover if source and destination occupations do not coincide. We thus conservatively count the following situation also as an occupational stay: the worker is simultaneously employed in two firms at the moment the worker becomes unemployed, and finds a job afterwards in an occupation that matches the occupation in one of the two previous jobs, even when it matches the job with less hours. The effect on the occupational mobility statistics of counting as occupational stays the unemployment spells with two simultaneous jobs at either side is small.

We construct the occupational mobility statistics from transitions of the form: at least a month in employment (with a non-imputed occupational code), followed by an unemployment spell which has a duration of at least a month, followed by at least a month in employment (with a non-imputed occupational code). We label these transitions as EUE transitions. We also consider transitions of the form: at least a month in employment (with a non-imputed occupational code), followed by a non-employment spell which has a duration of at least a month and involved at least one month of unemployment. We call these E-NUN-E transitions, or NUN-spells of nonemployment. Further convexifying the space between EUE and E-NUN-E, we also consider spells that started with a EU transition, i.e. employment directly followed by unemployment (though later the worker can report to stop looking for work), and those that ended with UE transition. We label these transitions as E-UN-E, E-NU-E, and if both restrictions apply, E-UNU-E transitions. We also tried other versions of the latter in which the full jobless spell was non-employed (ENE).

Occupational Classifications. The SIPP uses the Census of Population Occupational System, which relates closely to the Standard Occupational Code (SOC). The 1984-1991 panels use the 1980 Census Occupational classification, while the 1992-1996 and 2001 panels use the 1990 Census Occupational classifications. These two classifications differ only slightly between them. The 2004 and 2008 panels use the 2000 Census occupational classification, which differs more substantially from the previous classifications. We use David Dorn's recoding of the 1980 and 2000 Census Occupational Classification (Dorn, 2009, and Autor and Dorn, 2013) into the 1990 Census Occupational Classification to have a uniform coding system. In robustness exercises, we instead use the IPUMS crosswalk to map the 1980 and 1990 Census occupational system into the 2000 Census occupational system.

We aggregate the information on “broad” occupations  (3-digit occupations) provided by the SIPP into “minor” and “major” occupational categories.[In any of these classifications we have not included the Armed Forces. The 1980 and 1990 classifications can be found at https://www.census.gov/people/io/files/techpaper2000.pdf. The 2000 classification can be found in http://www.bls.gov/soc/socguide.htm. Additional information about these classifications can be found at http://www.census.gov/hhes/www/ioindex/faqs.html.] Measurement error in occupational codes might give rise to spurious transitions, as discussed for example in Kambourov and Manovskii (2008, 2009) and Moscarini and Thomsson (2007). We correct for coding error using the $\mathbf{\Gamma}$-correction method we propose in Supplementary Appendix A. The occupational categories of the classifications used can be found in Tables 4 and 5 of that appendix.

Time series construction. We construct monthly time series for the unemployment rate, employment to unemployment transition rate (job separation rate), unemployment to employment transition rate (job finding rate), occupational mobility rates and the other measures described in the main text. Job finding rates are simply $UE_{t}/U_{t}$, the proportion of unemployed at time $t$ that moves to employment at time $t+1$. Similarly, the separation rate is $EU_{t}/E_{t}$, the proportion of employed at time $t$ that moves to unemployment at time $t+1$. We start measuring job finding and separation rates at the first month were we have information for all four rotation groups.

For construction of de-trended time series we have to address the issue of gaps (missing observations) in time series. There are a few quarters that are not covered at all by the 1984-2008 SIPP panels: 2000Q2-2000Q3, and 2008Q1. The gaps around these times can become larger, and new gaps can be created, when censoring issues cause us to drop further quarters from the analysis. We discuss this in more detail below.

To cover the missing observations we interpolate the series using the TRAMO (Time Series Regression with ARIMA Noise, Missing Observations and Outliers) procedure developed by Gomez and Maravall (1999), with interpolation of missing observations through regression (“Additive Outlier Approac”).[See also Fujita, et al. (2007) for a similar procedure using the SIPP. Tramo/Seats is a parametric, ARIMA model based method (AMB), that works in two steps. In the first step (TRAMO) the series is interpolates missing observations and deals with outliers. The second step (SEATS), among other things, decomposes the time series resulting from step 1 into e.g. a trend-cycle, an irregular, and a seasonal component. Seasonal adjustment is broadly similar to the Census Bureau's X12 procedure, and is used e.g. by Eurostat. (Hood et al. (Hood, Ashley, Finley, US Census Bureau: “An Empirical Evaluation of the Performance of Tramo/Seats on Simulated Series”), who also argue that SEATS does better than X12-ARIMA with longer time series that have large irregular components.)] In our baseline cyclical timeseries, de-trended data series are produced with after HP-filtering the resulting (logged) time series, with smoothing parameter 1600.

Two aspects are especially important for the time series of the propensity of hires from U to start in a new occupation: (1) the code error correction, and (2) addressing censoring issues, to counter noise and bias due to shifts in the duration distributions between adjacent quarters that are orthogonal to the business cycle.

We select only observations of individuals who have been in sample for more than 14 interviews and are hired beyond wave 4. Given this, we correct for coding error at the level of quarter $\times$ (completed) spell duration. This means that, for all hires from unemployment with a given completed duration in a given quarter, we calculate the transition matrix of occupational mobility and correct it using the appropriate $\mathbf{\Gamma}$ (code-error) correction matrix. We then calculate the cumulative mobility of all hires with unemployment durations up to (and including) 12 months, and in an alternative measure, up to and including 14 months. We find little difference across these two measures, and hence report only the former measure. The same corrected observations at the level of duration$\times$quarter will subsequently be used to calculate the mobility-duration profile shift from times of low unemployment to times of high unemployment.

While time series gaps are small for many statistics, this issue is more important for the average occupational mobility of all hires from unemployment (of $\leq$ 12-14 months). The restriction to wave 5 and beyond means that, by default, we leave out those observed unemployment spells that arise from workers losing their jobs and being hired within the first 16 months of each panel. As argued above, this is to make sure that we capture the nearly full duration distribution of unemployment at each quarter considered in our analysis. This is especially important here as not doing so would generate a downward bias in the occupational mobility rates at the beginning of a panel, which is particularly strong in e.g. 2004. In addition, we take into account that, given the rotating survey design, some further quarters early in the panel do not have a seam in each month (this is also a relevant issue for the job finding and separation time series). We are also conservative on this issue and disregard those quarters as well.

We also note that censoring issues may arise at the very end of the panel, where again a quarter may contain varying proportion of monthly observations that are the final seam. We observe that in practice, some of these “ending” quarters have duration distributions that are very different from the duration distributions in previous quarters. As we show below, this variation is significantly larger and more abrupt than changes that appear to move with business cycle conditions. For this reason, we also exclude the last quarter of the 1988 panel and all panels from the 1993 panel onwards, with the exception of the 2001 panel. The quarters which are used for our analysis are then given in Table <ref>. These observations span a number of different moments during multiple business cycles and therefore the cyclical patterns can indeed be clear and significant, as is shown in Section 2.5 of the main text and this appendix.

SIPP Panels     Qtr from     Qtr to (inclusive)
1984,85,86,87,1988     1985q1     1989q3

1990,91,92,1993     1991q2     1995q3

1996     1997q3     1999q4

2001     2002q2     2003q4

2004     2005q1     2007q3

2008      2009q3     2013q3

Quarters included in time series for overall occupational mobility rate of all unemployed (with unemployment duration between 1-14 months

[1984-1988 SIPPs]
[1990-1993 SIPPs]

[1996 SIPP]
[2001 SIPP]

[2004 SIPP]
[2008 SIPP]

Completed Unemployment Duration of Hires from U, per quarter

To visualize the impact of censoring, Figure <ref> presents the duration distribution of hires from unemployment per quarter. Each horizontal bar specifies the unemployment durations of hires (from 1 month to 14 months, in order). On the y-axis are the quarters in which we have observations beyond wave. Those quarters that we exclude from our time series calculations, due to one of the reasons mentioned above, are prefaced by an $*$.[One can distinguish some cyclical patterns directly, clearest perhaps when comparing across panels, where the red bands (hires after 6 months of unemployment) in the early 1990s panels are to the left of those in the 1996 panel, and likewise for the 2001 SIPP vs the 1996 and 2004 SIPP, and the 2008 vs the 2004 SIPP, signifying more long-term unemployment in the “recession” 1990-1993, 2001, and 2008 SIPP panels.]

Here we can indeed observe the some of the last and first of the quarters considered are affected by these (censoring) issues and hence are excluded from our analysis. We also observe that the remaining duration distributions are relatively stable across individual quarters, with the amount of quarter-to-quarter variation dropping with the 1996 panel and thereafter. This occurs because the sample sizes per panel are larger, while the overlapping nature of panels beforehand imply that some additional noise is generated when panels are phased in and out, while other panels are ongoing. We also tried to further restrict the sample to exclude observations from these phase-in and -out, but the impact of this on time series patterns is minor.

Aggregate Series Productivity, Unemployment, Vacancy Series For output per worker, we take nonfarm business output from the Major Sector Productivity and Costs section of Bureau of Labor Statistics (series id: PRS85006043), and divide this by the sum of employment of private industry wage and salary workers (not in agriculture) and employment of self-employed unincorporated workers in these sectors (series LNS12032189Q and lns12032192q, respectively) Unemployment rates in the cyclical analysis are BLS Unemployment rates (UNRATE in the St. Louis Fred data). We take the vacancy data from extended Help-Wanted Index constructed by Regis Barnichon.

Repeat Mobility In our baseline repeat mobility measures, we concentrate on pure u-spells that follow pure u-spells within a SIPP panel, for workers that are at least 4 years in sample. To avoid including seasonal unemployment in our measures (which are not informative about attachment as interpreted in the model), we exclude construction and agricultural workers. We are left with 610 observations of repeat unemployment. For our NUN measures, we consider NUN-spell following NUN-spells with the same restrictions, leaving 1306 observations of repeat unemployment. We average the measures that consider the occupation of intermediate employment at the beginning and end of that spell. We correct coding errors using the procedure described in footnote 13 in the main text. Finally, to gauge occupational moving, we exclude any observations of return movers.

§.§ Panel Survey of Income Dynamics

Following Kambourov and Manovskii (2008) sample restrictions, we consider the 1968-1997 period as during these years the PSID interviews were carried out annually. We also consider males head of households between the ages of 23-61 years who were not self- or dual-employed and were not public sector works. This sample restriction then gives 1,643 employed individuals in the year 1968 and 2,502 employed individuals in 1997. In an alternative sample we also included women, younger workers and self or dual-employed workers with no meaningful change in our main results.

To construct the across-employer occupational mobility rate we compute the fraction of employed workers who's occupational code differs between years $t$ and $t+1$ and have reported an employer change between these years, divided by the number of employed workers in year $t$ that have reported an employer change between years $t$ and $t+1$. As Kambourov and Manovskii (2009), to identify employer changes in the PSID we use Brown and Lights' (1992) Partition T method. As robustness we also use Brown and Lights' (1992) Partition 24T method. To identify employer changes using these two methods we followed exactly the same procedure as specified in Appendix A1 of Kambourov and Manovskii (2009). In addition, we used Hospicio's (2015) method to identified employer changes in the PSID, as described in Hospido (2015), Section 3.2. We find that our conclusions are not affected by the method used.

When constructing the $ENE$ occupational mobility rate we consider (i) those workers who were employed at the interview date in year $t-1$, non-employed at the interview date in year $t$ and once again employed at the interview date in year $t+1$; and (ii) those workers employed at the interview dates in years $t$ and $t+1$ but who declared that they experienced an involuntary employer change during these two interviews. In an alternative specification we also added those workers who were employed at the interview date in year $t-2$, non-employed at the interview date in years $t-1$ and $t$ and once again employed at the interview date in year $t+1$. Given the small number of workers in the latter category the results hardly change.

We follow Stevens (1997) and Hospido (2015) and classify an involuntary employer change as those cases were the worker declared a job separation due to business or plant closing, due to being laid off or were fired or their temporary job ended. We also added an “other” category as a reason why workers left their employers to increase the number of observations. This category encompasses other reasons such as military draft. Pooling together the “involuntary” and “others” categories and computing the $ENE$ occupational mobility rate gives very similar results. Further, in constructing the $ENE$ occupational mobility rates we were not able to eliminate those workers in temporary layoff. The analysis of Fujita and Moscarini (2016), however, suggests that unemployed workers in temporary layoff will bias downwards our $ENE$ occupational mobility rates as these workers have a very high probability of re-gaining employment in the same occupation.

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Supplementary Appendix C: Theory

This appendix complements Section 3 of the paper. The first part investigates, using a simplified version of the model, (i) the conditions under which rest unemployment arises and (ii) the cyclical properties of workers' job separation and occupational mobility decisions. The second part presents the equations describing worker flows in a BRE. The third part shows that the sub-market structure we impose in paper endogenously arises from a competitive search model. The fourth part provides the definition of a BRE, the proof of Proposition 2 (in the main text), the proof of existence of the separation and reallocation cutoffs as well as of the results presented in Sections 1 and 3 of this appendix.

§ MODEL IMPLICATIONS AND COMPARATIVE STATICS

We start by exploring the main implications of our theory: the occurrence of rest unemployment and the cyclical properties of workers' decisions to search across occupations and to separate from jobs. To keep the intuition as clear as possible we study a simplified version of our model without occupational human capital accumulation, setting $x_{h}=1$ for all $h$, and without occupational-wide productivity shocks, setting $p_{o}=1$ for all $o$. These restrictions imply that all occupations are identical, worker mobility across occupations is fully undirected and purely driven by occupation-worker idiosyncratic shocks such that gross mobility equals excess mobility. Further, within an occupation workers differ only in their $z$-productivities and labor market segmentation is done along this dimension. Agents' value functions are still given by the Bellman equations described in Section 3.2 of the main text, but now with state space $(z,A)$ instead of $(z,x,o,A,p)$. The net value of searching across occupations then simplifies to

\begin{align}\label{a:r}
R(A) = -c + \int_{\underline{z}} ^{\overline{z}} W^U(A,\tilde{z})dF(\tilde{z}).
\end{align}

In Carrillo-Tudela and Visschers (2013) we show that in this setting and assuming $F(z'|z)<F(z'|\tilde{z})$ for all $z,z'$ if $z>\tilde{z}$, the value functions $W^U(A,z)$, $W^E(A,z)$, $J(A,z)$ and $M(A,z)$, exist, are unique and increase in $z$. This implies that $\theta(A,z)$ also exists, is unique and increases with $z$. Further we show that if $\delta +\lambda (\theta(A,z))<1$ for all $A,z$, in equilibrium there exists a unique cutoff function $z^s$ that depends only on $A$, such that $d(A,z)=\sigma(A,z)=1$ if and only if $z<z^s(A)$, and $d(A,z)=\sigma(A,z)=0$ otherwise. Since $R(A)$ is constant in $z$, $W^U(A,z)$ is increasing in $z$, and by the existence of a unique $z^s$ so is $\max\{\lambda(\theta(A,z^r(A)))(W^E(A,z^r(A))-W^U(A,z^r(A))),0\}$, there also exists a reallocation cutoff function $z^r(A)$ such that workers decide to search across occupations if and only if $z<z^r(A)$ for every $A$, where $z^r(A)$ satisfies

\begin{equation}\label{e:reallocation}
R(A)=W^U(A,z^r(A))+\max\{\lambda(\theta(A,z^r(A)))(W^E(A,z^r(A))-W^U(A,z^r(A))),0\}. \\
\end{equation}

Using this simplified framework we first study the relative positions of the job separation and reallocation cut-off functions and hence gain insights on the conditions under which rest unemployment arises. We then study the slopes of these cut-off functions and gain insights into the cyclicality of separations and (excess) occupational mobility in our model. Using our calibrated model we have verified that the same properties as derived below apply to the more general setup considered there.

§.§ The Occurrence of Rest Unemployment

We first analyse how the value of waiting in unemployment and in employment changes with $c$, $b$ and the persistence of the $z$-productivity process, and how these changes determine the relative position of $z^r$ and $z^s$. The simplest setting that captures a motive for waiting is one in which the $z$-productivity is redrawn randomly with probability $0<(1-\gamma)<1$ each period from cdf $F(z)$ and $A$ is held constant. Time-variation in $z$ is essential here because a worker can decide to stay unemployed in his occupation, even though there are no jobs currently available for him, when there is a high enough probability that his $z$-productivity will become sufficiently high in the future. All other features of the model remain the same, with the exception that we do not consider human capital accumulation or occupational-wide productivity differences. In Section C.3.2 of this appendix we formulate the value functions for this stationary environment and provide the proofs of Lemmas 1 and 2, below.

In this stationary setting, the expected value of an unemployed worker with productivity $z$, measured at the production stage, is given by

\begin{eqnarray}\label{e:w^u iid}
W^U(A,z)&=&\gamma \Big(b + \beta \max \Big\{R(A), W^U(A,z)+ \max \{\lambda(\theta(A,z))(1-\eta)(M(A,z)-W^U(A,z)), 0\}\Big\}\Big)  \notag \\
&+ &(1-\gamma)\Exp_{z}[W^U(A,z)].
\end{eqnarray}

Equation (<ref>) shows that there are two ways in which an unemployed worker with a $z<z^s$ can return to production. Passively, he can wait until his $z$-productivity increases exogenously. Or, actively, by paying $c$ and sampling a new $z$ in a different occupation. In the case in which the worker prefers to wait, the inner $\max \{.\}=0$ as the workers is below the separation cutoff and the outer $\max \{.\}=W^U(A,z)=W^U(A,z^s(A))$, where the latter equality follows as in this simplified environment the $z$-productivity process is assumed to be iid. In the case in which the worker prefers to search across occupations the outer $\max \{.\}=R(A)=W^U(A,z^r(A))$.
The difference $W^U(A,z^s(A))-R(A)$, then captures the relative gain of waiting for one period over actively sampling a new $z$ immediately. If $W^U(A,z^s(A))-R(A)\geq0$, then $z^s\geq z^r$ and there is rest unemployment. If $W^U(A,z^s(A))-R(A)<0$, then $z^r>z^s$, and endogenously separated workers immediately search across occupations.

Changing $c$, $b$, or $\gamma$ will affect the relative gains of waiting, in employment and in unemployment. In the following lemma we derive the direction of the change in $W^U(A,z^s(A)) - R(A)$, where we take fully into account the feedback effect of changes in $c$, $b$, or $\gamma$ on the match surplus $M(A,z)-W^U(A,z)$ that arises due to the presence of search frictions as discussed in Section 3.3 in the main text.

Changes in $c$, $b$ or $\gamma$ imply

\[\frac{d(W^U(A,z^s(A))-R(A))}{dc}>0, \frac{d(W^U(A,z^s(A))-R(A))}{db}>0, \frac{d(W^U(A,z^s(A))-R(A))}{d\gamma}<0.\]

It is intuitive that raising $c$ directly increases the relative gains of waiting as it make occupational mobility more costly. An increase in $c$, however, also leads to a larger match surplus because it reduces $W^U(A,z)$, making employed workers less likely to separate and hence reducing rest unemployment. The lemma shows that, overall, the first effect dominates. A rise in $b$ lowers the effective cost of waiting, while at the same time decreasing the match surplus by increasing $W^U(A,z)$, pushing towards more rest unemployment. An increase in $\gamma$, decreases the gains of waiting because it decreases the probability of experiencing a $z$-shock without paying $c$ and hence increases the value of sampling a good $z$-productivity. In the proof of Lemma 1 we further show that an increase in $W^U(A,z^s(A))-R(A)$ leads to an increase in $z^s(A)-z^r(A)$. This implies that for a sufficiently large $c$, $b$ or $1-\gamma$ rest unemployment arises.

Rest Unemployment and Occupational Human Capital Occupational human capital accumulation makes a worker more productive in his current occupation. This implies that workers are willing to stay longer unemployed in their occupations because, for a given $z$, they can find jobs faster and receive higher wages. At the same time, a higher $x$ makes the employed worker less likely to quit into unemployment, generating a force against rest unemployment. Taken together, however, the first effect dominates as the next result shows.

Consider a setting where $A$ is fixed, $z$ redrawn with probability $(1-\gamma)$, and production is given by $y=Axz$. Consider an unexpected, one-time, permanent increase in occupation-specific human capital, $x$, from $x=1$. Then

\begin{align*}
\frac{d(W^U(A,z^{s}(A,x),x)-R(A))}{dx}>0.
\end{align*}

This result implies that the difference $z^s(A,x)-z^r(A,x)$ becomes larger when human capital increases. Thus, more occupational human capital leads to rest unemployment.

§.§ The Cyclicality of Occupational Mobility and Job Separation Decisions

In Section 2.5 of the main text we documented that occupational mobility through unemployment (non-employment) is procyclical, while it is well established that job separations into unemployment (non-employment) are countercyclical (see Section 5 in the main text). In the model, the cyclicalities of workers' occupational mobility and job separation decisions are characterised by the slopes of the cutoff functions $z^r$ and $z^s$ with respect to $A$. As discussed in Section 3.3 of the main text, occupational mobility decisions are procyclical and job separation decisions are countercyclical when $dz^r/dA>0$ and $dz^s/dA<0$, respectively. We now explore the conditions under which such slopes arise endogenously in our model.

Occupational Mobility Decisions We start by investigating the impact of rest unemployment on the slope of $z^r$ using the simplified version of the model (i.e. $x_{h}=1$ for all $h$ and $p_{o}=1$ for all $o$). Using (<ref>) and noting that $R(A)-W^U(A, z^r(A))=0$, we obtain

\begin{equation}\label{e: dz^r/dp general}
\frac{dz^r}{dA}=\frac{\int_{z^r}^{\overline{z}} \left(\frac{\partial W^U(A,z)}{\partial A}-\frac{\partial W^U(A,z^r)}{\partial A}\right)dF(z) - \frac{\partial }{\partial A }\Big(\lambda(\theta(A,z^r))(W^E(A,z^r)-W^U(A,z^r))\Big)}{\frac{\partial W^U(A,z^r)}{\partial A}+ \frac{\partial}{\partial z^r }\Big(\lambda(\theta(A,z^r))(W^E(A,z^r)-W^U(A,z^r))\Big)}.
\end{equation}

Since workers who decided to search across occupations must sit out one period unemployed, the term

\[\lambda(\theta(A,z^r))(W^E(A,z^r)-W^U(A,z^r))\]
captures the expected loss associated with the time cost of this decision: by deciding not to search across occupations, the worker could match with vacancies this period. When $z^r>z^s$, $\lambda(\theta(A,z^r))(W^E(A,z^r)-W^U(A,z^r))>0$ and is increasing in $A$ and $z^r$. Therefore, an increase in $A$ in this case, increases the loss associated with the time cost of searching across occupations and decreases $dz^r/dA$. However, when rest unemployment occurs ($z^s>z^r$), $\lambda(\theta(A,z^r))(W^E(A,z^r)-W^U(A,z^r))=0$ and this effect disappears. This follows as during rest unemployment periods workers have a contemporaneous job finding rate of zero and hence by searching across occupations, the worker does not lose out on the possibility of matching this period. In this case, the cyclicality of occupational mobility decisions purely depends on the remaining terms, in particular $\int_{z^r}^{\overline{z}} \left(\frac{\partial W^U(A,z)}{\partial A}-\frac{\partial W^U(A,z^r)}{\partial A}\right)dF(z)$. Thus, the presence of rest unemployment adds a procyclical force to occupational mobility decisions.

Now consider the impact of search frictions on the slope of $z^r$. We focus on the more general case of $z^r>z^s$, which includes the additional countercyclical force discussed above. To gain analytical tractability we consider the stationary environment used in the previous subsection and set $\gamma=1$ such that both $A$ and workers' $z$-productivities are permanent. This allows us to link wages and labor tightness to $y(A,z)$ in closed form. We then analyse the effects of a one-time, unexpected, and permanent change in $A$ on $z^r$.[This approach follows Shimer (2005), Mortensen and Nagypal (2007), and Hagedorn and Manovskii (2008). Since the equilibrium value and policy functions only depend on $A$ and $z$, analysing the change in the expected value of unemployment and joint value of the match after a one-time productivity shock is equivalent to compare those values at the steady states associated with each productivity level. This is because in our model the value and policy functions jump immediately to their steady state level, while the distribution of unemployed and employed over occupations takes time to adjust.] To isolate the role of search frictions, we compare this case with one without search frictions in which workers (who are currently not changing occupations) can match instantaneously with firms and are paid $y(A,z)$. In both cases we keep in place the same reallocation frictions. Let $z^r_c$ denote the reallocation cutoff in the case without search frictions. The details of both cases, including the corresponding value functions and the proof of the following lemma can be found in Section C.4 of this appendix.

Consider a one-time, unexpected, permanent increase in $A$. With search frictions the cyclical response of the decision to search across occupations is given by

\begin{align}\label{dzr1}
\frac{d z^r}{d A}& = \frac{\beta \int_{z^r}^{\bar{z}}\Big(\big( \frac{C_s(A,z)}{C_s(A,z^r)}\big) y_A(A,z)- y_A(A,z^r)\Big)dF(z)-(1-\beta)y_A(A,z^r)} {(1-\beta F(z^r))y_z(A,z^r)},
\end{align}

where $y_i(A,z)=\partial y(A,z)/\partial i$ for $i=A,z$ and $C_s(A,z)=\frac{\beta \lambda(\theta(A,z))}{(1-\beta)(1-\beta+\beta \lambda(\theta(A,z)))}$.
Without search frictions the cyclical response is given by

\begin{align}\label{dzr2}
\frac{d z^r_c}{d A}& = \frac{\beta \int_{z^r_c}^{\bar{z}} (y_A(A,z)-y_A(A,z^r_c))dF(z)-(1-\beta)y_A(A, z^r_c)} {(1-\beta F(z^r_c))y_z(A,z^r_c)}.
\end{align}

The first term in the numerator of (<ref>) and (<ref>) relates to $\int_{z^r}^{\overline{z}} \left(\frac{\partial W^U(A,z)}{\partial A}-\frac{\partial W^U(A,z^r)}{\partial A}\right)dF(z)$ in equation (<ref>), while the second term captures the opportunity cost of the reallocation time. The proof of Lemma <ref> shows that $C_s(A,z)/C_{s}(A,z^r)$ is increasing in $z$ and $\frac{dz^r}{dA}>\frac{dz^r_c}{dA}$ at $z^r=z^r_c$. Thus, the presence of search frictions also adds procyclicality to the decision to search across occupations. Search frictions lead to a steeper reallocation cutoff function because, in this case, an increase in $A$ increases $W^U(A,z)$ through both the wage and the job finding probability. In contrast, an increase in $A$ in the frictionless case only affects $W^U(A,z)$ through wages (with a proportionally smaller effect on $w-b$), as workers always find jobs with probability one. The fact that $C_s(A,z)/C_{s}(A,z^r)$ is increasing in $z$ for $z>z^r$, reflects that the impact of $y(A,z)$ on $W^U(A,z)$ is increasing in $z$. Indeed, from the proof of Lemma <ref> one obtains $\frac{\partial W^U(A,z)}{\partial A}\Big/\frac{\partial W^U(A,z^r)}{\partial A}=\frac{C_s(A,z)y_A(A,z)}{C_s(A,z^r)y_A(A,z^r)}>\frac{y_A(A,z)}{y_A(A,z^r) }$ for $z>z^r$.

One can get further intuition by considering the planners' problem (for details see Carrillo-Tudela and Visschers, 2013). The envelope condition implies that the planner, at the optimum allocation, does not need to change labor market tightness at each $z$ for a infinitesimal change in $A$ to still obtain the maximum net increase in expected output. At a given $z$, this means that an increase of $dA$ creates
\[dW^U(A,z)=\frac{\beta \lambda(\theta(A,z))}{(1-\beta)(1-\beta +\beta \lambda(\theta(A,z)))}y_A(A,z)dA=C_s(A,z)y_A(A,z)dA\]
in additional life-time expected discounted output for the planner. Since our economy is constrained efficient, the change in $A$ also creates the same lifetime expected discounted income for an unemployed worker with such $z$.

In addition, (<ref>) and (<ref>) show that the cyclicality of $z^r$ in either case depends on the production function $y(A,z)$, in particular on the sign of $y_A(A,z)-y_A(A,z^r)$ for $z>z^r$. In the proof of Lemma <ref> we show that with search frictions, the decisions to search across occupations will already be procyclical when the production function is modular and $z^r$ is sufficiently close to $z^s$. This follows because the opportunity cost of searching across occupations becomes smaller as $z^r$ approaches $z^s$ from above. With rest unemployment this opportunity cost is zero. If $z^r$ is substantially above $z^s$, we will need sufficient complementarities between $A$ and $z$ in the production function to obtain a procyclical $z^r$. Without search frictions, in contrast, a supermodular production function is only a necessary condition to generate procyclicality in $z^r$ .

Job Separations The main aspect of having endogenous job separation and occupational mobility decisions is that the two can potentially interact. In particular, if $z^r>z^s$, workers separate endogenously to search across occupations and this could lead to $z^r$ and $z^s$ having the same cyclical behavior. For example, in the setting of Lemma <ref> were both $A$ and workers' $z$-productivities are permanent, we show at the end of Section C.3.2 of this appendix that $dz^s/dA$ depends directly on $dz^r/dA$. Namely,

\begin{align}  \label{e: z^s lemma main eq}
\frac{dz^s(A)}{dA}=-\frac{y_A(A, z^s(A))}{y_z(A, z^s(A))}+\frac{\beta \lambda(\theta(A, z^r(A)))}{1-\beta(1-\delta)+\beta \lambda (\theta(A, z^r(A)))}\frac{y_A(A, z^r(A))}{y_z(A, z^s(A))}\left(1+\frac{y_z(A,z^r(A))}{y_A(A, z^r(A))}\frac{dz^r(A)}{dA}\right). \nonumber
\end{align}

The second term makes explicit the interaction between the decisions to separate from a job and to search across occupations when there is no rest unemployment. It captures the change in the gains of search across occupations, $dR(A)/dA$, and shows that $z^r$ and $z^s$ can have the same cyclicality. When instead $z^s>z^r$, workers endogenously separate into a period of rest unemployment. In this case, $R(A)$ has a smaller impact on the value of unemployment at the moment of separation. This is because searching across occupations would only occur further in the future, and then only if a worker's $z$-productivity would deteriorate below $z^r$. Thus, the presence of rest unemployment weakens any feedback of a procyclical $z^r$ onto $z^s$. Indeed, by setting $\lambda(.)=0$ in the above expression we get that $z^s$ is always countercyclical.

§ WORKER FLOWS

In a BRE the evolution of the distribution $\mathcal{G}$ of employed and unemployed workers across labor markets $(z,x)$, occupations $o$ and employment status $es$ is a result of (i) optimal vacancy posting $\theta(.)$, job separation decisions $d(.)$ and occupational mobility decisions $\rho(.)$ and $\mathcal{S}(.)$, all depending on the state vector $\omega=(A,p,z,x,o)$; and (ii) the exogenous retiring probability $\mu$. To obtain the laws of motions of unemployed and employed workers it is then useful to derive the measure of unemployed and employed workers at each stage $j$ within a period, where $j=s,r,m,p$ represent separations, reallocations, search and matching and production as described in the main text. It is also useful to consider the following Markov Chain: in period $t$ an employed worker with human capital level $x_{h}$ increases his human capital to $x_{h+1}$ with probability $\chi^{e}(x_{h+1}|x_{h})$, where $\chi^{e}(x_{h+1}|x_{h})=1-\chi^{e}(x_{h}|x_{h})$, $x_{h}<x_{h+1}$, $h=1,\ldots,H$ and $x_{H}<\infty$. Human capital depreciation occurs during unemployment with probability $\chi^{u}(x_{h-1}|x_{h})$, where $\chi^{u}(x_{h-1}|x_{h})=1-\chi^{u}(x_{h}|x_{h})$, $x_{h}>x_{h-1}$, $h=1,\ldots,H$. Let $u_{t}^{j}(z,x_{h},o)$ denote the measure of unemployed workers in labor market $(z,x_{h})$ in occupation $o$ at the beginning of stage $j$ in period $t$. Similarly, let $e_{t}^{j}(z,x_{h},o)$ denote the measure of employed workers in labor market $(z,x_{h})$ in occupations $o$ at the beginning of stage $j$ in period $t$.

§.§ Unemployed Workers

Given the initial conditions $(A_0, p_{0}, \mathcal{G}^{p}_0)$, the measure of unemployed workers characterised by $(z,x_h)$ in occupation $o$ at the beginning of next period's separation stage is

\begin{eqnarray}
u_{t+1}^{s}(z,x_{h},o) dz & = & (1-\mu) \Bigg [ \chi^{u}(x_{h}|x_{h}) \int_{\underline{z}}^{\overline{z}}u_{t}^{p}(\tilde{z},x_{h},o)dF(z|\tilde{z})d\tilde{z} + \chi^{u}(x_{h}|x_{h+1}) \int_{\underline{z}}^{\overline{z}}u_{t}^{p}(\tilde{z},x_{h+1},o)dF(z|\tilde{z})d\tilde{z}  \Bigg] \notag \\
&+& \mu  \bigg [ \sum_{\tilde{o}=1}^{O} \sum_{\tilde{h}=1}^{H}  \int_{\underline{z}} ^ {\overline{z}} \Big [u_{t}^{p}(\tilde{z},x_{\tilde{h}},\tilde{o})  +  e_{t}^{p}(\tilde{z},x_{\tilde{h}},\tilde{o}) \Big] d\tilde{z}  \bigg]  \psi_{o}(\mathbf{1}_{h=1})dF(z). \label{uflow_s_t+1}
\end{eqnarray}

Conditional on not retiring from the labor market, the terms inside the first squared bracket show the probability that unemployed workers in labor markets $(\tilde{z},x_{h},o)$ and $(\tilde{z},x_{h+1},o)$ in the previous period's production stage will be in labor market $(z,x_{h},o)$ immediately after the $z$ and $x_{h}$ shocks occur. The term in the second squared bracket refers to the measure of new workers who entered the economy to replace those who left at the beginning of the period due to the $\mu$-shock. We assume that the population of workers is constant over time, making the inflow equal to the outflow of workers. New workers are assumed to enter unemployed with a $\tilde{z}$ randomly drawn from $F(.)$ and with the lowest human capital level $x_{1}$. The above equation considers the inflow who has been assigned productivity $z$ and occupation $o$, where $\psi_{o}$ denotes the probability that workers in the inflow are assigned occupation $o$ and $\mathbf{1}_{h=1}$ denotes an indicator function which takes the value of one when the labor market $(z,x_{h})$ is associated with $x_{1}$ and zero otherwise.

During the separation stage some employed workers will become unemployed. Since by assumption these newly unemployed workers do not participate in the current period's reallocation or search and matching stages, it is convenient to count them at the production stage. This implies that $u_{t+1}^{s}(z,x_{h},o)dz=u_{t+1}^{r}(z,x_{h},o)dz$. Similarly, we will count at the production stage those unemployed workers who arrived from other occupations during the reallocation stage, as they also do not participate in the current period's search and matching stage. This implies that the measure of unemployed workers characterised by $(z,x_h)$ in occupation $o$ at the beginning of the search and matching stage is given by
\[u_{t+1}^{m}(z,x_{h},o)dz=(1-\rho(A,p,z,x_{h},o))u_{t+1}^{r}(z,x_{h},o)dz.\]

Noting that $(1-\lambda(\theta(A,p,z,x_{h},o)))u_{t+1}^{m}(z,x_{h},o)dz$ workers remain unemployed after the search and matching stage, the above assumptions on when do we count occupational movers and those who separated from their employers imply that the measure of unemployed workers characterised by $(z,x_h)$ in occupation $o$ during the production stage is given by

\begin{align}
u_{t+1}^{p}(z,x_{h},o) dz & = \!(1\!-\!\lambda(\theta(A,p,z,x_{h},o)))u_{t+1}^{m}(z,x_{h},o)dz + d(z,x_{h},o, A,p)e_{t+1}^{s}(z,x_{h},o)dz \label{uflow_p_t+1} \\
&+  (\mathbf{1}_{h=1})  \bigg [ \sum_{\tilde{o} \neq o} \sum_{\tilde{h}=1}^{H} \Big [ \int_{\underline{z}} ^ {\overline{z}} \!\!\!\!\rho(\tilde{z},\tilde{x}_{h},\tilde{o},A,p)\alpha(s_{o}(\tilde{z},\tilde{x}_{h},\tilde{o},A,p),\tilde{o}) u_{t+1}^{r}(\tilde{z},x_{\tilde{h}},\tilde{o}) d\tilde{z} \Big]  \bigg] dF(z) \Bigg]. \nonumber
\end{align}

§.§ Employed Workers

Next we turn to describe the laws of motion for employed workers. Given the initial conditions $(A_0, p_{0}, \mathcal{G}^{p}_0)$, the measure of employed workers characterised by $(z,x_h)$ in occupation $o$ at the beginning of next period's separation stage is given by

\begin{eqnarray}
e_{t+1}^{s}(z,x_{h},o) dz & = & (1-\mu) \Bigg [ \chi^{e}(x_{h}|x_{h}) \int_{\underline{z}}^{\overline{z}}e_{t}^{p}(\tilde{z},x_{h},o)dF(z|\tilde{z})d\tilde{z} \notag \\
& +  &(\mathbf{1}_{h>1})\chi^{e}(x_{h}|x_{h-1}) \int_{\underline{z}}^{\overline{z}}e_{t}^{p}(\tilde{z},x_{h+1},o)dF(z|\tilde{z})d\tilde{z}  \Bigg] . \label{eflow_s_t+1}
\end{eqnarray}

Conditional on not retiring from the labor market, the terms inside the squared bracket show the probability that employed workers in labor markets $(\tilde{z},x_{h},o)$ and $(\tilde{z},x_{h-1},o)$ in the previous period's production stage will be in labor market $(z,x_{h},o)$ immediately after the $z$ and $x_{h}$ shocks occur. In this case, the indicator function $\mathbf{1}_{h>1}$ takes the value of one when the labor market $(z,x_{h})$ is associated with a value of $x_{h}>x_{1}$ and zero otherwise.

Since we count those employed workers who separated from their employers in the production stage and employed workers do not participate in the reallocation or the search and matching stages, it follows that $e_{t+1}^{s}(z,x_{h},o)dz=e_{t+1}^{r}(z,x_{h},o)dz=e_{t+1}^{m}(z,x_{h},o)dz$. This implies that the measure of employed workers characterised by $(z,x_h)$ in occupation $o$ during the production stage is given by

\begin{align}
e_{t+1}^{p}(z,x_{h},o) dz &= (1-d(z,x_{h},o, A,p))e_{t+1}^{s}(z,x_{h},o)dz + \lambda(\theta(\omega))u_{t+1}^{m}(z,x_{h},o)dz,\label{eflow_p_t+1}
\end{align}

where the last term describes those unemployed workers in labor market $(z,x_{h})$ who found a job in their same occupation $o$.

§ COMPETITIVE SEARCH MODEL

In the model described in the main text we exogenously segment an occupation into many sub-markets, one for each pair $(z,x)$. We assumed that workers with current productivities $(z,x)$ in occupation $o$ only participate in the sub-market $(z,x)$ in such an occupation. We now show that this sub-market structure endogenously arises from a competitive search model in the spirit of Moen (1997) and Menzio and Shi (2010). To show this property in the simplest way, we focus on the case in which all occupations have the same productivities (only excess mobility) and workers only differ in their $z$-productivities within an occupation. This is the same simplification we used in Section C.1 of this appendix. Adding differences in occupation productivities and occupational human capital is a straightforward extension. The full theoretical and quantitative analysis of this competitive search model can be found in our earlier working paper Carrillo-Tudela and Visschers (2013).

§.§ Basic Setup

As in the main text, we look for an equilibrium in which the value functions and decisions of workers and firms in any occupation only depend on the productivities $\omega=(A,z)$ and workers' employment status. Following Menzio and Shi (2010) and Menzio, Telyukova and Visschers (2016) we divide the analysis in two steps. The first step shows that at most one sub-market is active for workers with current productivity $z$. The second step shows that in equilibrium firms will post wage contracts such that a worker with current productivity $z$ does not find it optimal to visit any other sub-market other than the one opened to target workers of this productivity.

Assume that in each occupation firms post wage contracts to which they are committed. For each value of $z$ in an occupation $o$ there is a continuum of sub-markets, one for each expected lifetime value $\tilde{W}$ that could potentially be offered by a vacant firm. After firms have posted a contract in the sub-market of their choice, unemployed workers with productivity $z$ (henceforth type $z$ workers) can choose which appropriate sub-market to visit. Once type $z$ workers visit their preferred sub-market $j$, workers and firms meet according to a constant returns to scale matching function $m(u_{j},v_{j})$, where $u_{j}$ is the measure of workers searching in sub-market $j$, and $v_{j}$ the measure of firms which have posted a contract in this sub-market. From the above matching function one can easily derived the workers' job finding rate $\lambda (\theta_{j})=m(\theta_{j})$ and the vacancy filling rate $q(\theta_{j} )=m(1/\theta_{j})$, where labor market tightness is given by $\theta_{j} =v_{j}/u_{j}$. The matching function and the job finding and vacancy filling rates are assumed to have the following properties: (i) they are twice-differentiable functions, (ii) non-negative on the relevant domain, (iii) $m(0,0)=0$, (iv) $q(\theta)$ is strictly decreasing, and (v) $\lambda(\theta)$ is strictly increasing and concave.

We impose two restrictions on beliefs off-the-equilibrium path. Workers believe that, if they go to a sub-market that is inactive on the equilibrium path, firms will show up in such measure to have zero profit in expectation. Firms believe that, if they post in an inactive sub-market, a measure of workers will show up, to make the measure of deviating firms indifferent between entering or not. We assume, for convenience, that the zero-profit condition also holds for deviations of a single agent: loosely, the number of vacancies or unemployed, and therefore the tightness will be believed to adjust to make the zero-profit equation hold.

§.§ Agents' Problem

Consider the value function of an unemployed worker having productivity $z$ in occupation $o$ at the beginning of the production stage, $W^U(\omega)= b + \beta \mathbb{E}[ W^{R}(\omega^{\prime})]$. The value of unemployment consists of the flow benefit of unemployment $b$ this period, plus the discounted expected value of being unemployed at the beginning of next period's reallocation stage,

\begin{equation}
W^{R}(\omega^{\prime})=\max_{\rho(\omega^{\prime})} \{\rho(\omega^{\prime}) R(\omega^{\prime}) + (1-\rho(\omega^{\prime})) \mathbb{E}[S(\omega^{\prime})+W^U(\omega^{\prime})]\},\end{equation}

where $\rho(\omega^{\prime})$ takes the value of one when the worker decides to reallocate and take the value of zero otherwise and recall $R$ is the expected value of reallocation. In this case,

\[ R(\omega)= -c+\sum_{o'\neq o} \int W^{U}(A^{\prime },\tilde{z})\frac{dF(\tilde{z})}{O-1}. \]
The worker's expected value of staying and searching in his current occupation is given by $\mathbb{E}[S(\omega^{\prime})+W^{U}(\omega^{\prime})]$. In this case, $W^{U}(\omega^{\prime})=\mathbb{E}[W^{U}(\omega^{\prime})]$ describes the expected value of not finding a job, while $S(\omega^{\prime})$ summarizes the expected value added of finding a new job. The reallocation decision is captured by the choice between $R(\omega^{\prime})$ and the expected payoff of search in the current occupation.

To derive $S(.)$ note that $\lambda (\theta (\omega,W_{f}))$ denotes the probability with which a type $z$ worker meets a firm $f$ in the sub-market associated with the promised value $W_{f}$ and tightness $\theta(\omega)$. Further, let $\alpha (W_{f})$ denote the probability of visiting such a sub-market. From the set $\mathcal{W}$ of promised values which are offered in equilibrium by firms for a given $z$, workers only visit with positive probability those sub-markets for which the associated $W_{f}$ satisfies

\begin{equation}  \label{e:application_max}
W_{f}\in \arg \max_{\mathcal{W}}\lambda (\theta (\omega^{\prime},W_{f}))(W_{f}-W^{U}(\omega^{\prime}))\equiv S(\omega^{\prime}).
\end{equation}

When the set $\mathcal{W}$ is empty, the expected value added of finding a job is zero and the worker is indifferent between visiting any sub-market.

Now consider the value function at the beginning of the production stage of an employed worker with productivity $z$ in a contract that currently has a value $\tilde{W}_{f}(\omega)$. Similar arguments as before imply that

\begin{align}  \label{employed}
\tilde{W}_{f}(\omega)=&w_{f} + \beta \mathbb{E}\bigg[ \max_{d(\omega^{\prime})} \{ (1-d(\omega^{\prime}))\tilde{W}_{f}(\omega^{\prime})+ d(\omega^{\prime})W^U(\omega^{\prime})\}\bigg],
\end{align}

where $d(\omega^{\prime})$ take the value of $\delta$ when $\tilde{W}_{f}(\omega^{\prime})\geq W^U(\omega^{\prime})$ and the value of one otherwise. In equation (<ref>), the wage payment $w_{f}$ at firm $f$ is contingent on state $\omega$, while the second term describes the worker's option to quit into unemployment in the separation stage the next period. Note that $W^U(\omega^{\prime})= \mathbb{E}[W^U(\omega^{\prime})]$ as a worker who separates must stay unemployed for the rest of the period and $\tilde{W}_f(\omega^{\prime})=\mathbb{E}[\tilde{W}_f(\omega^{\prime})]$ as the match will be preserved after the separation stage.

Consider a firm $f$ in occupation $o$, currently employing a worker with productivity $z$ who has been promised a value $\tilde{W}_{f}(\omega)\geq W^{U}(\omega)$. Noting that the state space for this firm is the same as for the worker and given by $\omega$, the expected lifetime discounted profit of the firm can be described recursively as

\begin{eqnarray}  \label{filledjob}
J(\omega;\tilde{W}_{f}(\omega)) =\max_{w_{f}, \tilde{W}_{f}(\omega^{\prime})} \bigg\{y(A,z)-w_{f}+\beta \mathbb{E}\Big[\max_{\sigma (\omega^{\prime})}\Big\{(1-\sigma (\omega^{\prime}))J(\omega^{\prime};\tilde{W}_{f}(\omega^{\prime}))  +\sigma (\omega^{\prime})\tilde{V}(\omega^{\prime})\Big\}\Big]\bigg\},
\end{eqnarray}

where $\sigma (\omega^{\prime})$ takes the value of $\delta$ when $J(\omega^{\prime};\tilde{W}_{f}(\omega^{\prime}))\geq \tilde{V}(\omega^{\prime})$ and the value of one otherwise, $\tilde{V}(\omega^{\prime})=\max \big\{\overline{V}(\omega^{\prime}),0\big\}$ and $\overline{V}(\omega^{\prime})$ denotes the maximum value of an unfilled vacancy in occupation $o$ at the beginning of next period. Hence (<ref>) takes into account that the firm could decide to target its vacancy to workers of with a different productivity in the same occupation or withdraw the vacancy from the economy and obtain zero profits.

The first maximisation in (<ref>) is over the wage payment $w_{f}$ and the promised lifetime utility to the worker $\tilde{W}_{f}(\omega^{\prime})$. The second maximisation refers to the firm's layoff decision. The solution to (<ref>) then gives the wage payments during the match (for each realisation of $\omega$ for all $t$). In turn these wages determine the expected lifetime profits at any moment during the relation, and importantly also at the start of the relationship, where the promised value to the worker is $\tilde{W}_{f}$.

Equation (<ref>) is subject to the restriction that the wage paid today and tomorrow's promised values have to add up to today's promised value $\tilde{W}_{f}(\omega)$, according to equation (<ref>). Moreover, the workers' option to quit into unemployment, and the firm's option to lay off the worker imply the following participation constraints

\begin{equation}  \label{partconst}
\Big(J(\omega^{\prime};\tilde{W}_{f}(\omega^{\prime}))-\tilde{V}(\omega^{\prime})\Big)\geq 0 \text{\ \ \ and \ \ \ }  \Big(\tilde{W}_{f}(\omega^{\prime})-W^{U}(\omega^{\prime})\Big)\geq 0.
\end{equation}

Now consider a firm posting a vacancy in occupation $o$. Given cost $k$ and knowing $\omega$, a firm must choose which unemployed workers to target. In particular, for each $z$ a firm has to decide which $\tilde{W}_{f}$ to post given the associated job filling probability, $q(\theta (\omega,\tilde{W}_{f}))$. This probability summarises the pricing behaviour of other firms and the visiting strategies of workers. Along the same lines as above, the expected value of a vacancy targeting workers of productivity $z$ solves the Bellman equation

\begin{equation}  \label{unfilled job fe}
V(\omega)=-k+\max_{\tilde{W}_{f}}\Big\{q(\theta (\omega,\tilde{W}_{f}))J(\omega,\tilde{W}_{f})+(1-q(\theta (\omega,\tilde{W}_{f})))\tilde{V}(\omega)
\Big\}.
\end{equation}

We assume that there is free entry of firms posting vacancies within any occupation. This implies that $V(\omega)=0$ and $\tilde{W}_{f}$ that yield a $\theta(\omega,\tilde{W}_{f})>0$, and $V(\omega)\leq0$ for all those $\omega$ and $\tilde{W}_{f}$ that yield a $\theta(\omega,\tilde{W}_{f})\leq0$. In the former case, the free entry condition then simplifies (<ref>) to $k=\max_{\tilde{W}_{f}}q(\theta (\omega,\tilde{W}_{f}))J(\omega,\tilde{W}_{f})$.

§.§ Endogenous Market Segmentation

We now show that if there are positive gains to form a productive match with a worker of type $z$, firms offer a unique $\tilde{W}_f$ with associate tightness $\tilde{\theta}(A,z)$ in the matching stage. This implies that only one sub-market opens for a given value of $z$ and workers of productivity $z$ optimally chose to search in such a sub-market. Consider a value of $z$. For any promised value $W^E$, the joint value of the match is defined as $W^E+J(A,z,W^E)\equiv\tilde{M}(A,z,W^E)$. Lemma <ref> shows that under risk neutrality the value of a job match is constant in $W^E$ and $J$ decreases one-to-one with $W^E$.

The joint value $\tilde{M}(A,z, W^E)$ is constant in $W^E\geq W^U(A,z)$ and hence we can uniquely define $M(A,z)\overset{def}{=}\tilde{M}(A,z,W^E), \ \forall \ M(A,z)\geq W^E \geq W^U(A,z)$ on this domain. Further, $J_W(A,z,W^E)=-1, \ \forall \ M(A,z)>W^E>W^U(A,z)$.

The proof of Lemma <ref> is presented in Section C.4.4. It crucially relies on the firms' ability to offer workers inter-temporal wage transfers such that the value of the job match is not affected by the (initial) promised value. Note that Lemma <ref> implies that no firm will post vacancies for $z$ values such that $M(A,z)-W^U(A,z)\leq0$. Lemma <ref> now shows that for a given $z$, for which $M(A,z)-W^U(A,z)>0$, firms offer a unique $\tilde{W}_f$ in the matching stage and there is a unique $\theta$ associated with it.

If the elasticity of the vacancy filling rate is weakly negative in $\theta$, there exists a unique $\theta^*(A,z)$ and $W^*(A,z)$ that solve (<ref>), subject to (<ref>).

The proof of Lemma <ref> is also presented in Section C.4.4. The requirement that the elasticity of the job filling rate with respect to $\theta$ is non-positive is automatically satisfied when $q(\theta)$ is log concave, as is the case with the Cobb-Douglas and urn ball matching functions. Both matching functions imply that the job finding and vacancy filling rates have the properties described in Lemma <ref> and hence guarantee a unique pair $\tilde{W}_{f},\theta$. Consider a Cobb-Douglas matching function as it implies a constant $\varepsilon _{q,\theta }(\theta )$. Using $\eta$ to denote the (constant) elasticity of the job finding rate with respect to $\theta $, we find the well-known division of the surplus according to the Hosios' (1991) rule

\begin{equation}
\eta (W^{E}-W^{U}(A,z))-(1-\eta )J(A,z,W^{E})=0.  \label{hosios}
\end{equation}

Since for every value of $z$ there is at most one $\tilde{W}_{f}$ offered in the matching stage, the visiting strategy of an unemployed worker is to visit the sub-market associated with $\tilde{W}_{f}$ with probability one when $S(A,z)>0$ and to randomly visit any sub-market when $S(A,z)=0$ (or not visit any submarket at all). Let $\tilde{W}^{z}_{f}$ denote the unique expected value offered to workers of productivity $z$ in equilibrium.

The final step is to verify that if we allow firms to post a menu of contracts which specify for each $z$ a different expected value, then the above equilibrium allocation and payoffs can be sustained in this more general setting. To show this we closely follow Menzio and Shi (2010) with the addition of endogenous reallocation. Since the expected value of reallocation $R$ is a constant across $z$-productivities, it is easy to verify that the proof developed in Menzio and Shi (2010) applies here as well.

Intuitively, consider a BRE in which firms that want to only attract a worker of type $z$ offer a contract with expected value $\tilde{W}^{z}_{f}$, as derived above, to any worker of type $z$ or higher; and for lower worker types, the firm offers contracts with expected values strictly below these types' value of unemployment. Since $\tilde{W}^{z}_{f}$ and $\theta^{z}$ in this candidate equilibrium are increasing in $z$, this implies that only the type $z$ workers visit these firms. To show that this is indeed optimal for firms, first consider the deviation in which a firm opens a new sub-market that attracts workers of different types $z$ and $\hat{z}$ by offering expected values $W^{z}$ and $W^{\hat{z}}$. Without loss of generality assume that firms obtain higher profit from matching with a worker of type $z$ instead of type $\hat{z}$. Next consider an alternative deviation in which a firm opens a sub-market that only attract workers of type $z$ by offering $W^{z}$ and the value of unemployment to any other worker type. Given the off-equilibrium beliefs, a higher tightness is associated with this alternative deviation which makes it more attractive for type $z$ workers to visit relative to the original deviation. That is, if there is a strictly profitable deviation in which more than one type visits the same sub-market, there is always another strictly more profitable deviation in which a sub-market is visited by only one type. However, Lemma <ref> shows that the latter deviation cannot exist and therefore there also cannot be a profitable deviation in which more than type visits the same sub-market in equilibrium.

§ PROOFS

A Block Recursive Equilibrium (BRE) is a set of value functions $W^U(\omega)$, $W^E(\omega)$, $J(\omega)$, workers' policy functions $d(\omega)$, $\rho(\omega)$, $\mathcal{S}(\omega)$, firms' policy function $\sigma(\omega)$, tightness function $\theta(\omega)$, wages $w(\omega)$, laws of motion of $A$, $p$, $z$ and $x$ for all occupations, and laws of motion for the distribution of unemployed and employed workers over all occupations, such that: (i) the value functions and decision rules follow from the firm's and worker's problems described in equations (1)-(5) in the main text; (ii) labor market tightness $\theta(\omega)$ is consistent with free entry on each labor market, with zero expected profits determining $\theta(\omega)$ on labor markets at which positive ex-post profits exist; $\theta(\omega)=0$ otherwise; (iii) wages solve equation (6) in the main text; (iv) the worker flow equations map initial distributions of unemployed and employed workers (respectively) over labor markets and occupations into next period's distribution of unemployed and employed workers over labor markets and occupations, according to the above policy functions and exogenous separations.

Proposition 2
Given $F(z'|z)<F(z'|\tilde{z})\ $for all $z,z'$ when $z>\tilde{z}$: (i) a BRE exists and it is the unique equilibrium, and (ii) the BRE is constrained efficient.

§.§ Proof of Proposition 2

We divide the proof into two parts. In the first part we show existence of equilibrium by deriving the operator $T$, showing it is a contraction and then verifying that the candidate equilibrium functions from the fixed point of $T$ satisfy all equilibrium conditions. The second part shows efficiency of equilibrium.

§.§.§ Existence

Step 1:

Let $M(\omega)\equiv W^E(\omega)+J(\omega)$ denote the value of the match. We want to show that the value functions $M(\omega)$, $W^{U}(\omega)$ and $R(\omega)$ exist. This leads to a three dimensional fixed point problem. It is then useful to define the operator $T$ that maps the value function $\Gamma(\omega,n)$ for $n=0,1,2$ into the same functional space, such that $\Gamma(\omega,0)=M(\omega),$ $\Gamma(\omega,1)=W^{U}(\omega)$, $\Gamma(\omega,2)=R(\omega)$ and

\begin{align*}
T(\Gamma(\omega,0))&=y(A,p_{o},z,x)+\beta \mathbb{E}_{\omega'}\Big[\max_{d^{T}}\{(1-d^{T})M(\omega')+d^{T}W^{U}(\omega')\}\Big], \non \\
T(\Gamma(\omega,1))&=b+\beta \mathbb{E}_{\omega'}\Big[\max_{\rho^{T}}\{\rho^{T} R(\omega')+(1-\rho ^{T})(D^{T}(\omega')+W^{U}(\omega'))\}\Big], \non \\
T(\Gamma(\omega,2))&=\max_{\mathcal{S}(\omega)}\Big( \sum_{\tilde{o} \in \vec{O}^{-}} \alpha(s^{T}_{\tilde{o}}) \int_{\underline{z}}^{\overline{z}}W^{U}(\tilde{z},x_{1},\tilde{o},A,p)dF(\tilde{z}) + (1-\sum_{\tilde{o} \in \vec{O}^{-}} \alpha(s^{T}_{\tilde{o}}))[b+ \beta \mathbb{E}_{\omega'} R(\omega')] - c \Big),
\end{align*}

where the latter
maximization is subject to $s_{o}\in[0,1]$ and $\sum_{o \in \vec{O}^{-}} s_{o}=1$, and

\begin{align}\label{eq:Ddef}
D^{T}(\omega')\equiv \lambda (\theta (\omega'))(1-\eta) \Big(M(\omega')-W^{U}(\omega')\Big), \text{ with } \theta(\omega')= \left(\frac{\eta(M(\omega')-W^U(\omega'))}{k}\right)^{\frac{1}{1-\eta}}.
\end{align}

Lemma A.1: $T$ is (i) a well-defined operator mapping functions from the closed space of bounded continuous functions $\mathcal{E}$ into $\mathcal{E}$, and (ii) a contraction.

First we show that the operator $T$ maps bounded continuous functions into bounded continuous functions. Let $W^{U}(\omega)$, $R(\omega)$ and $M(\omega)$ be bounded continuous functions. It then follows that $\lambda (\theta(\omega) )$ and $D(\omega)$ are continuous functions. It also follows that $\max\{M(\omega),W^{U}(\omega)\}$ and $\max\{R(\omega),D(\omega)+W^{U}(\omega)\}$ are continuous. Further, since the constraint set for $\mathcal{S}$ is compact-valued and does not depend on $\omega$, functions $\alpha(.)$ are continuous, and the integral of continuous function $W^U$ is continuous, the theorem of the maximum then implies that the expression

\begin{equation}
\max_{\mathcal{S}(\omega)}\Big( \sum_{\tilde{o} \in \vec{O}^{-}} \alpha(s^{T}_{\tilde{o}}) \int_{\underline{z}}^{\overline{z}}W^{U}(\tilde{z},x_{1},\tilde{o},A,p)dF(\tilde{z}) + (1-\sum_{\tilde{o} \in \vec{O}^{-}} \alpha(s^{T}_{\tilde{o}})) [ b+ \beta \mathbb{E}_{\omega'} R(\omega') ] - c \Big)
\end{equation}

is continuous. Therefore $T$ maps continuous functions into continuous functions. Moreover, since the domain of $\omega$ is bounded, and $\alpha(.)$ and $\lambda(.)$ are bounded on bounded domains, $T$ maps the space of bounded continuous functions into itself.

Second we show that $T$ defines a contraction. Consider two functions $\Gamma,\Gamma^{\prime } \in \mathcal{E}$, such that $\Vert \Gamma-\Gamma^{\prime }\Vert_{\sup }<\varepsilon $. It then follows that $\Vert W^{U}-W^{U}{^{\prime }}\Vert _{\sup }<\varepsilon $, $\|R-R'\|_{sup}<\varepsilon$ and $\Vert M-M^{\prime} \Vert _{\sup }<\varepsilon $, where $W^{U}$, $R$ and $M$ are part of $\Gamma$ as defined above. We first establish that

\begin{align}\label{e:prop1_i_step1_eq1}
\|D+W^U-  D'-W^{U}{'}\|_{sup}<\varepsilon,
\end{align}

when function-tuples $D$ and $D'$ are derived from $(M, W^U)$, and $(M', W^U{'})$ respectively. With this aim consider, without loss of generality, the case in which $M(\omega)-W^U(\omega)>M^{' }(\omega)-W^U{' }(\omega)$ at a given $\omega$. Instead of $D(\omega)$, write $D(M(\omega)-W(\omega))$ to make explicit the dependence of $D$ on $M$ and $W$. To reduce notation we suppress the dependence on $\omega$ for this part of the proof and further condense $W^U=W$. From $M-W>M^{' }-W^{' }$, it follows that $\varepsilon >W^{' }-W\geq M^{' }-M>-\varepsilon $. Construct $M^{'' }=W^{' }+(M-W)>M^{' }$ and $W^{''}=M^{' }-(M-W)<W^{' }$. Equation (<ref>) implies $D(M-W)+W$ is increasing in $M$ and in $W$. To verify this, hold $M$ constant and note that

\begin{align}\label{e:incr_z}
\frac{d(D(M-W)+W)}{dW}=-\lambda(\theta(.))+1\geq 0
\end{align}

by virtue of $d(D(M-W))/d(M-W)=\lambda$ and that $\lambda\in[0,1]$, where the inequality in (<ref>) is strict when $\lambda\in[0,1)$ and weak when $\lambda=1$. Then, it follows that

\begin{align}
-\varepsilon <D(M^{' }-W^{''})+W^{'' }-D(M-W)-W &
\leq D(M^{' }-W^{' })+W^{' }-D(M-W)-W  \notag \\
& \leq D(M^{''}-W^{' })+W^{' }-D(M-W)-W<\varepsilon
\notag
\end{align}

where $D(M^{' }-W^{''})=D(M-W)=D(M^{''}-W^{' })$ by construction. Note that the outer inequalities follow because $M-M^{' }>-\varepsilon$ and $W^{' }-W<\varepsilon $. Given that $D$, $M$ and $W$ are bounded continuous functions on a compact domain and the above holds for every $\omega$, it then must be that $\Vert D+W^{U}-D^{' }-W^{U}{^{' }}\Vert_{\sup } <\varepsilon $. Since $\Vert \max \{a,b\}-\max \{a^{\prime},b^{\prime }\}\Vert <\max \{\Vert a-a^{\prime }\Vert ,\Vert b-b^{\prime}\Vert \}$, as long as the terms over which to maximize do not change by more than $\varepsilon$ in absolute value, the maximized value does not change by more $\varepsilon$, it then follows that $\Vert T(\Gamma(\omega, n))-T(\Gamma^{' }(\omega,n)\Vert <\beta \varepsilon $ for all $\omega$; $n=0,1$.

To show that $\Vert T(\Gamma(\omega, 3))-T(\Gamma^{' }(\omega,3)\Vert <\beta \varepsilon $ for all $\omega$, with a slight abuse of notation, define $TR(\omega, \mathcal{S})$ as

\begin{align}\label{eq:app_reall_s}
T R(\omega,\mathcal{S}) =&\sum_{\tilde{o} \in \vec{O}^{-}} \alpha(s_{\tilde{o}}(\omega)) \int_{\underline{z}}^{\overline{z}}\Big[b+\beta \mathbb{E}_{\omega'}\Big[\max_{\rho^{T}}\{\rho^{T} R(\omega')+(1-\rho ^{T})(D^{T}(\omega')+W^{U}(\omega'))\}\Big]\Big]dF(\tilde{z}) \\ \notag
+& (1-\sum_{\tilde{o} \in \vec{O}^{-}} \alpha(s_{\tilde{o}}(\omega)))[b+ \beta \mathbb{E}_{\omega'} R(\omega')] - c.
\end{align}

Let $\mathcal{S}_*(\omega)=\{s^*_1,\ldots, s^*_{o-1}, s^*_{o+1},\ldots, s^*_O\}$ be the maximizer of $TR$ at $\omega$ and $\mathcal{S}_*{'}(\omega)$ of $TR'$. Without loss of generality assume that $TR(\omega,\mathcal{S})> TR'(\omega,\mathcal{S}')$. Since our previous arguments imply that

\[
\Vert \Big[\max_{\rho}\{\rho R+(1-\rho)(D+W^{U})\}\Big] - \Big[\max_{\rho}\{\rho R'+(1-\rho)(D'+W')\}\Big] \|_{sup} <\beta \varepsilon
\]

it then follows that $\Vert TR(\omega,\mathcal{S}_*)-TR'(\omega,\mathcal{S}_*)\|_{sup}<\beta \varepsilon $. Further, since $TR(\omega,\mathcal{S}_*)> TR'(\omega,\mathcal{S}^{'}_*)> TR'(\omega,\mathcal{S}_*)$ we have that $0<TR(\omega,\mathcal{S}_*)-TR'(\omega,\mathcal{S}_*)<\beta \varepsilon$. Using the latter inequality to obtain $0< TR(\omega,\mathcal{S}_*)-TR'(\omega,\mathcal{S}^{'}_*)+TR'(\omega,\mathcal{S}^{'}_*)-TR'(\omega,\mathcal{S}_*)<\beta \varepsilon$ and noting that $TR'(\omega,\mathcal{S}^{'}_*)-TR'(\omega,\mathcal{S}_*) >0$, it follows that $0< TR(\omega,\mathcal{S}_*)-TR'(\omega,\mathcal{S}^{'}_*)<\beta \varepsilon$. This last step is valid for all $\omega$, then it implies that $\Vert T(\Gamma(\omega, 3))-T(\Gamma^{' }(\omega,3)\|_{sup}<\beta \varepsilon $ and hence the operator $T$ is a contraction and has a unique fixed point.

Step 2 (Linking the Mapping $T$ and BRE Objects):

From the fixed point functions $M(\omega)$, $W^{U}(\omega)$ and $R(\omega)$ define the function $J(\omega)=\max\{(1-\zeta)[M(\omega)-W^{U}(\omega)],0\}$, and the functions $\theta (\omega)$ and $V(\omega)$ from $0=V(\omega)=-k+q(\theta (\omega))J(\omega)$. Also define $W^{E}(\omega)=M(\omega)-J(\omega)$ if $M(\omega)>W^{U}(\omega)$, and $W^{E}(\omega)=M(\omega)$ if $M(\omega)\leq W^{U}(\omega)$. Finally, define $d(\omega)=d^{T}(\omega)$, $\sigma(\omega)=\sigma^{T}(\omega)$, $\rho (\omega)=\rho^{T}(\omega)$, $\mathcal{S}(\omega)=\mathcal{S}^{T}(\omega)$ and a $w(\omega)$ derived using the Nash bargaining equation in the main text given all other functions.

Given $1- \zeta=\eta$ and provided that the job separation decisions between workers and firms coincide, which they are as a match is broken up if and only if it is bilaterally efficient to do so according to $M(\omega)$ and $W^{U}(\omega)$, then equations (5) and (6) in the main text (describing $J(\omega)$ and surplus sharing) are satisfied. Further, equation (3) in the main text (describing $W^E(\omega)$) is satisfied by construction, $\theta(\omega)$ satisfies the free-entry condition and $w(\omega)$ satisfies equation (6) in the main text. Hence, the constructed value functions and decision rules satisfy all conditions of the equilibrium and the implied evolution of the distribution of employed and unemployed workers also satisfies the equilibrium conditions.

Uniqueness follows from the same procedure in the opposite direction and using a contradiction argument. Suppose the BRE is not unique. Then a second set of functions exists that satisfy all the equilibrium conditions. Construct $\hat{M}$, $\hat{W}^{U}$ and $\hat{R}$ from these conditions. Since in any equilibrium the job separation decisions have to be bilaterally efficient and the occupational mobility decisions ($\rho$ and $\mathcal{S}$) are captured in $T$, then $\hat{M}$, $\hat{W}^{U}$ and $\hat{R}$ must be a fixed point of $T$, contradicting the uniqueness of the fixed point established by Banach's Fixed Point Theorem. Hence, there is a unique BRE.

§.§.§ Efficiency

The social planner, currently in the production stage, solves the problem of maximizing total discounted output by choosing job separations decisions $d(.)$, occupational mobility decisions $\rho(.)$ and $\mathcal{S}(.)$, as well as vacancy creation decisions $v(.)$ for each pair $(z,x_{h})$ across all occupations $o \in O$ in any period $t$. The first key aspect of the planner's choices is that they could potentially depend on the entire state space $\Omega^j=\{z, x, o, A, p, \mathcal{G}^j\}$ for each of the four within period stages $j=s,r,m,p$ (separation, reallocation, search and matching, production) and workers' employment status, such that its maximization problem is given by

\begin{align}
\max_{\{d(\Omega^s),\rho(\Omega^r), \mathcal{S}(\Omega^r),v(\Omega^m)\}} \text {{\Large {$\mathbb{E}$}}}
& \sum_{t=0} ^ {\infty} \bigg( \sum_{o=1} ^ {O} \sum_{h=1}^{H}\int_{\underline{z}} ^{\overline{z}} \beta ^{t} \left[ u_{t}^{p}(z,x_{h},o) b + e_{t}^{p}(z,x_{h},o) y(A_{t},p_{o,t},z,x_{h}) \right] dz   \\
& \qquad -  \sum_{o'=1} ^ {O} \sum_{h'=1}^{H}\int_{\underline{z}} ^{\overline{z}} \beta ^{t+1} \Big[c\rho(z', x_h', o', A_{t+1}, p_{t+1}, \mathcal{G}^r_{t+1}) u_{t+1}^{r}(z',x'_{h},o') \non \\
& \qquad \qquad \qquad \qquad \qquad  + k v_{t+1}(z', x_h', o', A_{t+1}, p_{t+1}, \mathcal{G}^m_{t+1})\Big] dz' \bigg), \non
\end{align}

subject to the initial conditions $(A_0, p_{0}, \mathcal{G}^{p}_0)$, the laws of motion for unemployed and employed workers described in Section C.2 of this appendix (with corresponding state space $\Omega^j$ for the decision rules), and the choice variables $\rho(.)$ and $d(.)$ being continuous variables in $[0,1]$, as the planner can decide on the proportion of workers in labor market $(z,x_{h})$ to separate from their jobs or to change occupations.

Note that implicitly the social planner is constrained in the search technology across occupations: it faces the same restrictions as an individual worker (in occupation $o \in O$), on the proportion of time that can be devoted to obtain a $z$-productivity from occupation $\tilde{o}\neq o$. Namely, $s_{\tilde{o}}(.)\in[0,1]$, $\sum_{\tilde{o} \in \vec{O}^{-}_{o}} s_{\tilde{o}}(.)=1$ and $\sum_{\tilde{o} \in \vec{O}^{-}_{o}} \alpha(s_{\tilde{o}}(.),o) \leq 1$, where $\vec{O}^{-}_{o}$ denotes the set of remaining occupations relative to $o$. The latter notation highlights that, as in the decentralised problem, once the occupational mobility decision has been taken the new $z$-productivity cannot be obtain from the departing occupation.

Rewriting the planners' problem in recursive form as the fixed point of the mapping $T^{SP}$ and letting next period's values be denoted by a prime yields

\begin{align}
&T^{SP}W^{SP}(\Omega^p) =\max_{\substack{\big\{d(\Omega^{s'}),\rho(\Omega^{r'}), \\\mathcal{S}(\Omega^{r'}), v(\Omega^{m'})\big\}}} \sum_{o=1}^{O} \sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}}(u^{p}(z,x_{h},o)b+e^{p}(z,x_{h},o)y(A,p_{o},z,x_{h}))dz \label{e: SP_VF_contraction_mapping}\\
& \quad +\beta \mathbb{E}_{\Omega^{s}{'}|\Omega^{p}}\left[-\left( c \sum_{o=1} ^ {O} \sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}} \rho(\Omega^{r'})u^{r}{'}( z', x_{h}' ,o)dz' +k \sum_{o=1} ^ {O} \sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}} v(\Omega^{m'})dz' \right) \right.  +W^{SP}(\Omega^{p}{'})\Bigg] \non,
\end{align}

subject to the same restrictions described above. Our aim is to show that this mapping is a contraction that maps functions $W^{SP}(.)$ from the space of functions linear with respect to the distribution $\mathcal{G}^p$ into itself, such that

\begin{align}
W^{SP}(\Omega^p)=\sum_{o=1}^{O} \sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}}\!\!\!\!\left( u^{p}(z,x_h,o) W^{u,SP}(\omega)\!+e^{p}(z,x_h,o)M^{SP}(\omega)\right)dz,\label{e:linear_SP_VF_guess}
\end{align}

for some functions $W^{u,SP}(\omega)$ and $M^{SP}(\omega)$, where $\omega=(z,x_h,o, A, p)$, and $u^{p}(.)$ and $e^{p}(.)$ are implied by $\mathcal{G}^p$.

In Section C.2 of this appendix, we decomposed the next period's measure of unemployed workers at the production stage, $u^{p'}$, into three additive terms (see equation (<ref>)). The first term refers to those unemployed workers that were unsuccessful in matching, either due to not finding a posted vacancy in their submarket or because the planner chose not to post vacancies in their submarket. The second term refers to those workers who separated from employment and hence were restricted from the search and matching stage within the period. The third term refers to those who changed occupations and came from other markets into submarket $(z,x_h,o)$ and were restricted from the search and matching stage within the period. Likewise, we decomposed the next period's measure of employed workers at the production stage into two additive terms (see equation (<ref>)). The first one refers to the survivors in employment from the previous separation stage, while the second terms refers to new hires. Given a continuation value function $W^{SP}(\Omega^{p'})$ that is linear in $\mathcal{G}^p$, as in (<ref>), we now show that we can also decompose the expression in (<ref>) into different additive components.

Consider first those unemployed workers who are eligible to participate in the search and matching stage. Note that the matching technology implies $v(\Omega^{m})=\theta(\Omega^{m})(1-\rho(\Omega^{r}))u^{m}(z,x_{h},o)$. Therefore we can consider $\theta(\Omega^{m'})$ as the planner's choice in equation (<ref>) instead of $v(\Omega^{m'})$. Next we isolate the terms of $W^{SP}(\Omega^{p'})$ that involve workers who went through the search and matching stage: the first term of the unemployed worker flow equation (<ref>) and the second term of the employed worker flow equation (<ref>). We then combine these terms with the cost of vacancy posting in (<ref>). Noting that the dependence of $u^m(\Omega^{m'})$ captures a potential dependence of the planner's occupational mobility decisions made in the previous reallocation stage, we can express the terms in the mapping (<ref>) that involve $\theta(\Omega^{m'})$ as

\begin{align}
\sum_{o=1}^{O} \sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}} u^m{'}(\Omega^{m'}) \Big\{-k \theta(.) + \lambda(\theta(.))M^{SP}(\omega') + (1-\lambda(\theta(.))W^{u,SP}(\omega') \Big\}dz'.  \label{e:SP_max_searchstage}
\end{align}

Note that by backwards induction, the planner's optimal decisions for $\theta(\Omega^{m'})$ maximize (<ref>). Given that the terms within the curly brackets only depend on $\omega'$ when the continuation value is linear as in (<ref>), maximizing with respect to $\theta$ implies an optimal $\theta(\omega')$. Using this result we let $W^{m,SP}(\omega')$ denote the sum of the maximized terms inside the curly brackets in (<ref>) .

Next consider the search-direction choice across occupations for those workers who the planner has decided to move across occupations. The following expression summarizes the terms of (<ref>) that involve this choice,

\begin{align}
\sum_{o'} \sum_{h'=1}^{H} \int_{\underline{z}} ^ {\overline{z}} \bigg[ u_{t+1}^{r}(\Omega^{r'}) \rho(\Omega^{r'})\bigg\{\sum_{\hat{o} \neq o'}\alpha(s_{\hat{o}}(.),o') \int_{\underline{z}} ^ {\overline{z}} \Big[ W^{u,SP}(\hat{z}, x_1,\hat{o}, A', p') \Big]  dF(\hat{z}) \non \\
+ (1- \sum_{\hat{o} \neq o'}\alpha(s_{\hat{o}}(.),o'))[b+\beta \mathbb{E}_{\omega''} R^{SP}(\omega'')] \bigg\}\bigg] dz' , \label{e:SP_max_reallstage1}
\end{align}

where we have applied the properties of the $z$-productivity process to change the order of integration relative to the third term of flow equation (<ref>) and have integrated over all of next period's production stage states. Since we allow previous decisions to depend on the entire state space, $u_{t+1}^{r}(.)$ and $\rho(.)$ are written as a function of $\Omega^{r'}$ as they potentially depend on the distribution $\mathcal{G}^r$. Note, however, that the term inside the curly brackets in (<ref>) can be maximized separately for each $(z',x{_h}{'},o')$ and can therefore be summarized by $R^{gross,SP}(\omega')=R^{SP}(\omega')-c$, while the search intensity decision $\mathcal{S}(.)$ has solution vector $\{s(\omega', \tilde{o})\}$.

Noting that $u^s(.)=u^r(.)$ and $u^m(.)=(1-\rho(.))u^r(.)$ (see Section C.2 of this appendix), the above results imply we can express the term on the second line of equation (<ref>) as

\begin{align}
\beta \mathbb{E}_{\Omega^{s}{'}|\Omega^{p}}\Bigg[\Bigg(  \sum_{o=1} ^ {O} \sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}}
& \bigg[u^{s}{'}( z', x_{h}' ,o)\Big\{\rho(.)R^{SP}(\omega')+(1-\rho(.))W^{m,SP}(\omega')\Big\} \\
& \quad + e^{s}{'}( z', x_{h}' ,o)\Big\{(1-d(.))M^{SP}(\omega')+d(.)W^{u,SP}(\omega')\Big\} \bigg]dz  \Bigg) \Bigg], \non
\end{align}

where it follows that the maximizing decisions $\rho(.)$ and $d(.)$ are also functions of $\omega'$, as these are the only other dependencies within the curly brackets.

Finally, given the properties of the shock processes, note that $u^s_{t+1}(.)$ and $e^s_{t+1}(.)$ are linear functions of $u^p(.)$ and $e^p(.)$, as demonstrated by equations (<ref>) and (<ref>) in Section C.2 of this appendix. This implies that we can express $T^{SP}W^{SP}(\Omega^p)$ as

\begin{equation}
T^{SP}W^{SP}(\Omega^p)=\sum_{o=1} ^ {O}\sum_{h=1}^{H} \int_{\underline{z}} ^{\overline{z}} \left[TW_{max}^{U,SP}(z,x_{h},o, A, p) u(z,x_{h},o) +TM^{SP}_{max}(z,x_{h},o, A, p)e(z,x_{h},o) \right] dz, \label{e: SP_VF_contraction_mapping2}
\end{equation}

where $TW_{max}^{U,SP}$ is given by

\begin{align}
TW_{max}^{U,SP}(\omega) =& \max_{\rho(\omega'), \theta(\omega')}\bigg\{b+\beta \mathbb{E}_{\omega'|\omega}\bigg[\rho(\omega')\Big( \int_{\underline{z}} ^{\overline{z}} \max_{\mathcal{S(\omega')}} \Big[\sum_{\tilde{o}\neq o} \alpha(s_{\tilde{o}}(\omega', o))W^{U, SP}_{max}(\tilde{z},x_{1}, \tilde{o},A', p')dF(\tilde{z}) \non \\
& \ \ \qquad + (1-\sum_{\tilde{o}\neq o} \alpha(s_{\tilde{o}}(\omega', o)))[b+\beta \mathbb{E}_{\omega''|\omega'}R^{U, SP}_{max}(\omega'')\Big]-c\Big)  \label{e:TWmaxSP_contractionmapping} \\
& \ \ \qquad  +(1-\rho(\omega'))\Big[ \lambda (\theta(\omega'))( M^{SP}_{max}(\omega')-W^{U, SP}_{max}(\omega')) -\theta(\omega')k+W^{U,SP}_{max}(\omega')\Big]\bigg]\bigg\}, \non
\end{align}

and $TM_{max}^{SP}$ is given by

\begin{align}
TM_{max}^{SP}(p,x, z) & =\max_{d(\omega')}\bigg\{y(z,x_{h},A,p_o) +\beta \mathbb{E}_{\omega'|\omega}\bigg[\left( d(\omega')W^{U,SP}_{max}(\omega')+(1-d(\omega'))M^{SP}_{max}(\omega')\right) %
\bigg]\bigg\}. \label{e:TMmaxSP_contractionmapping}
\end{align}

Hence we have established that the mapping $T^{SP}$ described in equation (<ref>) maps functions $W^{SP}(.)$ from the space of functions linear with respect to the distribution $\mathcal{G}^p$ (of the form (<ref>)) into itself.

It is now straightforward to show that if $TW_{max}^{U,SP}$, $TR_{max}^{SP}$ and $TM_{max}^{SP}$ are contraction mappings, then (<ref>) (and thereby (<ref>)) is also a contraction mapping. Given the regularity properties assumed on the shock processes and following the proof of the decentralised case in Section C.4.1.1 of this appendix, we can show that a fixed point exists for (<ref>) and (<ref>). Using equation (<ref>), we then can construct the fixed point of the expression in $\eqref{e: SP_VF_contraction_mapping}$. It then follows from (<ref>) and (<ref>) that if the Hosios' condition holds, allocations of the fixed point of $T$ are allocations of the fixed point of $T^{SP}$, and hence the equilibrium allocations in the decentralised setting are also the efficient allocations.

§.§.§ BRE gives the unique equilibrium allocation
We now show that in any equilibrium, decisions and value functions only depend on $\omega=(z,x_h,o, A, p)$. We proceed using a contradiction argument. Suppose there is an alternative equilibrium in which values and decisions do not depend only on $\omega$, but also on an additional factor like the entire distribution of workers over employment status and $z$-productivities $\mathcal{G}$, or its entire history of observables, $H_{t}$. Consider the associated value functions in this alternative equilibrium, where the relevant state vector of the alternative equilibrium is given by $(\omega,H_{t})$. Our aim is to show that such an equilibrium cannot exist.

First suppose that in the alternative equilibrium all value functions are the same as in the BRE, but decisions differ at the same $\omega$. This violates the property that, in our setting, all maximizers in the BRE value functions are unique, leading to a contradiction. Now suppose that in the alternative equilibrium at least one value function differs from the corresponding BRE value function at the same $\omega$. It is straightforward to show that the expected values of unemployment must differ in both equilibria. Let $W^U(\omega,H_t)$ denote the value function for unemployed workers in the alternative equilibrium, and let $W^U(\omega)$ denote the corresponding value function in the BRE for the same $\omega$. Since in the proof of efficiency of a BRE we did not rely on the uniqueness of the BRE in the broader set of all equilibria, we can use the proved results of Section C.3.1.2 of this appendix here.

In particular, recall that the social planner's problem is entirely linear in the distribution of workers across states and hence $W^U(\omega)$ is the best the unemployed worker with $\omega$ can do (without transfers), including in the market equilibrium. Likewise, $M(\omega)$ is the highest value of the joint value of a match, including in the market equilibrium. Since value functions are bounded from above and from below and are continuous in their state variables, there exists a supremum of the difference between $W^U(\omega)$ and the candidate market equilibrium's $W^U(\omega, H_t)$, $\sup (W^U(\omega)-W^U(\omega,H_t))=\epsilon_u>0$. Similarly, there also exists a supremum for the difference between $M(\omega)$ and $M(\omega,H_t)$, $\sup (M(\omega)-M(\omega, H_t))=\epsilon_m>0$. In what follows, we will show that a difference in the value functions for unemployed workers (or the value functions for the joint value of the match) arbitrarily close to $\epsilon_u>0$ (or $\epsilon_m>0$) cannot occur. Otherwise, this will require that the difference in tomorrow's values to be larger than $\epsilon_u$ (or $\epsilon_m$). In turn, this implies that an alternative equilibrium cannot exist.

From the above definition of supremum, it follows that

\[\max\{M(\omega), W^U(\omega)\}-\max\{M(\omega, H_t), W^U(\omega, H_t)\}<\max\{\epsilon_u, \epsilon_m\}.\]

\[M(\omega)=y(z,x_{h},A,p_o)+\beta \Exp [\max\{M(\omega'),W^U(\omega')\}],\]
and likewise for $M(\omega, H_t)$ it follows that at any $(\omega, H_t)$,

\begin{equation}\label{e:app_towards_contradiction}
M(\omega)-M(\omega, H_t))<\beta \max\{\epsilon_u, \epsilon_m\}.
\end{equation}

Consider first the case in which $\epsilon_m\geq \epsilon_u$. For $(\omega, H_t)$ achieving a difference $M(\omega)-M(\omega, H_t)> \beta \epsilon_m$ is not possible since this will lead to a contradiction in equation (<ref>) when $\epsilon_m>0$.

Next consider the case in which $\epsilon_m<\epsilon_u$. Simplifying notation by dropping $\omega$ and using the prime instead of $(\omega, H_t) $, we first establish an intermediate step. At any $(\omega, H_t)$ it holds that

\begin{align}\label{e:app_towards_contriduction2}
\lambda(\theta)(1-\eta)M + (1-\lambda(\theta)(1-\eta))W<\lambda(\theta')(1-\eta)M' + (1-\lambda(\theta')(1-\eta))W'+\epsilon_u.
\end{align}

There are two cases to be analysed to show the above relationship.

Case 1: Suppose that $(M'-W')\geq M-W$, then $\lambda(\theta')\geq \lambda(\theta)$. Define $K=(1-\eta)(\lambda(\theta')-\lambda(\theta))(M'-W')\geq 0$. Combining the latter with $\lambda(\theta)(1-\eta)(M-M') + (1-\lambda(\theta)(1-\eta))(W-W') \leq \epsilon_u$, it must be true that

\begin{align}\label{e:app_towards_contriduction3}
\lambda(\theta)(1-\eta)(M-M') + (1-\lambda(\theta)(1-\eta))(W-W')-K\leq \epsilon_u,
\end{align}

from which (<ref>) follows.

Case 2: Suppose that $(M-W)>(M'-W')$, then $\lambda(\theta)>\lambda(\theta')$. From the derivative of $\frac{d}{d(M-W)}(\lambda(\theta(M-W))(1-\eta)(M-W))=\lambda(\theta(M-W))$, we can establish that, if $(M-W)>(M'-W')$,

\[\lambda(\theta)((M-W)-(M'-W'))>\lambda(\theta)(1-\eta)(M-W)-\lambda(\theta')(1-\eta)(M'-W')>\lambda(\theta')((M-W)-(M'-W')).\]

Since in any equilibrium (not only in the BRE) $\theta'$ depends only on $(M'-W')$ and constant parameters, we can use this relationship to establish that

\[W+\lambda(\theta)(M-W)-(W'+\lambda(\theta)(M'-W'))<\epsilon_u,\]

from which (<ref>) follows.

The final step is to consider an $(\omega, H_t)$ such that $\max\{\epsilon_m, \beta^{-1}\epsilon_u\}<W^U(\omega)-W^U(\omega,H_t)<\epsilon_u$, where such a $(\omega, H_t)$ exists by the definition of supremum. With this in hand it is straightforward to check that the difference in tomorrow's value (under the expectation sign), between $W^U(\omega)$ and $W^U(\omega, H_t)$ will not exceed $\epsilon_u$, since term-by-term, the difference is bounded by $\epsilon_u$. This also implies that today's difference, $W^U(\omega)-W^U(\omega,H_t)$, cannot be more than $\beta \epsilon_u>0$, which contradicts our premise. This establishes that a difference in the value functions for unemployed workers (or the value functions for the joint value of the match) arbitrarily close to $\epsilon_u>0$ (or $\epsilon_m>0$) cannot occur. Hence the BRE is the unique equilibrium.

This completes the proof of Proposition 2.

§.§ Proof of Existence of a Reallocation and Separation cutoff

§.§.§ Reservation property of occupational mobility decisions, $z^{r}$
Here we show that $M(\omega)$ and $W^U(\omega)$ as derived in the proof of Proposition 2 are increasing in $z$. If $M(\omega)$ and $W^{U}(\omega)$ are continuous and bounded functions increasing in $z$, $T$ maps them into increasing (bounded and continuous) functions. For employed workers $(T(\Gamma(\omega,0))$, this follows since both $\max \{M(\omega^{'}),W^{U}(\omega^{'})\}$ and $y(.)$ are increasing in $z$, while the stochastic dominance of the $z$-productivity transition law implies higher expected $z$'s tomorrow. For unemployed workers $(T(\Gamma(\omega,1))$, note that the value of changing occupations $R(\omega)$ does not depend on the current $z$ of the worker, while equation (<ref>) implies that $D(M(\omega)-W^U(\omega))+W^U(\omega)$ is increasing in $z$. Again, given stochastic dominance of the tomorrow's $z$ when today's $z$ is higher, $T(\Gamma(\omega,1))$ is also increasing in $z$. The reservation property follows immediately, since $R(\omega)$ is constant in $z$ and $D(M(\omega)-W^U(\omega))+W^U(\omega)$ is increasing in $z$.

§.§.§ Reservation property of job separation decisions, $z^{s}$
We now show that $M(\omega)-W^U(\omega)$ is increasing in $z$ when $\delta+\lambda(\theta(\omega))<1$ for the case of no human capital accumulation and occupational-wide shocks. In the calibration we show that this property holds also for the case of human capital accumulation and occupational-wide shocks.

Consider the same operator $T$ defined in the proof of Proposition 2, but now the relevant state space is given by $(A,z)$. Note that the value functions describing the worker's and the firm's problem, do not change, except for the fact that we are using a smaller state space. It is straightforward to verify that the derived properties of $T$ in Lemma A.1 also apply in this case. We now want to show that this operator maps the subspace of functions $\Gamma$ into itself with $M(A,z)$ increasing weakly faster in $z$ than $W^{U}(A,z)$. To show this, take $M(A,z)$ and $W^U(A,z)$ such that $M(A,z)-W^{U}(A,z)$ is weakly increasing in $z$ and let $z^s$ denote a reservation productivity such that for $z<z^s$ a firm-worker match decide to terminate the match. Using $\lambda (\theta)(M-W^{U})-\theta k=\lambda (\theta)(M-W^{U})-\lambda ^{\prime}(\theta)(M-W^{U})\theta=\lambda (\theta)(1-\eta)(M-W^{U})$, we construct the following difference

\begin{align}
T\Gamma&(A,z,0)-T\Gamma(A,z,1) = \label{e:appdiff}\\
& y(A,z)-b+ \beta \mathbb{E}_{A,z} \Big[(1-\delta )\max \{M(A^{\prime }, z^{\prime })-W^{U}(A^{\prime }, z^{\prime },0\} - \nonumber \\
&  \max \left\{ \int W^{U}(A^{\prime },\tilde{z})dF(\tilde{z})-c-W^{U}(A^{\prime }, z^{\prime }),\lambda (\theta)(1-\eta )\big(M(A^{\prime }, z^{\prime })-W^{U}(A^{\prime }, z^{\prime })\big)\right\} \Big]. \nonumber
\end{align}

The first part of the proof shows the conditions under which $T\Gamma(A,z,0)-T\Gamma(A,z,1)$ is weakly increasing in $z$. Because the elements of the our relevant domain are restricted to have $W^U(A,z)$ increasing in $z$, and $M(A,z)-W^U(A,z)$ increasing in $z$, we can start to study the value of the term under the expectation sign, by cutting a number of different cases to consider depending on where $z'$ is relative to the implied reservation cutoffs.

– Case 1. Consider the range of tomorrow's $z'\in [\underline{z}(A'),z^r(A'))$, where $z^r(A')<z^s(A')$. In this case, the term under the expectation sign in the above equation reduces to $\mathbf{-}\int W^{U}(A^{\prime },\tilde{z})dF(\tilde{z})+c+W^{U}(A^{\prime }, z^{\prime })$, which is increasing in $z'$.

– Case 2. Now suppose tomorrow's $z'\in [z^r(A'),z^s(A'))$. In this case, the term under the expectation sign becomes zero (as $M(A^{\prime },z^{\prime })-W^{U}(A^{\prime},z^{\prime })=0$), and is therefore constant in $z'$.

– Case 3. Next suppose that $z'\in [z^s(A'),z^r(A'))$. In this case, the entire term under the
expectation sign reduces to

\[(1-\delta)(M(A^{\prime }, z^{\prime})-W^{U}(A^{\prime},z^{\prime }))\mathbf{-}\int W^{U}(A^{\prime },\tilde{z})dF(\tilde{z})+c+W^{U}(A^{\prime }, z^{\prime }) ,\]
and, once again, is weakly increasing in $z'$, because by supposition  $M(A',z')-W^{U}(A',z')$ is weakly increasing in $z'$, and so is $W^U(A', z')$ by Lemma A.1.

– Case 4. Finally consider the range of $z'\geq \max\{z^r(A'), z^s(A')\}$, such that in this range employed workers do not quit nor reallocate. In this case the term under the expectation sign equals

\begin{equation}\label{e:appcase4}
(1-\delta )[M(A^{\prime }, z^{\prime })-W^{U}(A^{\prime }, z^{\prime})]-\lambda (\theta (A', z'))(1-\eta )[M(A^{\prime }, z^{\prime})-W^{U}(A^{\prime }, z^{\prime })].
\end{equation}

It is easy to show using the free entry condition that $\frac{d(\lambda (\theta ^{\ast }(A', z'))(1-\eta )[M(A^{\prime }, z^{\prime
})-W^{U}(A^{\prime }, z^{\prime })])}{d(M-W) }=\lambda(\theta(A',z'))$, and hence that the derivative of (<ref>) with respect to $z'$ is positive whenever $1-\delta-\lambda(\theta)\geq 0$.

Given $F(z'|z)<F(z'|\tilde{z})\ $for all $z,z'$ when $z>\tilde{z}$, the independence of $z$ of $A$, and that the term under the expectation sign are increasing in $z'$, given any $A'$, it follows that the integral in (<ref>) is increasing in today's $z$. Together with $y(A,z)$ increasing in $z$, it must be that $T\Gamma(A,z,1)-T\Gamma(A,z,0)$ is also increasing in $z$.

To establish that the fixed point also has increasing differences in $z$ between the first and second coordinate, we have to show that the space of this functions is closed in the space of bounded and continuous functions. In particular, consider the set of functions $\mathbb{F}\overset{def}{=}\{f\in \mathcal{C}|f:X\times Y\rightarrow \mathbb{R}^{2},|f(x,y,1)-f(x,y,2)|\text{ increasing in }y\},$ where $f(.,.,1),f(.,.,2)$ denote the first and second coordinate, respectively, and $\mathcal{C}$ the metric space of bounded and continuous functions endowed with the sup-norm.

The next step in the proof is to show that fixed point of $T\Gamma(A,z,0)-T\Gamma(A,z,1)$ is also weakly increasing in $z$. To show we first establish the following result.

Lemma B.1:

$\mathbb{F}$ is a closed set in $\mathcal{C}$

Consider an $f' \notin \mathbb{F}$ that is the limit of a sequence $\{f_n\}, f_n \in \mathbb{F}, \forall n \in \mathbb{N}$. Then there exists an $y_1< y$ such that $f'(x,y_1, 1)-f'(x,y_1, 2)>f'(x,y,1)-f'(x,y,2),$ while $ f_n(x,y_1,1)-f_n(x,y_1,2) \leq  f_n(x,y,1)-f_n(x,y,2) , $ for every $n$. Define a sequence $\{s_n\}$ with $s_n= f_n(x,y_1,1)-f_n(x,y_1,2) - (f_n(x,y,1)-f_n(x,y,2)) $. Then $s_n\geq 0,\ \forall n \in \mathbb{N}$. A standard result in real analysis guarantees that for any limit $s$ of this sequence, $s_n \to s$, it holds that $s\geq 0$. Hence $ f'(x,y_1, 1)-f'(x,y_1, 2) \leq  f'(x,y,1)-f'(x,y,2) $, contradicting the premise.

Thus, the fixed point exhibits this property as well and the optimal quit policy is a reservation-$z$ policy given $1-\delta -\lambda (\theta)>0$. Since $y(A,z)$ is strictly increasing in $z$, the fixed point difference $M-W^U$ must also be strictly increasing in $z$. Furthermore, since $\lambda(\theta)$ is concave and positively valued, $\lambda^{\prime}(\theta)(M-W^U)=k$ implies that job finding rate is also (weakly) increasing in $z$.

§.§ Proofs of the “Model Implications and Comparative Statics”

We now turn to the proofs of Lemmas 1, 2 and 3 presented in Section C.1. Recall that these results were derived using a simplified version of the model without occupational human capital accumulation, setting $x_{h}=1$ for all $h$, and without occupational-wide productivity shocks, setting $p_{o}=1$ for all $o$. These restrictions imply that the relevant state space is $(z,A)$. We further assumed that the $z$-productivity is redrawn randomly with probability $0<(1-\gamma)<1$ each period from cdf $F(z)$ and $A$ is held constant. In this stationary environment, the expected value of unemployment for a worker currently with productivity $z$ in occupation $o$ is given by

\begin{eqnarray}\label{a:wu}
W^U(A,z)&=&\gamma \Big(b + \beta \max \Big\{R(A), W^U(A,z)+ \max \{\lambda(\theta(A,z))(1-\eta)(M(A,z)-W^U(A,z)), 0\}\Big\}\Big)  \notag \\
&+ &(1-\gamma)\Exp_{z}[W^U(A,z)].
\end{eqnarray}

where the expected value of occupational mobility is given by $R(A) = - c + \Exp_{z}[W^U(A,z)]$.
The values of employment at wage $w(A,z)$ for a worker currently with productivity $z$ and a firm employing this worker are given by

\begin{equation}\label{a:we}
W^E(A,z)=\gamma \big[ w(A,z) + \beta[ (1-\delta) W^E(A,z) + \delta W^U(A,z)] \big] +(1-\gamma)\Exp_{z} [W^E(A,z)],
\end{equation}

\begin{equation}\label{a:j}
J(A,z)=\gamma \big[ y(A,z)-w(A,z) + \beta[ (1-\delta) J(A,z))] \big] +(1-\gamma)\Exp_{z} [J(A,z)].
\end{equation}

The joint value of a match is then

\begin{align}\label{a:m}
M(A,z)=\gamma \big[ y(A,z) + \beta[ (1-\delta) M(A,z) + \delta W^U(A,z)] \big] +(1-\gamma)\Exp_{z} [M(A,z)].
\end{align}

Proof of Lemma 1 We state the detailed version of this lemma as Lemma H.1. To simplify the analysis and without loss of generality, in what follows we let $\delta=0$. Since we do not vary aggregate productivity to proof this lemma, we let $A=1$ and abusing notation we refer to $z$ as total output. This is done without loss of generality as now output only depends (and is strictly increasing and continuous) on the worker-occupation match-specific productivity. To abbreviate notation defined $W^s\equiv W^U(z^s)$.

Lemma H.1: The differences $W^s-R$ and $z^r-z^s$, respond to changes in $c$, $b$ and $\gamma$ as follows

* (i) $W^s-R$ is strictly increasing in $c$ and (ii) $z^r-z^s$ is decreasing in $c$, strictly if $z^r>z^s$.

* (i) $W^s-R$ is strictly increasing in $b$ and (ii) $z^r-z^s$ is strictly decreasing in $b$.

*  (i) $W^s-R$ is strictly decreasing in $\gamma$ and (ii) $z^r-z^s$ is strictly increasing in $\gamma$.

We first consider the link between $W^s-R$ and $z^r-z^s$ as the parameter of interest $\kappa$ (i.e. $c$, $b$ or $\gamma$) changes. The reservation productivities for job separation and occupational mobility implicitly satisfy

\begin{align}
M(\kappa,z^s(\kappa))-W^s(\kappa)& =0  \label{e:l7_reservation_zs} \\
W^U(\kappa,z^r(\kappa))\!-\!R(\kappa)=  \lambda(\theta(\kappa,z^r(\kappa)))(1\!-\!\eta)(M(\kappa,z^r(\kappa))-W^s(\kappa))+(W^s(\kappa)\!-\!R(\kappa))&=0 \label{e:l7_reservation_zr1},
\end{align}

where (<ref>) only applies when $R(\kappa)\geq W^s(\kappa)$ and hence only when $z^r(\kappa) \geq z^s(\kappa)$. In the case of $W^s(\kappa)>R(\kappa)$, the assumed stochastic process for $z$ implies that $z^r(\kappa)=\underline{z}$. The latter follows as $W^U(z)=W^s(\kappa)$ for all $z<z^s$. Also note that we can use $W^s(\kappa)$ instead of $W(\kappa, z^r(\kappa))$ in (<ref>) as the assumed stochastic process for $z$ also implies that $W^U(z)=W^s(\kappa)$ for all $z<z^r$ when $R(\kappa)>W^s(\kappa)$.

Note that by defining

\begin{align*}
Z(\kappa) \equiv \Exp_z[\max\{M(\kappa,z),\!W^s(\kappa)\}-\max\{W^U(\kappa,z)\!+\!\lambda(\theta(\kappa,z))(1\!-\!\eta)(M(\kappa,\!z)\!-\!W^U(\kappa,\!z)),R(\kappa)\}],
\end{align*}

we can rewrite equations (<ref>) and (<ref>) as

\begin{align*}
M(\kappa,z^s(\kappa))-W^s(\kappa)=z^s-b+\beta(1-\gamma)Z(\kappa)+\beta \gamma (W^s(\kappa)-R(\kappa))=0.
\end{align*}

Further note that

\begin{align*}
M(\kappa,z^r(\kappa))-W^s(\kappa) &=z^r-b+\beta(1-\gamma)Z(\kappa)+\beta \gamma (1-\lambda(\theta(\kappa, z^r))(1-\eta))(M(\kappa,z^r(\kappa))-W^s(\kappa)),
\end{align*}

which is strictly positive and changes with $\kappa$. Using (<ref>) leads to

\begin{align*}
(1-\beta \gamma)(M(\kappa,z^r(\kappa))-W^s(\kappa))=z^r-b+\beta(1-\gamma)Z(\kappa)+\beta \gamma(W^s(\kappa)-R(\kappa))
\end{align*}

Leaving implicit the dependency on $\kappa$, we obtain that

\[
\frac{d(M-W^s)}{d(W^s-R)}= \frac{d(M-W^s)}{d(\lambda(\theta)(1-\eta)(M-W^s))}\cdot\frac{d(\lambda)(1-\eta)(M-W^s)}{d(W^s-R)},
\]

where the latter term equals $-1$, and the former term equals $\lambda^{-1}(\theta)$. This follows as Nash Bargaining and free-entry imply $q(\theta)\eta(M-W^U)=c$ and hence $\frac{d\theta}{dM-W^U}=\frac{\theta}{(1-\eta)(M-W^U)}$.
Then $\lambda'(\theta)\frac{d\theta}{dM-W^U}(1-\eta)(M-W^U)+\lambda(\theta)(1-\eta)$ reduces to  $\lambda(\theta)$.
It then follows that when $z^r(\kappa) \geq z^s(\kappa)$, the derivative of $z^r$ with respect to $\kappa$ is given by

\begin{align}   \label{e:l7_derivative_z^r}
\frac{dz^r(\kappa)}{d\kappa}&=-\frac{d}{d\kappa}\bigg[-b+\beta(1-\gamma)Z(\kappa)\bigg]-\left(\beta\gamma+ \frac{1-\beta\gamma}{\lambda(\theta(\kappa,z^r(\kappa)))}\right)\frac{d(W^s(\kappa)-R(\kappa))}{d\kappa}.
\end{align}

We can similarly obtain that when $z^r(\kappa) \geq z^s(\kappa)$, the derivative of $z^s$ with respect to $\kappa$ is given by

\begin{align} \label{e:l7_derivative_z^s}
\frac{d z^s(\kappa)}{d\kappa}&=-\frac{d}{d\kappa}\bigg[-b+\beta(1-\gamma)Z(\kappa)\bigg] -   \beta \gamma \frac{d(W^s(\kappa)-R(\kappa))}{d\kappa}. \end{align}

Equations (<ref>) and (<ref>) then imply that when $z^r(\kappa) \geq z^s(\kappa)$ (i.e. $z^r$ and $z^s(\kappa)$ are interior), the derivative of $z^r(\kappa)-z^s(\kappa)$ with respect to $\kappa$ has the opposite sign to the derivative of $W^s(\kappa)-R(\kappa)$ with respect to $\kappa$.

As mentioned above, in the case of $W^s(\kappa)>R(\kappa)$ our simplified model implies $z^r(\kappa)=\underline{z}$. Hence changes in $\kappa$ can only affect $z^s$, although will affect $W^s(\kappa)$ and $R(\kappa)$. Using (<ref>) and (<ref>) we obtain that for $W^s(\kappa)>R(\kappa)$ the reservation separation cutoff is given by

\begin{align*}
z^s(\kappa)=b-\frac{1-\gamma}{\gamma} \Exp_{z} [M(\kappa,z)-W^{U}(\kappa,z) ].
\end{align*}

Taking derivative with respect to $\kappa$ then lead to

\begin{align} \label{e:zs_rest}
\frac{d z^s(\kappa)}{d\kappa} &= \frac{d b}{d\kappa} - \frac{d ((1-\gamma)/\gamma)}{d\kappa} \Exp_{z} [M(\kappa,z)-W^{U}(\kappa,z) ] + \frac{1-\gamma}{\gamma} \frac{d \Exp_{z} [M(\kappa,z)-W^{U}(\kappa,z) ]}{d\kappa}.
\end{align}

To complete the lemma we now obtain the derivatives of $W^s(\kappa)-R(\kappa)$ and $\Exp_{z} [M(\kappa,z)-W^{U}(\kappa,z) ]$ with respect to $\kappa$.

Comparative Statics with respect to $c$:

Consider the difference $W^s-R$ and values of $c$ such that $R\geq W^s$. In this case we have that

\begin{align*}
W^s=(1-\gamma)(R+c)+\gamma(b+\beta R), \\
W^s-R=-\gamma(1-\beta)R+(1-\gamma)c+\gamma b.
\end{align*}

Suppose towards a contradiction that $d(W^s-R)/dc<0$. The above equations imply that $\frac{dR}{dc}>\frac{(1-\gamma)}{\gamma(1-\beta)}>0$. We will proceed by showing that under $d(W^s-R)/dc<0$ both the expected match surplus (after a $z$-shock) and the match surplus for active labor markets (those with productivities that entail positive match surplus) decrease with $c$, which implies that the value of unemployment decreases with $c$, which in turn implies $\frac{dR}{dc}<0$, which is our contradiction.

Consider an active labor market with $W^U(z)>R$, the surplus on this labor market is then given by

\begin{align} \label{e:l7_1}
M(z)-W^U(z)&=\gamma(z-b + \beta (1-\lambda(\theta(z))(1-\eta))(M(z)-W^U(z))) \non \\
& \qquad +(1-\gamma)(\Exp[M(z)-W^U(z)]+(z-\Exp[z])),
\end{align}

where $\Exp[M(z)-W^U(z)]$ describes the expected surplus after a $z$-shock (after the search stage). Note that

\begin{equation}\label{e:espen_rule}
\frac{d}{d(M(z)-W^U(z))}(\lambda(\theta(z))(1-\eta)(M(z)-W^U(z)))=\lambda(\theta(z)).
\end{equation}

As described before, this result follows as (dropping the $z$ argument for brevity) our assumptions of Nash Bargaining and free-entry imply $(1-\eta)(M-W^U)=\frac{(1-\eta)}{\eta} J = \frac{1-\eta}{\eta}\frac{k}{q(\theta)}$, and hence $\lambda(\theta)(1-\eta)(M-W^U)=\frac{1-\eta}{\eta} k \theta$. Further, since $\frac{d\theta}{d(M-W^U)}=\frac{\eta}{1-\eta}\frac{\lambda(\theta)}{k}$, the chain rule then yields a derivative equal to $\lambda(\theta)$.
This result implies that from (<ref>) we obtain

\begin{align}\label{e:l7_2a}
0<\frac{d(M(z)-W^U(z))}{d(\Exp[M(z)-W^U(z)])}=\frac{1-\gamma}{1-\gamma \beta (1-\lambda(\theta(z)))}<1.
\end{align}

We now show that under $d(W^s-R)/dc<0$, the expected match surplus decreases in $c$. Note that the expected match surplus measured after the search stage is given by

\begin{align}\label{e:l7_3}
\Exp[M(z)-W^U(z)]=&\int_{z^r} z-b + \beta(1- \lambda(\theta(z))(1-\eta))(M(z)-W^U(z))dF(z) \non \\
& \ + \int_{z^s}^{z^r} z-b + \beta (M(z)-R)dF(z) + \int^{z^s} z-b + \beta (W^s-R)dF(z),
\end{align}

where the $(1-\gamma)$ shock integrates out. Our contradiction supposition implies that the third term of this expression is decreasing in $c$. The second term, $\int_{z^s}^{z^r}[M(z)-W^U(z)]dF(z)$, can be rewritten as

\[M(z)-W^s=\gamma(z-b+\beta(M(z)-W^s+W^s-R))+ (1-\gamma)(\Exp[M(z)-W^U(z)]+z-\Exp[z]),\]

and rearranging yields

\[M(z)-W^s=\frac{\gamma}{1-\gamma \beta} (z-b+\beta (W^s-R))+ \frac{1-\gamma}{1-\gamma \beta} \Exp[M(z)-W^U(z)],\]

where $\frac{\gamma}{1-\gamma \beta} (z-b+\beta (W^s-R))$ is decreasing in $c$ under our contradiction supposition. In the case of the first term in (<ref>), note that (<ref>) implies $M(z)-W^U(z)$ depends on $c$ through $\Exp[M(z)-W^U(z)]$. Combining all the elements, (<ref>), (<ref>) and the last two equations, we find that

\begin{align}\label{e:7_exp[m-wu]}
\frac{d\Exp[M(z)-W^U(z)]}{dc}&=\int_{z^r} \frac{(1-\gamma)\beta(1-\lambda(\theta(z)))}{1-\gamma \beta(1-\lambda(\theta(z)))}dF(z)\frac{d\Exp[M(z)-W^U(z)]}{dc} \non \\
& +(F(z^r)-F(z^s))\left(\frac{\gamma\beta}{1-\gamma \beta}\frac{d(W^s-R)}{dc} +\frac{1-\gamma}{1-\gamma \beta} \frac{d\Exp[M(z)-W^U(z)]}{dc}\right) \non  \\
& +F(z^s)\beta\frac{d(W^s-R)}{dc} \non \\
\iff \frac{d\Exp[M(z)-W^U(z)]}{dc} & = C \cdot \frac{d(W^s-R)}{dc}<0,
\end{align}

where $C$ is a positive constant. Equation (<ref>) then implies that $\frac{d[M(z)-W^U(z)]}{dc}<0$.

Next consider $\frac{dW^U(z)}{dc}$ and $\frac{d\Exp[W^U(z)]}{dc}$. Note that for the case in which $z\leq z^r$, $W^U(z)=W^s=(1-\gamma)\Exp[W^U(z)]+\gamma (b+\beta \Exp[W^U(z)]-\beta c)$; while for $z>z^r$, $W^U(z)=(1-\gamma)\Exp[W^U(z)]+\gamma(b+\beta(\lambda(\theta(z))(1-\eta)(M(z)-W^U(z))+\beta W^U(z)))$. It then follows that

\[
\Exp[W^U(z)]=F(z^r)(b+\beta \Exp[W^U(z)]-\beta c)+\int_{z^r}(b+\beta \lambda(\theta(z))(1-\eta)(M(z)-W^U(z))+\beta W^U(z))dF(z),
\]

which combined with

\[
W^U=\frac{1-\gamma}{1-\beta \gamma}\Exp[W^U(z)]+\frac{\gamma }{1-\beta \gamma}(b+\beta\lambda(\theta(z))(1-\eta)(M(z)-W^U(z))),
\]

leads to

\begin{align*}
& \left(1-\beta F(z^r)-\beta \frac{1-\gamma}{1-\beta \gamma}(1-F(z^r))\right)\Exp[W^U(z)] \non \\
& \qquad=F(z^r)(b-\beta c) + \int_{z^r}\frac{b+\beta\lambda(\theta(z))(1-\eta)(M(z)-W^U(z))}{1-\beta \gamma}dF(z).
\end{align*}

Taking the derivative with respect to $c$, we find that both the first and second terms on the RHS of the last expression decrease with $c$, the latter because we have established before that $\frac{d(M(z)-W^U(z))}{dc}<0$. It then follows that $\frac{d\Exp[W^U(z)]}{dc}<0$ and hence $\frac{dR}{dc}=\frac{d\Exp[W^U(z)]}{dc}-1<0$, which yields our desired contradiction. Equations (<ref>) and (<ref>) then imply that $\frac{d (z^r(c)-z^s(c))}{dc}<0$ when $z^r>z^s$.

Now consider the difference $W^s-R$ and values of $c$ such that $R<W^s$. In this case rest unemployment implies $W^s=\gamma(b+\beta W^s)+(1-\gamma)\Exp[W^U(z)]$. Note that here $\frac{dW^s}{dc}=0$, since workers with productivities $z \leq z^s$ will never change occupations. Doing so implies paying a cost $c>0$ and randomly drawing a new productivity, while by waiting a worker obtains (with probability $1-\gamma$) a free draw from the productivity distribution. Hence, $d(W^s-R)/dc= -dR/dc$. Since workers with $z>z^s$ prefer employment in their current occupation, the above arguments imply that when $R<W^s$ the expected value of unemployment, $W^U(z)$, is independent of the value the worker obtains from sampling a new $z$ in a different occupation. It then follows that $\frac{dR}{dc}=\frac{d\Exp[W^U(z)]}{dc}-1=-1<0$, which once again yields are desired contradiction. Further note that in this case $\frac{d \Exp_{z} [M(c,z)-W^{U}(c,z) ]}{dc}=0$ and hence (<ref>) implies that $\frac{d z^s(c)}{dc}=0$ and $\frac{d (z^r(c)-z^s(c))}{dc}=0$ when $z^s>z^r$.

Comparative Statics with respect to $b$:
We proceed in the same way as in the previous case. Consider the difference $W^s-R$ such that $R\geq W^s$. Expressing $W^s$ and $W^U$, for $z>z^s$, as

\begin{align} \label{e:l7_7a}
W^s &= (1-\gamma)\Exp[W^U(z)]+\gamma(b+\beta (R-W^s)) + \gamma \beta W^s \\
W^U(z) &=  (1-\gamma)\Exp[W^U(z)]+\gamma(b+\beta (\lambda(\theta)(1-\eta)(M(z)-W^U(z)))) + \gamma \beta W^U(z), \label{e:l7_7b}
\end{align}

we find that $W^s - \Exp[W^U(z)]=\int_{z^r} (W^s-W^U(z))dF(z)$, which in turn implies

\begin{align}\label{e:l7_b_ws-r}
W^s-R=\frac{1}{1-\gamma \beta F(z^r)} \left(-\beta \gamma \int_{z^r} \lambda(\theta(z))(1-\eta)(M(z)-W^U(z))dF(z)+(1-\gamma \beta)c \right).
\end{align}

That is, the difference $W^s-R$ decomposes into (i) the forgone option of searching for a job in the new occupation next period (first term in the brackets) and (ii) a sampling cost that only has to be incurred next period with probability $\gamma$, and discounted at rate $\beta$ (second term in the brackets).

Next consider the relationship between $M(z)-W^U(z)$ and $\Exp[M(z)-W^U(z)]$. From (<ref>) and (<ref>), we find that

\begin{align}\label{e:l7_b_m-wu_e(m-wu)}
\frac{d(M(z)-W^U(z))}{db}=\frac{1-\gamma}{1-\gamma \beta (1-\lambda(\theta(z)))}\frac{d\Exp[M(z)-W^U(z)]}{db} - \frac{\gamma}{1-\gamma \beta (1-\lambda(\theta(z)))}.
\end{align}

Note that $\frac{d(M(z)-W^U(z))}{db}$ must have the same sign for all $z$, which is positive if and only if
\[\frac{d\Exp[M(z)-W^U(z)]}{db} >\frac{\gamma}{1-\gamma}.\]

Towards a contradiction, suppose $d(W^s-R)/db<0$. Then, we have $\frac{d(W^s-R)}{db}=\frac{d(W^s-\Exp[W^U(z)])}{db}$, which equals $\frac{d}{db} \left(-\int_{z^r} \max\{W^U(z)-W^s,0\}dF(z)\right)$. By the envelope condition, the effect $\frac{dz^r}{db}$ disappears. By the previous argument and (<ref>) subtracted by (<ref>), it follows that $\frac{d(W^s-R)}{db}<0$ implies $\frac{d(M(z)-W^U(z))}{db}>0$ and by (<ref>) that $\frac{d\Exp[M(z)-W^U(z)]}{db}>0$.

Using the the same arguments as in (<ref>) we find that

\begin{align}\label{e:7_exp[m-wu]2}
\frac{d\Exp[M(z)-W^U(z)]}{db}&=-1 + \int_{z^r} \frac{\beta(1-\lambda(\theta(z)))-\gamma\beta(1-\lambda(\theta(z)))}{1-\gamma \beta(1-\lambda(\theta(z)))}dF(z)\frac{d\Exp[M(z)-W^U(z)]}{db} \non \\
& -\int_{z^r} \frac{\gamma\beta(1-\lambda(\theta(z)))}{1-\gamma \beta(1-\lambda(\theta(z)))}dF(z)\frac{d\Exp[M(z)-W^U(z)]}{db} \non \\
&  +(F(z^r)-F(z^s))\left(\frac{\gamma\beta^2}{1-\gamma \beta}\frac{d(W^s-R)}{db} +\frac{\beta(1-\gamma)}{1-\gamma \beta} \frac{d\Exp[M(z)-W^U(z)]}{db}\right) \non  \\
& -(F(z^r)-F(z^s))\frac{\gamma \beta}{1-\gamma \beta} +F(z^s)\beta\frac{d(W^s-R)}{db} \\
\Longrightarrow \frac{d\Exp[M(z)-W^U(z)]}{db} & = C_2 \cdot \frac{d(W^s-R)}{db} -C_3<0,  \non
\end{align}

where $C_2$ and $C_3$ are positive constants. The fact that the last expression leads to $\frac{d\Exp[M(z)-W^U(z)]}{db}<0$ yields our desired contradiction. Equations (<ref>) and (<ref>) then imply that $\frac{d (z^r(b)-z^s(b))}{db}<0$ when $z^r>z^s$.

Next we consider the case in which $W^s>R$ and note that in this case equation (<ref>) becomes

\begin{align*}
W^s-\Exp[W^U(z)]=-\frac{\beta \gamma}{1-\beta \gamma} \int_{z^s} \lambda(\theta(z))(1-\eta)(M(z)-W^U(z))dF(z).
\end{align*}

As before, if we start from the premise that $d(W^s-R)/db<0$, this will imply, by virtue of (<ref>), that $\frac{d\Exp[M(z)-W^U(z)]}{db}>0$. Noting that in this case equation (<ref>) reduces to

\begin{align*}
\Exp[M(z)-W^U(z)]=&\int_{z^s} z-b + \beta \lambda(\theta(z))(1-\eta)(M(z)-W^U(z))dF(z),
\end{align*}

and (<ref>) reduces to

\begin{align*}
\frac{d\Exp[M(z)-W^U(z)]}{db}&=-1 + \int_{z^s} \frac{\beta(1-\lambda(\theta(z)))-\gamma\beta(1-\lambda(\theta(z)))}{1-\gamma \beta(1-\lambda(\theta(z)))}dF(z)\frac{d\Exp[M(z)-W^U(z)]}{db} \non \\
& \qquad -\int_{z^s} \frac{\gamma\beta(1-\lambda(\theta(z)))}{1-\gamma \beta(1-\lambda(\theta(z)))}dF(z)\frac{d\Exp[M(z)-W^U(z)]}{db},
\end{align*}

we obtain that $\frac{d\Exp[M(z)-W^U(z)]}{db}<0$, once again yielding our desired contradiction. Equation (<ref>) then imply that $\frac{d (z^r(b)-z^s(b))}{db}<0$ also when $z^s>z^r$.

Comparative Statics with respect to $\gamma$: Here we also proceed in the same way as before by assuming (towards a contradiction) that $d(W^s-R)/d\gamma>0$. We start with the case in which $R\geq W^s$. Using equation (<ref>) we find that

\begin{align*}
\frac{d(W^s-R)}{d\gamma}=&\frac{\beta F(z^r)}{1-\beta \gamma}(W^s-R)-\frac{1}{1-\beta \gamma F(z^r)}\left(\int_{z^r} \lambda(\theta(z))(1-\eta)(M(z)-W^U(z))dF(z)+\beta c\right) \non \\
& \qquad -\int_{z^r}\beta \gamma \lambda(\theta(z))\frac{d(M(z)-W^U(z))}{d\gamma}dF(z).
\end{align*}

It then follows that