## Unemployment and Vacancies with Sectoral Shifts

by
Arthur J. Hosios

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Title:

Unemployment and Vacancies with Sectoral Shifts

Author:

Arthur J. Hosios

Year:

1994

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The American Economic Review

Volume:

84

Issue:

1

Start Page:

124

End Page:

144

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Language:

English

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**Updated:**July 23rd, 2012

Abstract:

### Unemployment and Vacancies with Sectoral Shifts

Recent analyses of unemployment -vacancy series suggest that aggregate shocks, rather than sectoral shocks, are the primary factors responsible for unemploy- ment fluctuations. This inference follows from two widely held beliefs: sectoral shocks induce only positive unemployment -Liacancy comouements, while nega- tive comovements are necessarily the result of aggregate demand shocks. This paper describes an equilibrium matching model that identifies plausible circumstances in which neither assumption is correct, thus suggesting that unemployment -vacancy data are inconclusiue. Interestingly, this model's nor3el results are due to standard features in the contracting literature: firms experience relatice price shocks and negotiate contracts that prescribe temporary layoffs. (JEL E24, 560)

The sectoral-shifts hypothesis asserts that equilibrium rate of unemployment is not changes in the distribution of sectoral em- constant; rather, "[allowing] the variance of ployment demands, rather than aggregate individual market demands to vary over shocks, are the primary factors responsible time. ..leads to an equilibrium unemploy- for aggregate unemployment fluctuations. ment rate that itself varies as the quantity of This hypothesis is based on the following required labor reallocation within the econ- two-part explanation for unemployment omy changes" (David Lilien, 1982 p. 778). fluctuations. First, changes in the underly- Controversy surrounding the sectoraling joint probability distribution of tastes shifts explanation for unemployment flucand technologies change the distribution of tuations begins with Lilien's (1982) claim sectoral shocks so as to induce fluctuations that sectoral shifts were responsible for at in the degree of mismatch between the ac- least half of the cyclical variation in U.S. tual and desired allocations of labor among unemployment through the 1970's. Since sectors and, hence, in the pace of labor then, Katherine G. Abraham and Lawrence reallocation. Second, fluctuations in the F. Katz (1986) and Olivier J. Blanchard and pace of labor reallocation induce fluctua- Peter A. Diamond (1989) have described tions in aggregate unemployment in vacancy data that appear to dispute Lilien's economies where resource reallocation is interpretation of the unemployment data costly (see Fisher Black, 1982; Steven J. and favor the alternative aggregate-demand Davis, 1986, 1987; James D. Hamilton, hypothesis.' This conclusion is based on two 1988).' Thus, according to this view, the widely held beliefs: first, changes in the

distribution of reallocative shocks induce only positive comovements of unemploy

*university of Toronto, Toronto, ON, Canada, M5S 1Al. Helpful comments were provided by Steve Davis at an early stage and by four anonymous referees and seminar participants at various institutions. 2~obert J. Barro (19861, Blanchard and Stanley

'1n the stationary version of this model, where the Fischer (19891, N. Gregory Mankiw (19891, Bennett overall degree of mismatch between the actual and McCallum (19891, and Janet Yellen (1989) concur. desired allocations of resources remains unchanged Some additional sources for evidence concerning the (even though individual sectors' demands fluctuate ran- sectoral-shifts hypothesis, or variants thereof, are domly), the equilibrium level of unemployment is con- Prakash Loungani (19861, Kevin M. Murphy and Robert stant; Robert E. Lucas and Edward Prescott (1974) Topel (19871, Lougani and Richard Rogerson (1989), provide the best-known formal model of this type of Davis and John Haltiwanger (1990, 19921, and S. Lael unemployment. Brainard and David Cutler (1993).

124

ment and vacancies; and second, negative comovements are necessarily the result of aggregate demand shocks.

In this paper I describe an equilibrium matching model of unemployment and va- cancies that identifies plausible circumstances in which changes in the distribution of reallocative shocks induce negatice unemployment-vacancy comovements. In these circumstances, therefore, aggregate unemployment-vacancy data fail to be dis- criminatory, and a more modest interpreta- tion is appropriate. Before describing this model, I begin with a brief review of the relevant empirical work.

I. A Quick Review and Overview

Lilien (1982) argues that an increase in the cross-sectoral dispersion of desired em- ployment growth rates has two key implica- tions: bad times become worse in contract- ing sectors, and the overall quantity of required labor reallocation increases. Hence the cross-sectoral dispersion of desired em- ployment growth rates should be positively correlated with both aggregate layoffs and aggregate unemployment. His empirical confirmation of these conjectures supports the sectoral-shifts hyp~thesis.~

Abraham and Katz (1986) describe a set of empirically plausible conditions in which aggregate demand disturbances also induce countercyclical movements in Lilien's (1982) measure of cross-sectoral dispersion (see also Lawrence Weiss, 1986). They go on to argue that vacancy data should be discrimi- natory. Consider a multisector economy and suppose that in each sector, following Bent

kilien (1982) uses the cross-sectoral dispersion of actual employment growth rates as a proxy for the cross-sectoral dispersion of desired growth rates. He regresses the unemployment rate on current and lagged values of his dispersion measure and on current and lagged values of a monetary disturbance. The coeffi- cient of the current dispersion measure in this equa- tion is positive and significant. Using this regression model, he shows that unemployment over the 1970's is particularly well explained by current and lagged val- ues of the dispersion measure, whereas unemployment in the late 1950's and 1960's is largely accounted for by the unanticipated-monetary-growth terms.

Hansen (1970) and others, excess labor de- mand (supply) induces high (low) vacancies and low (high) unemployment. In this set- ting, an increase in the variance of sectoral demand shocks causes unemployment and vacancies to increase together as some mar- kets experience greater excess demand while others experience greater excess supply. On the other hand, changes in size of the aggre- gate demand shock induce a negative corre- lation, as all markets experience demand changes of the same sign. It then follows that, if sectoral (aggregate demand) shocks in fact account for the positive correlation between Lilien's dispersion measure and unemployment, this dispersion measure should also be positively (negatively) corre- lated with vacancies.

Abraham and Katz show that a help-wanted index (proxying for the job-vacancy rate) is negatively correlated with Lilien's dispersion mea~ure.~

This is interpreted as supporting the aggregate-disturbance hy- pothesis and is taken to be inconsistent with the alternative hypothesis. More recently, Blanchard and Diamond (1989) have exam- ined the same issue using the same identify- ing assumptions, but from a very different per~pective.~

Their basic findings are consis- tent with those of Abraham and Katz. In the short and medium terms, aggregate shocks have by far the largest effects on unemployment and vacancies, while struc- tural shocks have only small effects. In the long term, the effects of aggregate shocks

4~pecifically, they regress their help-wanted index on current and lagged values of Lilien's (1982) disper- sion measure and on current and lagged values of the same monetary disturbances. The coefficient of the current dispersion measure in this equation is negative and significant.

hey estimate the dynamic process characterizing the joint movement of unemployment, vacancies, and the labor force using a vector autoregression. The resulting reduced-form innovations to these VAR's are taken to be linear combinations of the unobservable innovations to the processes generating cyclical, struc- tural/sectoral, and labor-force shocks. To recover the latter innovations from the former, it is assumed that cyclical innovations affect unemployment and vacancies in the opposite direction, while structural innovations affect them in the same direction.

disappear, while the effects of structural shocks persist.

All together, these studies make two im- portant contributions. First, all parties agree that Lilien's (1982) dispersion measure is positively correlated with layoffs and unem- ployment and negatively correlated with va- cancies, and that unemployment and vacan- cies are negatively correlated in the short term and positively correlated in the long terra6 Any formal model of unemployment and vacancies should account for these ob- servations. Second, on the assumptions that sectoral shocks induce positive comovements of unemployment and vacancies and that negative comovements are the result of aggregate demand shocks, the Abraham-Katz and Blanchard-Diamond papers demonstrate that aggregate, rather than sectoral, shocks are the key factors respon- sible for aggregate unemployment fluctua- tions.

This leaves only one question to be resolved: are there any empirically plausible circumstances in which sectoral shocks can induce negative short-term comovements of unemployment and vacancies? The informal and intuitive multisector Hansen-like model sketched above suggests that the answer is negative. As yet, however, no formal model of sectoral shifts has been produced that has testable implications concerning the rel- evant sets of measured variables. The main contribution of this paper is the develop- ment of such a model.

The model in this paper extends earlier research on job matching with Nash bar- gaining to the case in which firms' output prices are random.' Each firm in this setup represents a single job; and each job either is matched with a worker or is searching for an employee. Each matched worker-job pair negotiates a long-term employment con-

6~eorgeJohnson and Richard Layard (1986) pre- sent evidence confirming a negative short-run unemployment-vacancy correlation in most OECD countries.

7~hestandard refetences in the matching literature include Diamond (1981, 1982, 1984), Dale Mortensen (1982), and Christopher A. Pissarides (1985, 1987).

tract. With relative price shocks, this con- tract may prescribe temporary layoffs which allow laid-off employees to return eventu- ally to their original employers. Temporary separations are standard features of optimal contracting models but have been ignored in previous matching models, which assume that all separations are permanent.

A change in the distribution of firm-specific shocks, which is the driving force underlying the sectoral-shifts hypothesis, is modeled here as a change in one or both of the following two parameters between states: (i) the permanent separation rate applied to employed workers and filled jobs; and (ii) the variance of relative output prices among ongoing worker-job matches. This formulation is a departure from earlier work, wherein reallocative shocks have been modeled exclusively by changes in the sepa- ration rate. In this paper I have chosen to highlight the effects of changes in the dis- persion of firms' relative output prices sim- ply because they better capture the conventional demand-dispersion notion of reallocative shocks stressed in the literature.

The main result of the paper is as follows. Greater reallocative shocks cause layoffs and unemployment to increase, but depending on the type of shock, vacancies may either increase or decrease. Specifically, an increase in the separation rate causes contem- poraneous unemployment and vacancies to increase together, whereas an increase in the variance of relative output prices causes unemployment to increase and vacancies to decrease. The latter result is novel and is clearly at odds with the identifying assump- tions imposed in previous empirical work.

The explanation for why price-variance shocks induce negative unemployment-vacancy comovements begins with the ob- servation that a job opening represents a resource commitment, often in the form of physical capital. This implies that the supply of job openings to an expanding sector rep- resents either a reallocation of existing jobs in the economy or the creation of new jobs through entry (or both). In all instances, however, an increase in output-price disper- sion among firms lowers the relative price at contracting firms and thereby increases the

number of workers on temporary layoff. Un- der the assumption that (some) temporarily laid-off workers search for alternative em- ployment, the total number of searching workers is an increasing function of output- price dispersion. In these circumstances, consider the following very different descriptions of aggregate job supply.

Suppose that the aggregate supply of job openings is fixed in the short term. This implies that the number of job openings attracted to a high-demand sector, though positive, cannot increase with that sector's mice draw. In this situation. an increase in price dispersion increases the temporary layoff rate, which increases the searching- worker to searching-job ratio, which makes it harder for individual workers to find jobs but easier for job openings to find workers. With a lower job-finding probability for workers, unemployment increases; and with a higher worker-finding probability for job openings, vacancies decrease.

At the other extreme, suppose that jobs' entry and exit decisions are made after all output prices are realized. In this case, ag- gregate supply is elastic and state-contin- gent in the short run. An increase in price dispersion again increases temporary layoffs but now decreases net job entry and hence decreases aggregate supply. The number of job openings decreases because an increase in the proportion of searching workers who are on temporary layoff increases the proba- bility that a job will be matched with a temporarily laid-off worker, and this decreases the expected profit from entry be- cause temporarily laid-off workers are paid more than unattached workers; that is, tem- porarily laid-off workers can always return to their old employers and therefore have stronger bargaining positions. Now, with more searching workers and fewer job open- ings, it is even more difficult (easier) for a worker (job) to find a trading partner; once again, unemployment and vacancies move in opposite directions.

Thus, for the class of economies studied here, the unemployment-vacancy pattern induced by price-variance shocks does not depend on the short-term elasticity of the aggregate supply of job openings.

The paper proceeds as follows. Sections 11, 111, and IV describe the model, the opti- mal layoff rule, and the market equilibrium, respectively. Section V describes the effects of separation and price-variance shocks on unemployment and vacancies under three alternative hypotheses concerning job creation in the short term. While the plausibil- ity of any one of these models of job creation can be disputed, the key fact is that, as a group, they are comprehensive; and with all three models, the corresponding unemployment-vacancy patterns are the same. The results in question thus seem to be rob~st.~

Section VI confirms that the model can generate short- and long-term comovements of unemployment and vacancies that are broadly consistent with the data; it also shows how the present model's implications for the comovements of gross job creations and destructions can explain the seemingly contradictory findings of Blanchard and Diamond (1989) and Davis and Haltiwanger (1990). Section VI also provides concluding remarks.

11. The Basic Model

This section describes a matching model of unemployment and vacancies. Following Diamond (1982, 1984), Mortensen (1982)) and Pissarides (1985, 1987), I assume that jobs and firms are synonomous, that employment contracts are determined by Nash bargains between matched workers and jobs, and that the aggregate outcome of search is represented by a matching function. I de- part from earlier work by allowing each firm's relative output price to be random. While aggregate demand shocks are absent from this model, the conclusions drawn later concerning the impact of reallocative shocks do not depend on this omission.

'~lsewhere,I have shown that these results depend neither on the length of employment contracts nor on the availability of symmetric information between em- ployers and employees (Hosios, 1991).

A. Preferences, Technologies, and Prices

All agents are neutral to real income risk, are infinitely-lived, and employ the common discount factor 6. Each worker inelastically supplies one unit of labor per period and bears no disutility from either work or job ~earch.~Each firm can employ only one worker, and each worker-job pair together produce one unit of a nondurable output per period. Search costs are zero for work- ers and firms.''

There are 2 possible states in each pe- riod, and in each state there are three possible outcomes at each firm with an es- tablished workforce: either the firm stops operating or, if it continues, its relative out- put price is either high or low. Let p,, and P,! denote the possible relative prices of output at ongoing firms in state s = 1,.. . ,C. These prices satisfy p,, <p,, and p,, + p,,

= 1.

State s occurs with probability p,; and, when s is the current state, a firm draws price P,, (p,,) with probability p,, (p,,), and stops operating with probability p,, = 1-p,, -p,,. The state, each established firm's viability, and all output prices become known at the beginning of period t. It is assumed that p,, > 0, so that there are al- ways some unattached workers searching for employment."

91n the present model, the assumption that labor supply is a zero-one decision precludes work-sharing and introduces a nonconvexity that is necessary for a positive layoff probability. When labor supply is a con- tinuous decision variable, however, additively separable search costs can be introduced to provide the noncon- vexity that may again make a positive layoff probability optimal (see Richard Arnott et al., 1988).

''while agents can forgo search, remain idle, and enjoy an expected present discounted value of lifetime utility equal to zero, it will always be more profitable to undertake costless search. Note that while explicit costs of participating in the matching process described be- low are zero, search remains costly in the sense that, with positive finite numbers of buyers and sellers, the probability of finding a trading partner in any given period will be less than 1.

11Alternatively, a model with workers randomly dy- ing and new workers entering would achieve the same effect.

B. Output

At the beginning of each period, a firm is either matched with a worker and said to have an established work force or is unmatched and referred to as a job opening. To model the product-market side of the transition between these states, I assume that (i) each worker-job pair produces the same type of output for the duration of its match (although different worker-job pairs can produce different types of output) and

(ii) an unmatched job is mobile and not specific to any particular type of output. It then follows that the change in status from being unmatched to matched requires a de- cision concerning which type of output to produce. This decision is made after the state is realized, but it can be made either before or after the worker and job meet.12

Since prices are independently and iden- tically distributed in each period, a newly formed worker-job match always chooses to produce one of the currently high-priced outputs to maximize the joint surplus cre- ated by the match. In each subsequent pe- riod of that match, however, the output price is passively drawn from the price dis- tribution corresponding to the initial output choice. The distribution described earlier is common to all types of output.

C. The Matching Process

At the beginning of period t, a random fraction of firms with established work forces stop operating, and the corresponding worker-job pairs permanently separate. In effect, firms that stop operating permanently lay off all of their employees. These firms exit. Output prices are then realized among the remaining ongoing establishments. As shown later below, firms with

12The matching process introduced below has the property that the probability that a job will find a worker depends only on the numbers of searching jobs and workers and does not depend on an output type or price. Hence it is inconsequential whether, in any pe- riod, a job opening chooses an output before or after it finds an employee in that period.

,,,,and N ,,,,Nwhere

high prices retain their employees, while those with low prices may temporarily lay off some workers. Unattached workers and temporarily laid-off workers then search for alternative employment.

Workers.-At the end of period t -1, there are NIP' unattached workers and L -Nt ' workers with employment con

tracts. The total number of workers, L, remains constant. Since N' and all other endogenous variables are functions of time, I suppress the time-period superscript for convenience.

All unattached workers and temporarily laid-off workers search for jobs.13 Hence the number of searching workers in period t, S,, is given by

denote, respectively, the number of permanently and temporarily laid-off workers in period t.14 Let 45 denote the probability that a searching worker finds a job in period t. Since matched workers negotiate new employment contracts and commence production, while unmatched workers remain unemployed, the number of unemployed workers in period t equals

13 To allow for the possibility that some workers choose to forgo search and remain unemployed, I need only assume that each worker's search cost is uncertain prior to participating in the market; ex post, workers who draw relatively high search costs will prefer to remain unattached. As long as the number of temporarily laid-off workers who choose to search is an increasing function of the total number of temporarily laid-off workers, all of the results in this paper remain valid.

14 I

have not yet discussed quits. A quit and a permanent layoff have identical contractual implica- tions (i.e., both permanently sever the relationship between the worker and firm). Since a temporary layoff leaves a worker with the option to return to her origi- nal employer, quits and temporary layoffs are distinct. To simplify, I model quits as synonymous with contract rejections. In this situation, workers will never quit, as contract negotiations always leave the worker with a nonnegative surplus. There is no on-the-job search in this matching model.

At the end of period t, after output is produced, the (1-+)NtemPworkers on tem- porary layoff who failed to find alternative employment return to their initial employ- ers. Finally, the number of unattached workers at the end of period t, N, equals the number of unattached workers who searched unsuccessfully; that is,

Jobs.-At the end of period t -1, there are J-' job openings. Since each firm-cum- job can employ only one worker, there are also L -NP1 firms with established work forces at the end of period t -1. Each firm that is permanently separated from its work force in period t immediately exits. Initially, I adopt the steady-state characterization of entry employed by Ariel Rubinstein and Ashcr Wolinsky (1985) and likewise assume that each job that exits induces a new job opening to enter. In this case, the total number of jobs remains constant.15 Let K denote the total number of jobs, so that K = L -N-' + J-'. Alternative models of job entry and exit are considered later, in Section V.

The number of searching jobs in period t,

equals the number of job openings carried over from period t -1 plus the number of new job openings; the latter number is sim- ply the number of newly separated jobs. Let $ denote the probability that an empty job finds an employee. Since matched jobs ne- gotiate new employment contracts and com-

15 In effect, the model of entry introduced here is one in which expected entry profits are independent of the current state of the economy and are instead dominated by long-term (steady-state) considerations. The advantage of this model of entry is not its realism, but that it allows one to highlight the processes gener- ating unemployment and vacancies without being side- tracked at this stage by more complex entry issues.

mence production, while empty jobs remain vacant, the number of vacancies in period t equals

Discussion of the definition of a vacancy in (lb) is postponed to the end of this subsec- tion.

The number of firms that temporarily lay off their employees and remain idle during pcriod t equals N,,,,.It follows that the total number of job openings at the end of period t, J. equals the number of job open- ings that searched unsuccessfully in period t plus the number of firms whose workers chose to accept alternative employment,

16

J = v + +N,,,,.

Matching.-The outcome of search in each period is represented by a matching function. It is assumed that S, workers searching for jobs and Sf firms searching for workers together generate M(S,, Sf) matched worker-firm pairs in period t. This matching function exhibits constant returns to scale with respect to both arguments, requires strictly positive numbers of search- ing workers and firms to generate matches, and satisfies MI > 0, MI, < 0, M, > 0, and M,, <0." The probability that a worker will find a job in period t is an increasing func- tion of the number of jobs per worker in period t,

while the probability that a job will find a

"~~~endix

A shows that temporarily laid-off work- ers who find jobs choose to not return to their original employers.

17Constant returns to scale precludes multiple equi- libria and is a necessary condition (at least locally) for a constrained efficient allocation of resources (Hosios, 1990); none of the main results relies on this simplify- ing assumption. Blanchard and Diamond (1989) esti- mate a matching function for the U.S. manufacturing sector that suggests constant or mildly increasing re- turns to scale.

worker is a decreasing function of the same ratio,

It is further assumed that 4(~)= +(0) = 1 and 4(0) = +(m) = 0.

Remark: An uncmployed worker and a va- cant job are defined symmetrically in (1) as agents who are unable to find trading part- ners. Thus, job openings are an input to the matching process at the beginning of the period, while vacancies are an outcome of that process at the end. Alternatively, one could define a vacancy as a searching job opening, so that vacancies (measured by Sf) then become an input into the matching function at the beginning of the period. From a theoretical perspective, this distinc- tion is not critical, as my main results remain valid under either definition of a vacancy. To address the data, however, the definition in (lb) is prefcrred.

Recognizing that matching is actually a continuous rather than a discrete process, it follows that for any given volume of job openings at the beginning of a period, the number of unmatched jobs will decline con- tinuously over that period at a rate that depends on the matching process. The ap- propriate vacancy measure is actually the aL1erage number of unmatched jobs over the period, rather than the initial or final num- bers, Sf or V in (lb). Since the matching function is an analytical device for summa- rizing the outcome of this type of complex, dynamic search process, the avcrage num- ber of unmatched jobs is not explicitly mod- eled here, and a proxy is requircd.

While Sf, V, and the average number of unattached jobs are positively correlated with each other, V is the prefcrred proxy for the average number of unattached jobs. This is because these three measures evolve sequentially. That is, an event that influ- ences Sf will have the same effects on the average number and on V, whereas an event that influences the average number will have

111. Employment Contracts

An unattached worker and a job opening that meet in period t and state s negotiate a lifetime employment contract.'' This con- tract specifies the current real wage, w:, and a sequence of future state-contingent real wages and layoff rates, {(W;~,I;~)IU=

is the

1,.,. ,C; j = t,h}:=,+,, where I;, probability of being temporarily laid off in period r when the firm's period-r output- price draw is p,, and w;, is the correspond- ing wage paid to retained workers. Unlike permanently separated workers, workers temporarily laid off at the beginning of pe- riod r (> t) can return to their old employer at the end of that period.19 A worker on temporary layoff who finds an alternative job will likewise negotiate a new lifetime employment contract. The assumption that a worker and firm negotiate a single lifetime employment con- tract when they first meet simplifies the exposition but is not essential. All of the paper's results hold when they instead ne- gotiate a sequence of finite-length contracts (Hosios, 1991); in fact, these results hold even when contract length varies cross-sectionally and over time. The optimal employment contract between a worker and a job is given by the solution to the following Nash bargaining problem: rnax[~]*[FI~-*,

where W and F respectively denote the worker's and the firm's ex~ected net surdus from the contract, and a is a satisfying o < a

< 1 which the negotiating parties take as given. This simple characterization of con- tract negotiation applies when a worker and

''A time superscript is needed in this section to distinguish the contract parameters prescribed for the current and every future period of a contract.

'ks the contractual availability of severance or lay- off pay to either permanently or temporarily laid-off workers has no effect on the results below, such trans- fers are simply ruled out from the start.

job first meet, independent of whether the worker is unattached or temporarily laid off.

The following feature of this contracting problem is unique and noteworthy. When a worker and job first meet in period t to negotiate a contract, they choose wages for periods r 2t taking full account of the an- ticipated effect of these wages on the worker's subsequent bargaining position elsewhere. That is, if this worker is tem- porarily laid off and finds a new job in a future period t + k (k 2l), the expected present discounted value of the wages pre- scribed for periods t + k + 1, t + k + 2,. . . by the worker's original contract then repre- sents her threat point in negotiations with the new job. Because of this external effect of contract wages on subsequent bargaining outcomes between the worker on temporary layoff and some alternative employer, the wages negotiated in period t affect both the size and division of the initial surplus cre- ated by the match; without temporary lay- offs, this external effect is absent, and wages play only a distributive role.

The solution to the worker-job bargain- ing problem is described in Appendix A, where the following two results are estab- lished: First, it is always more profitable to hire an unattached worker than a temporar- ily laid-off worker, because the latter worker's reservation utility is higher; that is, following the rejection of a contract offer, a temporarily laid-off worker can always re- turn to her original employer, whereas an unattached worker must continue to search.

Second, independent of a worker's employment history, the optimal layoff probability at high-price firms is zero, while the optimal probability at low-price firms in period r in state s, denoted by I,', is as follows:

where 4; is the probability that a worker will find a job in period r in state s. In words, layoffs are individually efficient whenever the expected value of the worker's marginal product from participating in the matching process is no less than her current value of marginal product.

The critical property of (3) is the follow- ing: taking the job-finding rate, 4:, as given, an increase in price dispersion (whereby pSh rises and p,, falls) increases the layoff rate. The next section shows that this positive relationship between price dispersion and temporary layoffs at individual firms explains why changes in price dispersion in- duce a negative correlation between unem- ployment and vacancies in the aggregate. It is important to recognize, however, that the relationship between price dispersion and temporary layoffs prescribed by (3) will arise in most other contractual settings as well. Indeed, it is difficult to conceive of a con- tracting model that generates temporary layoffs and does not also predict that the layoff rate is an increasing (decreasing) function of workers' alternative (own) value of marginal product. The latter properties are sufficient to generate the negative unemployment-vacancy correlation of interest.

IV. Equilibrium

An equilibrium consists of a sequence of job-finding probabilities and contractual lay- off rates satisfying (2) and (3). The details follow.

Recall that N-' and J-' denote, respec- tively, the numbers of unattached workers and empty jobs at the end of period t -1. Taking N-' as given, the number of search- ing workers in period t and state s, S,,, is an increasing function of l,, the layoff rate in t:

where p,,( L -N- ') and psplS(L-N-') equal, respectively, the number of perma- nently and temporarily laid-off workers in period t. Taking J-' as given, the number of searching jobs in period t and state s, Sf,, is an increasing function of the number of newly separated matches which, in turn, equals the number of permanently laid-off

workers in period t; that is,

Therefore, the equilibrium layoff proba- bility in period t and state s is determined, from (3) given {N1, J'), as follows:

Otherwise, 1, is given by the solution to

Expressions (2) and (4) imply that the equi- librium layoff probability is a nondecreasing function of the number of searching jobs, Sf,. That is, since 4'> 0, increasing the number of job openings increases the prob- ability that a worker will find a job (in any state) which, in turn, increases the return to participating in the ex post labor market. Since t,b(x), the probability that a job will find a worker when the job-to-worker ratio is x, satisfies xt,b(x) = +(XI, (4) also deter- mines the probability that a job will find a worker in equilibrium.

V. Unemployment and Vacancies

This section identifies circumstances in which the model generates a negative corre- lation between unemployment and vacancies and then compares them with those in which a positive correlation results. In either case, the basic process generating unemployment and vacancies is the same. A quick review follows.

The state realization in period t determines the fraction, p,,, of established worker-firm pairs that break up. Starting with stocks of J-' job openings and N-' unattached workers (and hence L -N-' established firms), the number of searching jobs in period t equals J-+ p,,( L -N-'). Temporary layoff decisions are then made at ongoing firms which draw the low output price. The number of searching workers in period t is equal to the sum of last period's unattached workers plus this period's permanently and temporarily laid-off workers, N' + p,,( L -N-')+ pSel,(L-N-'1.

The matching process determines the numbers of searching workers and search- ing jobs which find trading partners and those which remain idle. The latter groups of workers and jobs respectively comprise the unemployed workers and vacant jobs in period t. From (I),

where 1 -4, is the probability of not find- ing a job in period t and state s, and 1-+, is the corresponding probability of not find- ing a worker.

The corner solution where all workers are temporarily laid off is theoretically uninter- esting and is therefore ignored.*(' The alter- native corner solution where all workers are retained is empirically uninteresting and is also ignored; temporary layoffs account for approximately one-third of those unemployed (Murphy and Topel, 19871, and as indicated in the Introduction, temporary layoffs are essential for this paper's distin- guishing results. Attention is restricted to situations in which the equilibrium layoff probability satisfies 0 < I, < 1 and is deter- mined, from (4~1, by

where

In words, when the job-finding probability exceeds the price ratio p,, /p,,, the num-

20 For example, with a neoclassical production tech- nology, it is never optimal for a firm to lay off its entire work force, because, as the number of retained work- ers goes to zero, the marginal product of labor will eventually strictly exceed workers' outside option.

ber of temporarily laid-off workers increases, thereby decreasing the searching job-to-worker ratio, which in turn decreases the probability of finding a job; conversely, whenever the job-finding probability is less than p,! /p,,, the number of temporarily laid-off workers decreases, and the job-finding probability increases.

The key implications of (6) are: (i) the searching job-to-worker ratio in state s, P,, is an increasing function only of the price ratio p,, /p,,; (ii) the job-finding probabil- ity, 9, = $(P,), is an increasing function of PSe /psh, while the worker-finding probabil- ity, $, = 4, /P ,!.. is a decreasing function of PSe /psh; and (HI) the ratio of unemployed workers to vacant firms,

is a decreasing function of pSe/psh.

Finally, to facilitate disentangling the ef- fects of different reallocative shocks, it is assumed that the fraction of ongoing estab- lishments that draw price p,, in state s is constant, that is, pSp /(ps, + psh) is fixed for all s. Since p,,+p,, =1 and p,,>p,,, the price variance among ongoing establishments is then simply an increasing function of the price ratio p,, /p,,. I adopt us = p,, /pSt as the measure of demand disper- sion among firms. Each state is now charac- terized completely by two parameters, us and p,,, describing the level of demand dispersion and the permanent separation rate.

A. The Basic Model

This subsection examines the basic model in which the number of jobs is effectively fixed because separated jobs that exit in- duce entry on a one-for-one basis. It will be shown that an increase in price dispersion among firms increases layoffs and unemployment and decreases vacancies, while a higher separation rate increases layoffs, un- employment, and vacancies.

The present model implies that unemployment and vacancies in period t can be written as functions of the contemporary realizations of the reallocative shocks and the number of inherited job openings:

The derivatives of F and G are signed as follows. Suppose that the separation rate is fixed, p,, = p,. The number of searching jobs in period t then equals Sf= J-' + p,(K -.I-').

Thus, taking J-' and hence Sf as given, implications (ij-(iij of (6) imply that increasing demand dispersion in period t increases both the number of layoffs and the number of searching workers in period t and hence increases the searching-worker- to-job ratio. In turn, increasing the latter ratio decreases a worker's probability of finding a job and increases a job's probabil- ity of finding a worker. This gives F, > 0 and G, < 0.

Suppose that the price variance is fixed, a, = a. From implications (ij-(iii) of (6), note that, for a given relative output-price ratio, the searching job-to-worker ratio is fixed, as are the job- and worker-finding probabilities and the unemployment-vacancy ratio. Thus, fixing us= p,,, /pse implies that the num- ber of vacant jobs is directly proportional to the number of searching jobs and that the number of unemployed workers is directly proportional to the number of searching workers, which in turn is directly propor- tional to the number of searching jobs. Hence the derivatives of F and G with respect to p,, and J-' have the same sign. In particular, since an increase in J-' increases the number of searching jobs, and since a greater separation rate increases the number of permanent separations and hence increases the number of searching jobs, Fi> 0 and Gi > 0, i = 2,3.

B. Ex Ante Job Entry

So far, job entry that is responsive to the current state of the economy has been pre- cluded. This subsection and the next relax this assumption, endogenize the total num- ber of jobs, and then derive the counter- parts to (7). It is thereby shown that the main results do not depend on the degree of job mobility in the economy. In this subsec- tion I assume that firms' voluntary entry-exit decisions are made, ex ante, before separa- tion shocks and prices are realized; the next

subsection allows ex post entry and exit.

Exit and entry are modeled as follows. At the end of period t -1, old job openings decide either to exit or to participate in next period's matching process. Concurrently, new job openings can commit to enter and participate in t's matching process. Entry and exit decisions are thus made under un- certainty, prior to the realization of the state. Nevertheless, as before, job openings that enter and establish a work force will produce output with current price p,,.

The expected present discounted value of lifetime profits available elsewhere in the economy is assumed to be constant. Also, there is a fixed cost per job per period of participating in the matching process.

Letting AK denote the net addition to the stock of job openings at the end of period t -1,the total number of job open- ings available prior to the realization of the state in period t is J-' + AK, where A K satisfies w 2 A K 2 -J-l. Finally, it is assumed that an established firm that experiences a separation necessarily exits, as before, but that these exits do not induce entry and so have no effect on the current number of searching jobs.21 Hence the number of searching jobs in period t also equals J-' + A K.

Appendix B shows that, in equilibrium, the number of searching jobs in period t is an increasing function of the number of

"1f, in addition to ex ante entry decisions, each newly separated job that exits induces a job opening to enter (as in the previous subsection), the main results here would be identical to those in Subsection V-A; that is, the number of permanent layoffs would be added to the right-hand side of (81, and (9) would then be identical to (7) in terms of the effects of reallocative shocks.

unattached workers at the end of period t -1:

Recalling that it is always more profitable to hire an unattached worker than a temporar- ily laid-off worker, the intuition underlying

(8) is clear: increasing the number of unattached workers at the end of period t -1 increases the probability of finding an unattached worker among those searching for jobs in period t which, in turn, increases the expected profit from participating in period t's matching process.

The number of unemployed workers in period t is again described by (5a). The number of vacancies in period t is now given by

Focusing again on interior solutions, the equilibrium job-to-worker ratio is given by

and satisfies (6).

With ex ante entry, unemployment and vacancies in period t can be written as functions of the current realization of the dispersion measure and N-':

Taking N-' as given, and hence taking the number of searching jobs as given, an increase in price dispersion increases layoffs and the ratio of searching workers to jobs, which raises the worker-finding probability and lowers the job-finding probability, yield- ing F; > 0 and GT < 0. Taking us as given fixes the worker per job ratio as well as the worker- and job-finding probabilities, so that increasing Npl increases the numbers of searching firms and workers, which increases vacancies and unemployment. Hence, Ft , GT > 0.

Observe that U and V do not depend on p,,. This is because the separation rate in any period, which determines the number of permanently laid-off workers in that period, has no effect on the number of searching jobs [see (8)]. Since p, is fixed given us, the decrease in the number of temporarily laid- off workers exactly equals the increase in the number of permanently laid-off workers, and so the numbers of searching workers and jobs are unchanged.

C. Ex Post Entry

Suppose that firms' entry-exit decisions are made after the state is known and that, as in Subsection V-B above, newly sepa- rated jobs exit and do not influence entry directly. In this case, Appendix C shows that, in equilibrium, the number of search- ing jobs in period t and state s equals

The intuition underlying (10) is straightfor- ward.

Given p,, and N-', greater price disper- sion increases the temporary layoff rate, making it relatively more difficult for jobs to find unattached workers in the pool of searching workers. Since it is always more profitable to hire an unattached worker than a temporarily laid-off worker (the latter have better alternatives), a job opening's expected profit from participating in the matching process decreases, job entry is dis- couraged, and so J-+ A K decreases. On the other hand, given p,, /pst, increasing the separation rate or the number of unattached workers from the previous pe- riod increases expected profits, by making it relatively easier to find an unattached worker, and thereby encourages entry.

It follows from +(/3,) = Pse/psh and (10) that increasing price dispersion increases (decreases) the number of searching work- ers (jobs). With more workers searching for employment among fewer jobs, the proba- bility of finding a job (worker) decreases (increases), and hence unemployment increases and vacancies decrease. Also, by (lo), a higher separation rate or larger N-' value increases the number of searching jobs, which increases the number of search- ing workers in the same proportion (as the ratio of searching jobs to workers and their matching probabilities are all hed given us>.

Therefore, with ex post entry,

(lla) U= F**(U,,~~~,

##### N-')

(llb) V= G**(u,, p,,, N-I)

VI. Discussion

In the Introduction, I identified certain observations which a model of unemploy- ment and vacancies should be able to explain. Especially noteworthy are the short- term negative correlation and positive long-term correlation between unemployment and vacancies. I now verify that the model in this paper can account for these

observation^.^^ Recall that (7), (9), and (11) indicate that an increase in price dispersion increases

"1n addition to those features of the data already described in the text, the model explains (or is at least consistent with) the following: layoffs and unemploy- ment move together (Lilien, 1982); the rate of rehiring of laid-off workers by their original employers is coun- tercyclical (Lilien, 1980); and the fraction of unemployed workers who are temporarily laid off can be relatively small, with most of the remaining fraction being permanently unattached (Murphy and Topel, 1987).

unemployment and decreases vacancies; that

(7) and (11) indicate that an increase in the separation rate increases both unemployment and vacancies; and that (9) and (11) indicate that an increase in the number of unattached workers also increases both. Since the number of unattached workers in any period is an increasing function of the labor-force size, (9) and (11) also indicate that an increase in the labor force will in- crease both unemployment and vacancies.

Now, in all versions of the model, the key simplifying assumption concerning the prob- ability distribution over states is that output prices at individual firms are independently distributed over time. Neither prices nor separation rates need be identically distributed over time. Moreover, the paper's results all go through when either the sepa- ration rate is autocorrelated or the labor- force size changes over time.

It follows that, if the distribution of possi- ble output-price distributions is stationary, while either the separation rate is positively autocorrelated or the labor force is growing over time, the model will indeed generate a short-term negative correlation and positive long-term correlation between unemployment and vacancies. This explanation is not too different from the conventional one (transitory aggregate demand shocks and permanent structural shocks, respectively) because, as established here, price-variance shocks and aggregate demand shocks have similar implications for unemployment and vacancies.

A. Job Creation and Destruction

In this subsection I show that the model's implications for the comovements of gross job creations and destructions provide a simple explanation for the seemingly con- tradictory findings of Blanchard and Diamond (1989) and Davis and Haltiwanger (1990).

Davis and Haltiwanger (1990, 1992) intro- duce the following definitions: gross job cre- ation is the sum of employment growth at expanding and new establishments; gross job destruction is the sum of employment losses at shrinking or dying establishments; and gross job reallocation is the sum of gross job creation and destruction.

In their first (1990) paper, Davis and Haltiwanger adapt the methodology of Blanchard and Diamond (1989) to characterize empirically the responses of job creation and destruction to aggregate and reallocative innovations. Davis and Haltiwanger estimate the joint dynamics of job creation and destruction and use a set of identifying assumptions (analogous to those employed in Blanchard and Diamond [1989]) to recover the innovations to the underlying aggregate and reallocative shocks. In particular, they assume that an adverse aggregate shock increases job destruction and decreases job creation, while a greater reallocative shock increases job de- struction and (eventually) increases job cre- ation.

Davis and Haltiwanger (1990) find that reallocative shocks account for about half of the time-series variation in job creation and destruction. By contrast, Blanchard and Diamond (1989) found that they explain al- most none of the short- and medium-term variation in either unemployment or vacan- cies. One explanation for these seemingly contradictory findings is to attribute them to general differences in the researchers' data sets and specific differences in the potential responses of the relevant stock variables (i.e., unemployment, vacancies) and flow variables (i.e., job creation, job destruction) to reallocative and aggregate shocks (Abraham, 1990).

Alternatively, one can argue that the Davis-Haltiwanger and Blanchard-Diamond findings are in fact compatible to the extent that reallocative shocks induce both nega- tive comovements of unemployment and va- cancies and positive comovements of job creations and destructions. To complete this argument, I now examine the pattern of job creation and destruction in the model.

According to the Davis and Haltiwanger (1990, 1992) definitions, job destruction in any period equals the number of perma- nently and temporarily laid-off workers in that period, N,,,, plus N,,,,, which de- scribes employment losses at dying and

(temporarily) shrinking establishments. Job

creation equals the number of worker-firm

matches in period t, M(S,, Sf), which de-

scribes employment growth at new estab

lishments, plus any recalls, (1-qb)N,,,,, where qb is the job-finding probability. The model in this paper therefore implies that job destruction should respond immediately

to a change in the state, whereas job cre

ation should respond with a slight lag.

As shown below, a change in the disper- sion of relative output prices will generally cause (i) job creations and job destructions to move together and (ii) job reallocations and aggregate unemployment to move together. Both of these results are consistent with findings reported by Davis and Haltiwanger (1990, 1992).

For those versions of the model in which the number of searching jobs in period t is determined prior to the realization of the state in period t, as summarized by (7) or (9), properties (i) and (ii) follow immediately. An increase in price dispersion increases temporary layoffs and hence in- creases both job destructions and the num- ber of searching workers. Taking as given the number of searching jobs, it follows that the number of matches increases while the job-finding rate decreases; with more tem- porary layoffs and a lower job-finding rate, the number of recalls increases as well.23 Job creation therebv increases.

For the version with ex post entry, as summarized by (111, there is some ambigu- ity. Greater price dispersion increases lay- offs, which increases job destructions but decreases the number of searching jobs. With a greater number of searching workers but fewer searching jobs, the number of matches may either increase or decrease,

23~nincrease in price dispersion increases N,,,,, and hence increases N,,,, + N,,,, as well as S,. Given Sf, M(S,, Sf) increases and 4(Sf, S,) decreases as S, increases; hence (1 -4)Nt,,, increases. Since job de- structions and creations both increase, the sum (job reallocations) also increases. Of course, an increase in price dispersion increases unemployment.

even though recalls again increase. Hence there should be some parameterizations of the model with ex post entry such that greater price dispersion increases job creations.

Since a change in the dispersion of rela- tive output prices also causes unemployment and vacancies to move in opposite directions, it follows that this type of real- locative shock may be responsible for some nontrivial fraction of the short-term comovements both of unemployment and va- cancies and of job creations and destruc- tions. Estimating this fraction should be an interesting area for future empirical research.

B. Concluding Remarks

This paper describes an equilibrium matching model of unemployment and va- cancies in which firms experience both tem- porary relative price shocks and permanent separations. Of special interest are the re- sults which show that changes in the vari- ance of the distribution of firm-specific rela- tive price shocks cause unemployment and vacancies to move in opposite directions. These results imply that aggregate unem- ployment-vacancy data, in isolation, are un- able to resolve the question of whether sec- toral or aggregate shocks are the primary factors responsible for aggregate unemploy- ment fluctuations.

An interesting by-product of the present paper is that it is now possible to identify the set of assumptions that is needed to formalize the multisector Hansen-like model which underlies the maintained hypotheses of Abraham and Katz (1986) and Blanchard and Diamond (1989). If, in the event of a price-variance shock, temporarily laid-off workers do not search (or alternatively, job openings can avoid these workers altogether), and if entry and exit decisions are made after the realization of the current state (so that the supply of job openings to expanding sectors is elastic in the short term), then an increase in price variance can indeed increase unemployment and va- cancies together. .In these circumstances, temporarily laid-off workers have no effect on the expected return to job entry, and so the number of job openings in high-demand sectors is an increasing function of their current price realizations.

To determine whether or not aggregate unemployment-vacancy data can be used to test the sectoral-shifts hypothesis, one thus first needs to answer the following ques- tions: Do some temporarily laid-off workers search for alternative employment? More specifically, does the total number of searching workers increase with the total volume of temporary layoffs'? Of course, positive answers to these questions imply that firms with job openings do not avoid meeting and hiring workers on temporary layoff (even though these workers are more expensive to hire because they can credibly threaten to return to their original employ- er~).~~ answers

More interestingly, positive to these questions also imply that this pa- per's principal claim, that sectoral shifts of employment demand generate negative unemployment-vacancy comovements, does not hinge upon the short-term elasticity of the aggregate supply of job openings to ex- panding sectors.

The Hansen model sketched above and the earlier analysis in Subsection V-C both assume that the supply of job openings dur- ing the ex post period is perfectly elastic at some fixed entry fee. Contrary to the Hansen model, however, I have argued that demand-dispersion shocks induce negative unemployment-vacancy comovements because greater price variance increases the number of searching workers on temporary layoff from low-demand sectors, which in turn decreases the number of job openings available to high-demand sectors. Tempo- rary layoffs have a negative effect on entry because temporarily laid-off workers have stronger bargaining positions than unattached workers, are less profitable to hire, and so reduce the expected return to entry.

24~fter all, if firms with job openings refuse to negotiate with temporarily laid-off workers, these workers would have no incentive to search.

The possibility of a negative impact of temporary layoffs on job openings is certainly novel. However, the question of whether or not this externality is operative and empirically important presupposes that the aggregate supply of job openings is in fact highly elastic. As a practical matter, it seems more likely that aggregate job supply will be relatively inelastic in the short term. Once one accepts the view that a job open- ing involves a commitment of physical capi- tal, it is to be expected that the excess job capacity of firms within high-demand indus- tries will be hed in the short term, as will those resources that can be reallocated from elsewhere. In effect, an inelastic short-run aggregate job-supply function is only a spe- cial case of the more general and accepted proposition that the stock of capital is rela- tively fixed and immobile in the short run. In these circumstances, it is not surprising that a demand-dispersion shock that increases the number of searching workers will increase unemployment and decrease vacancies.

An unattached worker and a job opening that meet in period t and state snegotiate a lifetime employment contract. This con- tract specifies the current real wage, o:, and a sequence of future state-contingent real wages and layoff rates, {(w;~, l:j)la = 1,.. . ,2; j = 1,h}F=,+ l:, is the probability of being temporarily laid off in period T when the firm's current output-price draw is p,,, and w;j is the corresponding wage paid to retained workers. This appendix deter- mines the layoff rate prescribed by the opti- mal lifetime employment contract.

To start, let E(uT) and E(7rT) denote, respectively, the expected present discounted value (EPDV) of lifetime wages and profits in period T of a matched worker and firm. Let E(vT) and E(zT) denote, re- spectively, the EPDV of wages and profits in period T of an unattached worker and an empty job. Let E(aT)= max{E(uT+ rTj} denote the maximum joint surplus created by a match in period T;E(QT) is indepen- dent of the personal histories of the worker and job that form the match. All expecta- tions are conditioned on information avail- able when the contract in question is being negotiated.

The negotiated first-period wage follow- ing a match in period t, o:, between an unattached worker and job opening that will jointly produce output with current price p,,, satisfies2'

(Al) w4 + 6E(u1+')

That is, the EPDV of wages from the cur- rent and expected future sequence of em- ployment contracts, o: + SE(U'+'), equals the payoff from rejecting the contract (and match) this period and searching again next period, SE(vt+'), plus a fraction a of the maximum net joint surplus created by the match.

Similarly, a worker who is temporarily laid off in period T,prior to the end of her current contract, and finds a job will negoti- ate a first-period wage at this alternative job that satisfies

(A "A" is used to distinguish variables de- scribing the alternative employment oppor- tunities of laid-off workers.) Notice that the payoff from rejecting the contract and remaining unemployed in period T, SE(uT+'),

25~hefirst-period wage is chosen to maximize [W]"[FI1-", where W equals wf+ 8E(ur+')8E(ut+'), F equals p,,, -w: + SE(T'+')-8E(z1+'11, and E(@'+')equals max(~(u'+')+ E(T'")).

equals the EPDV of wages from returning to her original employer at the end of period T and continuing with the current contract. It follows that when the latter contract is initially negotiated, the effects of its wages on the worker's future bargaining positions with other employers will be fully internalized.

To determine the optimal layoff rates, I need to describe E(u7) and E(.rrT). Let E(u;,) and E(T;) denote, respectively, the EPDV of lifetime wages and profits in pe- riod T of a matched worker and firm when the firm's price draw is psi. Let E(vJ) and E(z,~) denote, respectively, the EPDV of wages and profits in period T and state s of an unattached worker and an empty job. Hence,

Given price draw p,, in period T, a worker under contract is retained with probability 1-1,; and temporarily laid off with probability I;; 4,' is the probability that a temporarily laid-off worker finds a job, and 1-4,' is the probability that the worker remains unemployed. Assuming, for the moment, that laid-off workers who find jobs will opt to stay with their new employers, then

For each price draw p,,, I,', is chosen to maximize the joint surplus E(u;,) + E(rS5).

Let E(cPS;) = ma~{E(u:~)+ E(T;)}, so that the maximum joint surplus in (Al)-(A2), E(@'), equals C,\P,\[ P,~~E(@,;~)

+ pShE(@,',)+ ps,E(u; + z:)]. From (A3),

Now, for any given pattern of current and future layoff rates, (A4) implies that wages will be set to maximize

and hence to maximize 9:+ SE(i:+').

The wages paid in period T + 1 and all subsequent periods of the contract will de- termine E(uT+').From (A2), &a + GE(Li:+') is an increasing function of E(uT+') on E(uT+')2 psh+ 6E(Q7+l)-SE(zT+l); if E(u7+') > pSh+ SE(@'+') -6E(z7+'), the net joint surplus from the match is negative, and so the match is immediately broken. Therefore, the wages paid in period T + 1 and all subsequent periods of the contract will be set so that E(uTf'1 = p,, + 6E(cPT+')-SE(zTf'1. Substituting this ex- pression into (A2) gives

which confirms that laid-off workers who find jobs will opt to stay with their new employers. Equation (A4) can now be writ- ten as

Therefore, independent of a worker's employment history, the optimal layoff prob- ability at high-price firms is zero, while the optimal probability at low-price firms in period r in state s equals

Suppose that at the end of period t -1, prior to the realization of the state in period t, old job openings decide either to exit or to participate in next period's matching pro- cess, and new job openings commit to enter and participate in the matching process in period t. The EPDV of lifetime profits per job available elsewhere equals O. Without loss of generality, I set O = 0. There is a fixed cost per job per period, c, of partici- pating in the matching process; hence the expected net return from searching for a worker in period t is E(zr)- c. Let E(Qr)

= max[E(zt) -c, 01; E(Qr) is a job opening's EPDV of profits in period t.

Letting AK' denote the net addition to the stock of job openings at the end of period t -1, the total number of job open- ings available prior to the realization of the state in period t is Jf-' + A Kt, where A Kt satisfies 2 AKt 2 -Jf-'. It is assumed that all established firms that separate nec- essarily exit, and so the number of searching jobs in period t equals S: = J'-'+ A K '.

Recall that $4 denotes the probability that a job will find a searching worker. Let yf denote the fraction of unattached work- ers among those who are searching for em- ployment in period t in state s, so that a job finds an unattached worker with probability $iy,'. Since the expected net surplus from hiring a temporarily laid-off workers is zero (see Appendix A), a job's expected profit from participating in the matching process in period t in state s is

where 6E(Qri') represents the job's threat point in negotiations with an unattached worker.

From the perspective of period t -1, searching for a worker will be profitable in period t if and only if

To establish an equilibrium with entry, 1 assume that firms in all future periods are indifferent between entering and exiting and then show that equilibrium in the current period entails indifference as well.

Suppose that E(zT) = c for all T 2 t + 1, so that E(QT) = 0 for all r 2 t + 1. Substi- tuting E(Q'+ '1 = 0 into the expression for E(zf) gives E(zsf) = $;y,'P,', where P,' denotes the maximum EPDV of profits from hiring an unattached worker in period t in state s; that is,

Equation (Bl) then implies that entry will occur if and only if

I will now show that 4,' is strictly posi- tive and can be written as a function of the current state and the distribution of future job-finding probabilities. Recall that E(QT)= CpsE(@,;) and E(cT) = Cp,E(c.d). From (A4'1, E(@;) = mad P,!, 4,?p,,I + 6E(@'+') . Also, for an unattached worker, the EPDV of the returns from searching in period T and state s is

For any given sequence of possible future

job-finding probabilities, (41,. ..,4',}:=, + ,, recursive substitution establishes that 6~(@'+')- 6E(vr+') is itself a strictly posi- tive function of the distribution of prices and {4;, . . .,42}:=, + ,.The same result thus holds for any nondegenerate distribution of possible sequences (41,...,4$.}:=,+ ,. Letting D,"' denote the probability distribution over all future job-finding probabilities, given the current state s, one obtains P,T = P,(D~+'); for later use, define P* to be the minimum value for P,' (i.e., P* = min P,' > 0).

From (2a) and (4), the equilibrium layoff probability in period t is a nondecreasing function of the number of searching jobs in period t. The layoff rate can thus be written as

14 = L',(AKt) s = 1,...,2.

Now, taking AKt as given, the probability that a job will find a searching worker in state s is given by

Likewise, the fraction of unattached work- ers among those who are searching for em- ployment in period t in state s can also be written as a function of kt;that is,

To establish the existence of an equilib- rium in which the number of job openings is positive and finite, one need only consider the following two extreme cases:

(i) Suppose that AK' = -Jt-' (i.e., all of last period's job openings exit, and the num- ber of job openings this period drops to zero). In this case, the probability that a temporarily laid-off worker will find a job falls to zero, as 4(O) = 0, and so the optimal layoff rate in each state is also zero [see (3)]; that is, L',( -Jr-') = 0. From the perspec- tive of an individual job opening contemplating entry, however, there are no competing jobs and plenty of unattached workers. Hence the probability of finding a worker equals 1, as $(O)= 1, while the probability that this worker is unattached also equals 1, as temporary layoffs are ab- sent, and therefore y,'( -Jtpl)= 1.

(ii) Suppose that AK' =a.In this case, the layoff probability rises to 1, as +(m) = 1. Thus, while 0 5 y:(m) 5 1, the probability that a job opening will find a worker falls to zero as the ratio of workers to jobs falls to zero; that is, $(a) = 0.

Summarizing (i) and (ii), A Kt = -Jt-' yields +;y,I = 1 while A Kt =w yields +fy: =

0. Therefore, if firms' search cost c is less than P*, there exists an equilibrium sequence of entry decisions, {AKT};=,, corresponding to the initial conditions (Nt-', Jt-'1, and satisfying -J'-' < AK7 <m, which are found by solving

Suppose, in particular, that the equilib- rium layoff rate lies strictly between 0 and 1 and is determined by (412). In this case, the distribution Dt+' of future job-finding probabilities is degenerate and stationary (i.e., 4,' = p,, /pSh), and so P,' = P, is inde- pendent of 7. Since $SfSffS = +f ShS and +j =

~s!/~sh,

II,%(AK7)~,.(AK7)

In this case, (B3) can be written as

where A=&, PS(ps,/psh)p,, L /c and B = CpspS(ps( /pSh)(l -pSx)/~'

APPENDIXC: EXPOSTENTRY REFERENCES

Suppose that firms' entry/exit decisions are made after (rather than prior to) the realization of the state in period t. In this case, the model goes through as described in Appendix B except that the expected profits from entry must be exhausted on a state-by-state basis. That is, in equilibrium,

The effects of a change in the distribution of relative prices, holding p,, = p, fixed, are determined as follows.

Observe that an increase in relative price dispersion implies that p,,/p,, falls, p,, decreases, and p,, increases. Hence an in- crease in price dispersion implies that P, in (B2a) rises, as it is more profitable to hire an unattached worker when p,, is higher. Assuming that (4c) holds, so that layoffs adjust to maintain equality between the job-finding probability and price ratio, it follows that 4; falls and 9: rises as p,, /p,, falls. Since 9: and P, both rise, (Cl) implies that y,' must fall; hence an increase in price dispersion causes temporary layoffs to increase. At the same time, however, job openings must decrease, because the increased profitability of hiring an unattached worker is more than offset by the decline in the probability of finding an unattached worker; that is, the product of p,, /pSh and

ps,

declines as p,, falls or p,, rises, and so (Cl) implies that AK' declines. Hence (C1) can be rewritten as

Ji-' + AK'

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