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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 1053, August 2012 --- Screen Reader
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NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

In the United States, the aggregate vacancy-unemployment (V/U) ratio is strongly procyclical, and a large fraction of its adjustment associated with changes in productivity is sluggish. The latter is entirely unexplained by the benchmark homogeneous-agent model of equilibrium unemployment theory. I show that endogenous search and worker-side horizontal heterogeneity in production capacity can be important in accounting for this propagation puzzle. Driven by differences in unemployed and on-the-job seekers' search incentives, the probability that any given firm with a job opening matches with a worker endowed with a comparative advantage in that job exhibits a stage of procyclical slow-moving adjustment. Consequently, so do the expected gains from posting vacancies and, hence, the V/U ratio. The model has channels through which the majority of both the V/U ratio's sluggish-adjustment properties and its elasticity with respect to output per worker can be accounted for.

Keywords: Amplification, comparative advantage, endogenous search, heterogeneity, market tightness, mismatch, on-the-job search, propagation, search and matching, search intensity, unemployment, vacancies

JEL classification: E25, J24, J64

The aggregate vacancy-unemployment (V/U) ratio broadly
summarizes the state of the labor market, as it reflects the ease
with which individuals can exit unemployment. Empirically, in the
United States, the V/U ratio is strongly procyclical, and a large
fraction of its adjustment given changes in productivity is
sluggish. Hence, for instance, an increase in productivity is
associated with an upward jump in the V/U ratio, followed by a
protracted stage over which the V/U ratio continues to slowly rise
(at a decreasing rate). The benchmark, homogeneous-agent model of
equilibrium unemployment theory has no channels through which such
slow-moving adjustment can be accounted for: given a change in
productivity, the model predicts that full adjustment of the V/U
ratio occurs instantly. This limitation is additional to the
well-known fact that under standard calibrations, the benchmark
model can account for less than half of the elasticity of the V/U
ratio with respect to productivity.^{1} Understanding the V/U
ratio's stage of sluggish adjustment is important. Indeed, a
slow-moving deterioration of the V/U ratio reflects the degree to
which the labor market responds persistently, for example, in the
wake of a recession. Moreover, as can be inferred from Fujita and
Ramey (2007), around 60% of the total change in the V/U ratio that
occurs given a change in productivity takes place during the V/U
ratio's stage of sluggish adjustment.^{2}

The objective of this paper is to develop an understanding of the extent to which endogenous search and horizontal worker-side heterogeneity in production capacity can have an impact on shaping the dynamic adjustment process of the V/U ratio relative to changes in productivity. I capture horizontal heterogeneity by considering a labor force composed of individuals who have a comparative advantage in a particular job, but are still able to work in jobs in which they have a comparative disadvantage. I assume no worker has an absolute advantage in production, and I endogenize the search behavior of all job seekers, both employed and unemployed, across all available job opportunities. The model I develop is not competing, but rather, complementary to the benchmark/standard model, which I show is nested within the present paper's framework.

Quantitative analysis reveals that the impact of horizontal worker-side heterogeneity and endogenous search can be substantial. Indeed, results imply that accounting for such factors can potentially help explain both the majority of the V/U ratio's slow-moving adjustment properties and the majority of its elasticity with respect to output per worker.

In the model, both workers and firms prefer
comparative-advantage employment (matches between the same worker-
and firm-type) over comparative-disadvantage employment (that is,
matches between workers and firms whose type is different), since
the former generates the highest surplus. Nonetheless, comparative
disadvantage employment generates valuable surplus also; therefore,
it represents an appealing alternative through which workers can
exit unemployment, as well as an additional channel through which
firms can fill positions and, accordingly, incur lower expected
vacancy-posting costs. Given this environment, incentives are such
that unemployed individuals search across all available job
opportunities, and on-the-job (OTJ) search emerges naturally as the
result of individuals who are employed in jobs in which they have a
comparative disadvantage (alternatively, *skill-mismatched
employment*), but search for comparative-advantage
(alternatively, *skill-matched*) employment. The intensity of
search that any given individual devotes to any given job
opportunity is endogenous and contingent on: an individual's
comparative advantage in production, the state of the economy,
search costs, and an individual's employment state.

Given worker-side heterogeneity, vacancy-posting decisions are
based on the expected value of a match, which depends on the
slow-moving masses of unemployed and OTJ searchers. Consider, for
example, an increase in productivity.^{3} This induces a sudden
increase in the expected gains from posting vacancies, triggering a
jump in the V/U ratio. Since unemployed individuals have a lower
outside-search option compared to OTJ seekers, following the
increase in productivity, as unemployment declines the ratio of OTJ
searchers to unemployed individuals rises slowly. Consequently, the
fraction of job seekers who direct search (exclusively) toward
comparative-advantage employment opportunities increases. This
leads to a slow-moving rise in the probability that any given firm
with a job opening matches with a worker endowed with a comparative
advantage in that job (that is, the probability that any given firm
with a vacancy matches with a worker of its own type). Hence, the
expected gains from posting vacancies exhibit a stage of sluggish
increase, inducing the same in the V/U ratio. The opposite occurs
in a contraction.

It follows that the process leading to slow-moving adjustment of the V/U ratio originates from endogenous changes in the composition of the pool of individuals searching for any particular type of job. Endogenous job-seeking magnifies this process and aids in accounting for the amplification of shocks by generating feedback between firm and worker-side decisions. In particular, this allows workers to respond optimally to relative changes in employment surpluses across job opportunities, which has a direct impact on cyclical changes in the composition of searchers, and hence, on firms' match-quality expectations.

As related to the role of worker-side heterogeneity, Albrecht and Vroman (2002), Gautier (2002), Chassamboulli (2009), and Dolado, Jansen, and Jimeno (2009) develop models that explore the impact of vertical worker-side skill differentiation on idiosyncratic differences in unemployment and wages, while Pries (2007) shows how vertical differentiation can help amplify aggregate productivity shocks. In turn, Bils, Chang, and Kim (2009) focus on understanding differences in unemployment and work hours across labor-force participants. In their analysis, worker-side heterogeneity operates in a context in which "comparative advantage" refers to individuals who have high market productivity relative to their home productivity. Furthermore, labor markets are segmented: although the labor force is heterogeneous, conditional on idiosyncratic characteristics individuals seek employment in only one production sector. In all of the previous, workers' search behavior is determined exogenously.

By accounting for horizontal worker-side differentiation and endogenous directed search, the analysis in the present paper complements the literature on three main fronts. First, it reveals the critical labor-market role of (directed) OTJ search. In the absence of this, workers' ability to refocus search given changes in productivity is limited to the extent that the model's channel for generating slow-moving adjustment of the V/U ratio is effectively shut down. Thus, comparative disadvantage employment emerges as necessary, but not sufficient for the V/U ratio to exhibit sluggish adjustment in response to changes in productivity. Second, analysis shows that the combination of worker-side heterogeneity and optimal search can generate amplification of changes in productivity broadly in line with the data even for relatively small values of net unemployment flow benefits. This stands in contrast to the fact that, as noted, for instance, in Hagedorn and Manovskii (2008), the standard model requires net unemployment flow benefits to be almost as large as output per worker in order to match the amplification of productivity shocks in the data. Finally, accounting for worker-side heterogeneity implies that output per worker (OPW) is endogenous. The theory reveals that, conditional on whether they stem from changes in productivity throughout or between job opportunities, otherwise identical changes in OPW can be associated with adjustment in aggregate labor-market variables of considerably different magnitude. Intuitively, changes in OPW that stem from changes in productivity between job opportunities are associated with greater changes in relative employment surpluses, and therefore with greater endogenous readjustment in the pool of individuals searching for any particular type of job.

This paper proceeds as follows. Section 2 develops the theory; although my ultimate interest lies in understanding the joint implications of worker-side heterogeneity and endogenous directed search, I develop the model sequentially, building it by initially focusing on the case in which search decisions are exogenous. Then, Section 3 details my methodology for numerical analysis, and Sections 4 and 5 present results. Finally, Section 6 concludes.

I consider an economy with two types of workers and two types of production sectors/firms, where worker-side differentiation is horizontal, and the environment is symmetric. This facilitates focus on developing an understanding of the extent to which arbitrarily small degrees of heterogeneity can have an impact on aggregate labor-market fluctuations relative to the standard, homogeneous-agent model of equilibrium unemployment theory.

I embed horizontal differentiation through assumptions on
production. Let workers and production sectors/firms be indexed by
. In the notation subscripts
refer to workers and superscripts to sectors/firms. *Comparative
advantage employment*, which I alternatively refer to as
*skill-matched employment*, occurs when the worker and firm
type coincide. The output generated by a type-
individual employed by a type- firm is
, where is an economy-wide (exogenous) productivity parameter.
For , the output generated by a
type- individual employed by a type- firm (*comparative disadvantage employment*, or,
alternatively, *skill-mismatched employment*) is
, where
is an (exogenous) penalty
parameter that captures the degree of comparative disadvantage of a
type- individual employed by a type- firm. Unless noted otherwise, henceforth when and appear together in some expression,
assume .

Symmetry implies that in all periods any type-specific worker/firm variable is equal to half of its aggregate counterpart, and that all model parameters are symmetric across worker and firm types. Given symmetry, whenever helpful I present the model from the point of view of type-1 economic agents. All statements carry over to type-2 agents by simple re-indexing. As in Hagedorn and Manovskii (2008) the model is cast in discrete time, but I assume that the time period is small enough to be a close approximation to continuous time. All economic agents discount the future at rate , and is the discount factor. In addition, all variables are normalized by the aggregate labor force.

Job seekers are in employment state
, where
means " unemployed" and means "skill-mismatch employed." Unemployed workers
receive net unemployment flow benefits , which are
equal to the difference between time-invariant gross unemployment
flow benefits and time-invariant (for now) search
costs . Unemployed individuals direct their
search across all employment opportunities.^{4}

The value of unemployment for a type-1 individual is given by . All job-finding probabilities are endogenous, and denotes the probability that a type-1 individual searching for a job in sector finds a job in that sector, given that he or she is in employment state . Moreover, is the value of employment for a type-1 individual who is matched with a type firm. Letting denote the expectation operator, it follows that

. | (1) |

Comparative advantage in production implies that in any period . By assumption, it is always true that .

Let
denote the wage of a type-1
individual who is matched with a type- firm. As
noted in Shimer (2006), the standard Nash bargaining solution to
wage determination in matching models is invalid given on-the-job
search. Thus, in line with Pissarides (1994), Chassamboulli (2009),
and Dolado, Jansen, and Jimeno (2009), I assume that wages are such
that a constant fraction of the surplus of a match
goes to the worker, where
is the bargaining
power of workers, goes to the firm, and
also that wages can be continuously revised and that long-term
contracts are not possible.^{5} Following the literature, there is an
exogenous job-destruction probability
with which any type of job, whether skill-matched or -mismatched,
is destroyed. Since the value of skill-mismatch is less than that
of skill-matched employment, it is optimal for an individual who is
skill-mismatched to engage in on-the-job (OTJ) search directed
toward the sector in which he or she has a comparative advantage
(conditional on the type of match, the production functions of all
firms within a sector are identical, so there are no gains in
moving from one skill-mismatched employment relationship to
another). It follows that the value of comparative disadvantage
employment for a type-1 individual is

, | (2) |

where represents (for now, time invariant) OTJ search costs. Note that a job-to-job transition occurs with probability , meaning that there is an endogenous job-destruction component associated with skill-mismatched employment.

Using analogous reasoning, the relevant value of comparative advantage employment is

. | (3) |

In this case, there is no endogenous job-destruction component, since skill-matched individuals have already achieved their best possible match. Hence, they do not engage in OTJ search.

Let , and denote, respectively, the mass of type- individuals who are unemployed, skill-mismatch employed, and skill-match employed; each of these masses is determined endogenously. Then,

, | (4) |

where, by symmetry, is the economy's fraction of type- individuals.

Each period, the number of matches formed in sector is determined by the sectoral matching function , which is increasing in sector- vacancies and searchers . Following related literature, such as Gautier (2002), I assume that has constant returns to scale. In particular, let , where is the elasticity of sectoral matches with respect to sectoral vacancies, and is the matching efficiency parameter.

The job-finding probability
is given by
(sector- matches per sector-
searchers), weighted by the worker-type/employment-state specific
technological component
: *effective
search*. For now, I assume that effective search is determined
exogenously. Thus, in any period
. The
technological component of the search process summarizes the
effectiveness with which all of an individual's job-seeking
activities lead to a job offer given his or her employment state
and the sector in which the individual is searching.^{6}
Effective search includes different kinds of search activities and
methods, the intensity with which search methods are used, etc. It
follows that sectoral searchers are a weighted sum of all
individuals searching in that sector, where the weights are
effective search. Thus, for instance,

. | (5) |

Note that sector-1 searchers do not include the weighted mass of
skill-mismatched type-2 individuals
, since
type-2 individuals who are employed in sector 1 only search for
sector-2 jobs.^{7} Given constant returns to scale
, where
denotes
*sectoral (market) tightness*, and
.

With the earlier development in mind, it is straightforward that the evolution of the mass of unemployed type- workers satisfies

, | (6) |

and the aggregate unemployment rate is . Moreover, the evolution of the mass of type- workers who are skill mismatched is given by

, | (7) |

with the aggregate rate of skill-mismatch satisfying . Finally, letting denote the employment rate, is the aggregate rate of skill-mismatched employment.

Turning toward firms, let the value of a job for a sector-1 firm that is matched with a type- worker be denoted by , and the value of a vacancy by . The firm's value of skill-matched employment is

, | (8) |

and its value of skill-mismatched employment is

. | (9) |

Comparative advantage in production implies that .

Following the literature, while a firm has a vacancy it incurs the time-invariant flow cost . The probability with which a sector- vacant job is filled is (sector- matches per sector- vacancies). Given constant returns to scale, this can be stated as , where . The probability that a sector- vacant job is filled with a worker who has a comparative advantage in that sector is , and otherwise. If follows that

, | (10) |

and given the earlier development

. | (11) |

That is,
is the effective fraction of
type-2 individuals looking for jobs in sector 1. This probability
is endogenous, and given its dependence on the slow-moving masses
of unemployed and OTJ searchers, slow moving as well. For short,
with some slight abuse of terminology, I henceforth refer to
as the *probability of
skill-mismatch.*

For , is the surplus generated by an employment match between a type- worker and a sector- firm. The earlier noted assumptions on wage negotiations jointly imply that wages ultimately satisfy conditions identical to the standard Nash bargaining solution, implying the surplus-sharing rule

. | (12) |

Since (and ), it follows that (and ). Therefore, the wage of an individual who is skill-mismatched is lower than his or her skill-matched wage, and the wage of an individual who is skill-mismatched in any given sector is lower than that of individuals who are skill-matched in that same sector.

Free entry into vacancy creation implies the zero-profit
condition
. Implementing this condition in
equation (10) along
with the definition of surplus and rearranging yields the
*vacancy/job-creation condition*:

. | (13) |

This is the model's fundamental equilibrium equation. Changes in
economy-wide productivity induce changes in
the expected gains from posting vacancies (the left-hand side
(LHS)). These changes must be balanced out in terms of changes in
expected costs (the right-hand side (RHS)). Such balancing occurs
through changes in , which is a decreasing
function of
. It follows that
and
are the model's fundamental
equilibrium variables.^{8}

Henceforth, I refer to the model as developed so far, that is,
with fixed effective search, as the *multi-agent (MA)
model*.

The MA model is not solvable analytically; however, the role of
worker heterogeneity can be understood intuitively by initially
focusing on the impact of the *absence* of heterogeneity.
Assume all workers are identical and normalize all production to
. Then,
,
, and symmetry implies that
, where
, and
additionally , where denotes
aggregate vacancies. Thus,
, where
: the ratio of aggregate vacancies
to aggregate unemployment (alternatively, the V/U ratio). Hence,
, and, more
generally, super and subscripts become unnecessary since sectors
and individuals are now entirely identical. Within this context,
the job-creation condition reduces to

, | (14) |

which is, in fact, the standard (homogeneous agent) model's job
creation condition.^{9} Thus, the standard model is a
special case of the MA model in which heterogeneity is done away
with.

Consider a permanent increase in economy-wide productivity . This leads to a one time, permanent increase in the expected gains from posting vacancies (the LHS of equation ) that is balanced out by a one-time increase in the expected costs of posting vacancies (the equation's RHS). Since the job-filling probability is a decreasing function of the V/U ratio, this balancing occurs through a one-time increase in the V/U ratio, which is driven by an instantaneous increase in aggregate vacancies. Hence, given a change in , the V/U ratio does not exhibit post-shock slow-moving adjustment.

Now, return to the MA model, and once again consider a permanent increase in . At the moment of the shock, the expected gains from posting vacancies jump up (the LHS of equation (13), as do the expected costs (the equation's RHS). The latter is driven by an instantaneous increase in sectoral market tightness , which itself is driven by a jump in sectoral vacancies. However, unlike the standard model, all adjustments do not end there. This is because the probability of skill-mismatch is slow moving, and therefore, only after the change in productivity has occurred will this probability begin to adjust. By extension, the expected gains from posting vacancies will also continue to (slowly) adjust after the change in .

When economy-wide productivity rises, as the pool of unemployed individuals declines, type- workers take relatively longer to exit than type- searchers. This is because upon becoming skill-mismatched, type- individuals become OTJ searchers, and therefore continue to form part of ; however, type- workers exit whether they become skill-matched or -mismatched. Such relatively faster drainage of type- workers maps into a decrease in the probability of skill-mismatch , which occurs slowly given its dependence on the slow-moving masses of unemployed and OTJ searchers. This leads to a slow-moving increase in the expected gains from posting vacancies, which is balanced out through a slow-moving increase in the expected costs, driven by a slow-moving increase in sectoral market tightness. By extension, the slow-moving increase in the availability of sectoral vacancies per searchers will lead to a slow-moving increase in the availability of aggregate vacancies per unemployed individual. Thus, in the MA model, an increase in economy-wide productivity results in a stage of slow-moving increase in the V/U ratio, with the reverse being true given a decline in .

More technically, note that given symmetry equation (11) can be stated as

. | (15) |

As such, the expression for
clearly shows that in an
expansion it is a slow-moving increase in the ratio of
skill-mismatched employment to unemployment that serves to foster a
slow-moving decline in the probability of skill-mismatch. Note, in
addition, that in the absence of OTJ search
, and therefore
reduces to being a constant,
meaning that the model's channel for generating sluggish adjustment
of the V/U ratio is effectively shut down.^{10}

The employment surpluses associated with skill-matched and -mismatched employment can be expressed, in steady state, respectively as

(16) |

and

. | (17) |

The term
in equation
(16) and its
analog in equation
(17) capture,
respectively, the opportunity costs of skill-matched and
-mismatched employment.^{11} As detailed in the appendix, a
permanent increase in *relative productivity* induces an on impact decrease in the expected gains
from posting vacancies. Intuitively, this reflects the relative
importance of skill-matched surplus in firms' vacancy-posting
decisions, and therefore, the extent to which a weighing down of
the highest surplus-generating employment arrangement in the
economy - due to higher opportunity costs - is particularly
damaging for overall vacancy-posting incentives. It follows that in
the MA model a decrease in will induce the
economy to adjust opposite to an increase in economy-wide
productivity , ultimately triggering a decline in
the V/U ratio, part of which will be slow-moving. The reverse will
occur given a decline in .

The costs of effective search directed toward a sector are
simply the costs of generating job offers in that sector. As noted
in Krueger and Mueller (2008), the time that unemployed individuals
spend searching is small, which suggests that time constraints are
not binding in optimal search decisions.^{12} Given this, an
intuitive reason for which unemployed individuals might limit the
effective search that they devote to any given type of job
opportunity is that search costs are sector specific. In turn,
sector-specific search costs are a natural motivation for
individuals to broaden their search to include jobs in which they
do not have a comparative advantage. To capture this intuition, I
assume that individuals bear the additively separable
effective-search disutility function

, | (18) |

where .

Of course, the only value functions that must be updated are a workers' value of skill-mismatched employment and unemployment. Quite simply, and are as before, except that now they are maximized, respectively, over , and and . Moreover, net unemployment flow benefits become endogenous, since is now equal to the difference between and . As before, I assume that in all states of the economy it is optimal for unemployed individuals to search for jobs across sectors. Given the surplus-sharing rule in equation (12) , the first-order conditions for optimal search can be stated as

(19) |

when skill-mismatched, and for

(20) |

when unemployed.^{13} In each of these first-order
conditions the RHS represents the expected gains from search. Note
that effective search is a jump variable.

The intuitive nature of the cost function is reflected on several fronts. By symmetry . Therefore, since , given equation (20) unemployed individuals will always devote greater effective search toward skill-matched employment: . This implies self selection. Furthermore, if non-symmetric environments were considered, the chosen cost function provides an additional and natural motivation for skill-mismatched employment to exist. Suppose . Then, it is optimal to set , but as long as the expected gains from skill-mismatched search are positive .

I henceforth refer to the MA model extended to account for
endogenous effective search as the *multi-agent optimal
search* (MA-OS) model. This is the paper's central model
of interest.

The MA-OS model nests three models. These are the MA model
(obtained by fixing effective search), the standard model (obtained
by fixing effective search and setting ), and a
version of the standard model in which effective search is
endogenous (*standard optimal search (standard-OS)* model; set
, but allow unemployed individuals to
choose effective search). Given the absence of worker-side
heterogeneity (and OTJ search), as is the case with the standard
model, the standard-OS model has no channels through which sluggish
adjustment of the V/U ratio can be generated.^{14}

Related literature focuses on the response of models' endogenous variables relative to changes in output per worker (OPW). In the case of the standard and standard-OS models, OPW corresponds in straightforward fashion to the exogenous economy-wide productivity parameter . In contrast, in the MA and MA-OS models OPW is determined endogenously, and given by

. | (21) |

Equation (21) calls explicit attention to the role of both economy-wide productivity and the skill-mismatch parameter in the determination of OPW. This highlights the fact that observed empirical changes in OPW need not stem from a unique source.

Although the MA-OS model is not solvable analytically, the
impact of endogenous effective search can still be gauged. Since
employment surpluses are procyclical in , then so are
the expected gains from search, and therefore, effective search as
well (recall equations
(19) and
(20)).
Intuitively, in an expansion jobs are easier to find and employment
surpluses are higher. This means that the opportunity cost of not
having a job, and for that matter, of being skill-mismatched,
increases. Hence, individuals react to above-average economic
conditions by supplying above-average effective search. In a
recession, the opposite occurs. For instance, think of discouraged
workers as an extreme example of this; these are individuals who
have set effective search equal to zero.^{15}

Endogenous effective search enhances the amplification of economy-wide productivity shocks because it generates feedback between firm- and worker-side decisions. For instance, when the expected gains from search increase, effective search rises, which decreases expected vacancy-posting costs (all else equal, declines). This raises the expected gains from posting vacancies, therefore increasing vacancies, which raises the expected gains from search, and so on and so forth.

Endogenous effective search also enhances the magnitude of the model's sluggish adjustment properties. This is so because it induces greater adjustment in the ratio of skill-mismatched employment . Indeed, since employed job seekers have a more attractive outside option than unemployed ones, which is employment itself, it follows that unemployed effective search is more procyclical than OTJ search (mathematically, contrast equation (20) to equation (19)). Hence, in response to a rise in economy-wide productivity the probability of entering skill-mismatched employment will increase relatively more than the probability of exiting skill-mismatched employment. This will magnify the post shock increase in relative to the MA model, and, accordingly, the post shock decline in the probability of skill-mismatch . Note that regardless of any jump in , it is the slow-moving changes of this variable that matter for sluggish adjustment of the V/U ratio. Therefore, the magnified slow-moving decrease in that occurs under endogenous search will accordingly enhance the slow-moving adjustment of the V/U ratio. Of course, in terms of both amplification and sluggish adjustment, a decline in will induce opposite effects to those detailed above.

Now, consider instead the effects of a permanent increase in . Recall from analysis of the MA model that this will induce a reduction in the expected gains from posting vacancies. In the MA-OS model, this effect will be amplified because on impact of the the increase in there will be an accompanying strong instantaneous increase in the relative effective search that unemployed individuals devote to comparative disadvantage employment. This, of course, will lead to an instantaneous increase in the probability of skill-mismatch, further depressing the expected gains from posting vacancies.

However, the extent to which greater relative effective search
devoted to skill-mismatched employment leads to (slow-moving)
increases in the ratio of skill-mismatched employment to
unemployment implies that following an
increase in the probability of skill-mismatch
will slowly *decrease* (recall, once more, equation
(15)). In
response to this, the V/U ratio will slowly rise. Hence, once
effective search is endogenized, an increase in can in fact lead to an overall increase in the ratio
of aggregate vacancies to unemployment, reversing the associated
implications noted earlier for the MA model.

In addition, in the MA-OS model, because an increase in
induces a relatively greater increase
in effective search devoted toward skill-mismatched employment than
an increase in (which has a direct and broad
impact across all effective search), the ratio will (slowly) increase more following the former than
following the latter. Given equation
(15),
this implies that changes in will tend to
induce greater slow-moving adjustment of the V/U ratio than changes
in . Moreover, it also follows that changes in
will tend to have less of an impact
on OPW than changes in . Indeed, note from equation
(21) that
while an increase in , all else equal, tends
to increase OPW, a relative increase in , all else
equal, tends to decrease OPW. In contrast, an increase in
works broadly toward increasing OPW
through increases in *both* skill-matched and -mismatched
productivity. Of course, a decline in induces
opposite effects to those stemming from an increase in relative
productivity.

My analysis will focus on the impact of permanent changes in
models' exogenous variables, as this substantially simplifies the
explanation and interpretation of results while fully addressing
the issues of central interest.^{16} In order to gain a full
understanding of the MA-OS model, in analogous fashion to earlier
in the paper, numerical analysis will contrast results to those
stemming from nested models. There are no empirical time-series
counterparts to the parameters and ; therefore, following related literature, results are
put in context by highlighting changes in endogenous variables
relative to changes in output per worker, when relevant. All
henceforth cited tables and figures can be found in the
appendix.

The choice of parameter values for each model is summarized in
Table 1. I assume that the time period is equal to one week.
Accordingly, I set the discount factor to
0.999, which is consistent with a quarterly
interest rate of 0.012. I use the matching function efficiency
parameter and the flow cost of vacancy posting
to target the equilibrium aggregate
unemployment rate and the equilibrium
V/U ratio
; this is in line with averages
of US data spanning the last six decades. Using US unemployment
data and the methodology described in Shimer (2005), I obtain the
job-finding probability of an average unemployed individual. At
monthly frequency, the mean of this is equal to 0.43. The associated job-finding probability at weekly
frequency is given by
, which is
equal to 0.131; I take this as the relevant
steady-state value. Using this and the target equilibrium
unemployment rate, solving for the exogenous job-destruction
probability implies
.^{17} In all cases, the
matching function exponent is chosen so
that the partial elasticity of aggregate matches with respect to
aggregate unemployment is in line with the corresponding evidence
from Petrongolo and Pissarides (2001).^{18}

The parameters and are specific to the MA-OS and standard-OS models. Numerical analysis reveals that for each there is a value of that will hit the target equilibrium unemployment rate, but nothing else changes. Thus, I normalize by setting this parameter equal to one. In addition, I assume quadratic effective search disutility, meaning that .

The skill-mismatch parameter is specific to the MA and MA-OS models. McLaughlin and Bils (2001) argue that average within-industry wage differentials between individuals who remain in an industry and those who switch can be interpreted as the result of equilibrium self-selection. They show that, empirically, the wages of industry switchers are, on average, 16 lower than those of industry non-switchers. I take this number as a reference point. Therefore, I use the skill-mismatch penalty parameter to set the equilibrium ratio of wages of skill-mismatched individuals to average wages in a sector equal to 0.84.

In the case of the MA model, effective search is assumed to be fixed at the equilibrium values implied endogenously by the MA-OS model. In order to further tighten the comparability of results, I purge the analysis from cross-model imbalances in bargaining power by setting the parameter equal to 0.5. In addition, I anchor all models around a common value for net unemployment flow benefits , which I set to 0.5. This is the average of the values advanced in Shimer (2005) and Hall and Milgrom (2008), assuming, in the latter, the lowest suggested replacement rate. Anchoring around is in line with the fact that it is the value of , not , which matters directly for the determination of the value of employment surpluses (recall equations (16) and (17)).

In all cases, the initial steady state of models is calculated
at . Numerical analysis reveals that in the
MA and MA-OS models the fraction of skill-mismatched employment is
always small, making the equilibrium value of output per worker
(OPW) arbitrarily close to one. Thus, across models, equilibrium
net unemployment flow benefits are approximately 50% of
OPW.^{19}

Figures 1 through 6 focus on the response of key endogenous variables to a 1 permanent (unanticipated) increase in economy-wide productivity . As shown in Figure 1, in the standard model on impact of the shock instantaneously jumps to its new equilibrium value, while in the MA-OS model initially jumps, and thereafter continues to slowly increase over 4 months. This slow-moving increase in the V/U ratio is driven by the slow-moving post-shock decline in the probability of skill-mismatch shown in Figure 2, which stems from the slow-moving post-shock increase in the ratio of skill-mismatched to unemployed individuals shown in Figure 3 (recall equation (15)).

In the MA-OS model
. The jump in
noted in Figure 2 is driven by
the greater procyclicality of effective search devoted to
skill-mismatched jobs (U
skill-mismatch,
, )
relative to unemployed and OTJ effective search devoted to
skill-matched search (U
skill-match,
, and OTJ search,
, respectively); this is
shown in Figure 4, and stems from the expected gains from search
for skill-mismatched employment always being relatively
lower.^{20} The relatively greater
procyclicality of
also accounts for the
relative adjustments in the slow-moving components of the
probability of skill-mismatch, as shown in Figure 5 (given symmetry
the percent changes in and are the same as those in their aggregate
counterparts), and that of the fraction of skill-mismatched
employment is shown in Figure 6.

Figures 7 and 8 show the individual responses of aggregate vacancies, , and unemployment, . In both models, on impact of the shock vacancies overshoot. In terms of the response of the V/U ratio, the key difference is that while in the standard model after the shock takes place vacancies decline at the same rate that unemployment does, in the MA-OS model the post-shock slow-moving decline in the probability of skill-mismatch maintains incentives for vacancy posting higher than otherwise. Hence, in the MA-OS model, after their initial jump vacancies decrease, but at a slower rate relative to unemployment than in the absence of a post-shock decline in the probability of skill-mismatch.

Of course, in the standard model the elasticity of OPW with respect to is one. In the MA-OS model, the 1% change in economy-wide productivity under consideration induces a 1.003% change in OPW. As shown in Figure 6, in the MA-OS model a permanent increase in ultimately induces a decline in the fraction of skill-mismatched employment . Hence, in terms of OPW, the change in is slightly amplified.

Figures 9 through 16 focus on the response of key endogenous variables to a 1% (unexpected) permanent increase in in the MA-OS model. As shown in Figure 9, while on impact of the shock the V/U ratio declines, thereafter it slowly increases across a period of roughly 15 months over which it fully reverses its initial decline, ultimately increasing relative to its starting value. Figures 10 and 11 show the corresponding changes in aggregate vacancies and unemployment. Of course, the on-impact decline in is driven by an on-impact decline in vacancies, which now exhibit a stage of sluggish adjustment. Indeed, after their initial jump, vacancies slowly increase in the direction of change of the exogenous driving force.

Figure 12 shows adjustment in the probability of skill-mismatch . Its initial jump, along with its persistent post-shock direction of change, is driven by the initial response of endogenous effective search. As shown in Figure 13, on impact of the increase in the associated increase in skill-mismatched employment surplus triggers a substantial increase in effective search devoted to such jobs, partially at the expense of that devoted to skill-matched employment. Figure 14 details the ensuing changes in the rate of skill-mismatch and the ratio of skill-mismatched employment to unemployment . Overall, it follows that the lack of propagation slow-moving vacancies response given changes in economy-wide productivity , as shown earlier, is simply a reflection of accompanying changes in the probability of skill-mismatch not being sufficiently large.

As shown in Figure 15, unsurprisingly, the increase in induces a substantial increase in the fraction of skill-mismatched employment, . In terms of OPW, the relative increase in skill-mismatched employment acts opposite to the increase in relative productivity . This leads, in particular, to the elasticity of OPW with respect to to be a meager 0.002. Overall, the analysis reveals that once endogenous effective search is accounted for, changes in relative productivity have the potential to be a much more powerful driving force than economy-wide productivity in terms of both amplification and propagation of shocks in the model's key endogenous variables.

Following related literature, I now focus on changes in models'
endogenous variables relative to changes in OPW.^{21} In
addition, I extend the analysis to account for the impact of joint
changes in and . In particular,
I focus on joint shocks in which both economy-wide and relative
productivity move in the same direction. This is intuitive given
that in the MA-OS model the (unique) skill-mismatched employment
opportunity is a stand-in for *all jobs* other than the one in
which a worker is most productive. Therefore, to the extent that an
increase in represents an increase in a
worker's outside option relative to skill-matched employment, an
increase in occurring jointly with an
increase in is broadly analogous to it being the
case that in an expansion individuals have relatively more
*viable* work opportunities available than otherwise.

Row R1 of Table 2 summarizes information stemming from empirical
US data. The elasticities of the V/U ratio ,
aggregate vacancies , and aggregate unemployment
with respect to OPW are, respectively,
7.79, 3.88, and -3.93 (columns C1, C4, C7). Moreover, the
elasticity of vacancies with respect to unemployment, which
implicitly captures the slope of the Beveridge curve, that is, the
empirical negative relationship between aggregate vacancies and
unemployment, is -0.86 (column C9).^{22} Turning toward the
propagation of productivity shocks, some broadly applicable
stylized facts can be inferred from the detailed analysis in Fujita
and Ramey (2007). These are summarized in columns C2, C3, C5, C6,
and C8. Empirically, in the United States, an impulse in OPW of
around 0.7% is associated with: 1) an on-impact jump in the V/U
ratio that is followed by a stage of slow-moving increase (that
occurs at a decreasing rate) during which approximately 60% of the
total rise in the V/U ratio takes place (the V/U ratio peaks around
12 months after the increase in OPW occurs), 2) a (decreasing rate)
decline in the aggregate unemployment rate that lasts approximately
15 months before bottoming out, and 3) sluggishness in the
adjustment of vacancies; on impact vacancies jump, and thereafter
they continue to rise (at a decreasing rate) for about 12 months
over which approximately 60% of their total increase occurs. It
follows that, empirically, slow-moving adjustment of vacancies is
an important contributing factor to the sluggish adjustment of the
V/U ratio.

Rows R2 through R5 in Table 2 show model-specific responses to a
-induced on-impact increase in OPW of 0.7%,
broadly serving to summarize results already shown in Section 4, and also, for comparison, incorporating
results pertaining to the standard-OS and MA models.^{23} The
standard model has limited explanatory power, accounting, in
particular, for 27% of the empirical elasticity of the V/U ratio
with respect to OPW, but none of its sluggish adjustment. The MA
model accounts for a similar fraction of the empirical elasticity
of the V/U ratio, but its dynamic adjustment is fundamentally
different than in the standard model, as it is characterized by a
stage of slow-moving response induced by vacancies adjusting at a
slower rate than unemployment. Comparatively similar are the
relative responses in the standard-OS and MA-OS models, with the
former accounting for 55% of the empirical elasticity of
with respect to OPW, and the latter
for 64%. The last column of Table 2 (C10) shows the elasticity of
OPW with respect to the *initial* impulse of the relevant
exogenous driving force(s).

It follows that limiting changes in OPW to being driven by changes in , relative to all other models MA-OS makes the most important gains in accounting for sluggish adjustment of , with its post-shock stage of slow-moving adjustment lasting for 4 months over which 12% of its total increase occurs. Thus, in response to a -induced increase in OPW, the MA-OS model's slow-moving adjustment properties are approximately 25% of its empirical counterparts. Moreover, as noted above, the model accounts for around 2/3 of the empirical elasticity of the V/U ratio with respect to OPW. Overall, results highlight the extent to which endogenous effective search aids in accounting for the amplification of productivity shocks, worker-side heterogeneity aids with propagation, and the combination of both magnifies their individual effects.

Figures 16 through 20 show responses in the MA-OS model's key endogenous variables to joint increases in and that (in all cases) induce an on-impact increase in OPW of 0.7%, where the size of the shock to is, alternatively, half of the shock in , equal to the shock in , and two, three, four, and five times the shock in (the legend in Figure 16 applies to Figures 17 through 20). For reference, the noted set of figures also shows the case corresponding to row R5 of Table 2, which is the situation in which only a shock in occurs.

Figure 16 highlights that the greater the contribution of relative productivity to the shock, the greater the total response of the V/U ratio, as well as the magnitude of the extent to which it adjusts sluggishly. Moreover, when combined with an increase in economy-wide productivity, an increase in does not necessarily induce an on-impact decrease in the V/U ratio, as shown earlier to be the case given a stand-alone increase in . As shown in Figure 17, this carries over to vacancies; moreover, while vacancies do not exhibit sluggish adjustment for relatively small changes in , they do for higher ones. Figure 18 shows the corresponding changes in unemployment, and Figure 19 shows changes in ; greater and longer increases in are associated with greater and longer post-shock decreases in the the probability of skill-mismatch, which induces greater and longer post shock increases in the V/U ratio. Finally, Figure 20 shows the relevant adjustments in the fraction of skill-mismatched employment .

Rows R6 through R11 of Table 2 put in context the information
shown in Figures 16 through 20. As the relative contribution of
to the rise in OPW increases, the
MA-OS model makes substantial progress in broadly accounting for
the entirety of the US empirical data, both in terms of
amplification and propagation of productivity shocks. This is
particularly important on four fronts. First, the fact that the
model is capable of generating sluggish adjustment of both the V/U
ratio and vacancies implies substantial improvement relative to the
standard model, which has no channels through which this can occur.
Second, it is noteworthy that the MA-OS model can yield
amplification broadly in line with the data, as captured by the
elasticities noted in Table 2, even for relatively small values of
net unemployment flow benefits. This stands in contrast to the fact
that, as noted, for instance, in Hagedorn and Manovskii (2008), the
standard model requires net unemployment flow benefits to be almost
as large as OPW in order to match the amplification of productivity
shocks in the data.^{24} Third, results highlight that, given
heterogeneity, not all productivity shocks induce equivalent
dynamic adjustments. In particular, the model suggests that changes
in relative productivity are a key factor driving observed changes
in aggregate labor-market variables. Finally, it is particularly
noteworthy that even in the absence of changes in relative
productivity, the MA-OS model already makes substantial progress in
accounting for both amplification and propagation of productivity
shocks relative to the standard model, as noted earlier in row R5
of Table 2.

In terms of the relative response of vacancies to unemployment, as noted in column C9 of Table 2, compared to the data the standard model errs on the downside, whereas the MA-OS model errs on the upside. However, when joint shocks are considered, as the relative contribution of increases, results from the MA-OS model move increasingly in line toward the data. Overall, the results shown in Table 2 suggest that conditional on the response of vacancies relative to unemployment increasing, the MA-OS model can indeed potentially account for much, if not all, of the data. In that sense, it is important to keep in mind that once heterogeneity comes into play, the MA-OS model is in fact a vast simplification of reality, and there are a host of additional issues that could help readjust the response of vacancies relative to unemployment in the degree needed for the entirety of the data to be precisely matched. Intuitively, two reasons for which the effects of changes in the probability of skill-mismatch could be magnified in terms of the relative dynamic adjustment of vacancies are greater heterogeneity in both job types and/or the labor force, and job-training costs that are decreasing in the quality of an employment match. The former is sensible, given that heterogeneity is the basis for the model's propagation channel. The latter would imply that slow-moving increases in the firm-side probability of skill-matched employment would induce declines in expected training costs, therefore reinforcing increases in vacancy-posting incentives.

This paper explores the effects that worker-side heterogeneity and optimal job-seeking behavior have on aggregate labor-market fluctuations. The context of the analysis is one in which individuals have a comparative advantage in a particular job/sector, but are still able to work in jobs/sectors in which they are at a comparative disadvantage. Given this, firms with vacancies have expectations regarding match quality that are a function of the slow-moving masses of unemployed and on-the-job (OTJ) searchers: this provides a channel for slow-moving adjustment of the V/U ratio.

In an expansion, for instance, endogenous changes in the composition of the pool of individuals searching for any particular type of job lead to a stage of slow-moving increase in the firm-side probability of comparative advantage employment. This induces a slow-moving increase in the expected gains from posting vacancies, which keeps vacancy-posting incentives higher than otherwise. Coupled with declining unemployment, this leads to sluggish adjustment of the V/U ratio. Endogenous job-seeking magnifies this process and aids in accounting for the amplification of shocks by generating feedback between firm and worker-side decisions. Intuitively, allowing workers to respond optimally to relative changes in employment surpluses across job opportunities bears direct impact on cyclical changes in the composition of searchers, and hence, on firms' match-quality expectations. Sluggish adjustment of the aggregate vacancy-unemployment (V/U) ratio given changes in output per worker (OPW) is a key feature of the data that the standard, homogenous-agent model of equilibrium unemployment theory cannot account for. This limitation is additional to the well known fact that under standard calibrations, the benchmark model can account for less than half of the elasticity of the V/U ratio with respect to productivity.

Comparative-disadvantage employment emerges as necessary, but not sufficient for slow-moving adjustment of the V/U ratio to occur in response to changes in productivity. In the absence of this, workers' ability to refocus search given changes in productivity is limited to the extent that the model's channel for generating slow-moving adjustment of the V/U ratio is effectively shut down. In addition, the theory reveals that, conditional on whether they stem from changes in productivity throughout or between job opportunities, otherwise identical changes in OPW can be associated with adjustment in aggregate labor-market variables of considerably different magnitude. In particular, changes in OPW that stem from changes in productivity between job opportunities induce greater changes in relative employment surpluses. This induces greater readjustment in the pool of individuals searching for any particular job, and therefore, in firms' expectations regarding the probability of comparative advantage employment.

Overall, quantitative analysis shows that accounting for horizontal worker-side heterogeneity and optimal search can potentially help explain both the majority of the V/U ratio's slow-moving adjustment properties and the majority of its elasticity with respect to output per worker. Results hold for relatively small values of net unemployment flow benefits. This stands in contrast to the fact that, as noted, for instance, in Hagedorn and Manovskii (2008), the standard model requires net unemployment flow benefits to be almost as large as output per worker in order to match the amplification of productivity shocks in the data.

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Table 1: Model Calibrations (Weekly Frequency)

Model | Parameter = b | Parameter = α | Parameter = A | Parameter = Γ | Parameter = c | Parameter = φ | Parameter = ε | Parameter = | Parameter = | Parameter = |
---|---|---|---|---|---|---|---|---|---|---|

Standard | 0.50 | 0.50 | 0.16 | - | 0.62 | - | - | - | - | - |

Standard-OS | 0.72 | 0.50 | 0.19 | 1.00 | 0.62 | - | 1.00 | - | - | - |

MA | 0.72 | 0.60 | 0.20 | 1.00 | 0.72 | 0.12 | 0.00 | 0.64 | 0.13 | 0.51 |

MA-OS | 0.72 | 0.60 | 0.20 | 1.00 | 0.72 | 0.12 | 1.00 | - | - | - |

In all cases δ = 0.0081, η = 0.50, β = 0.999)

Notes (reasons for parameter choices): : - see text for details; : Petrongolo and Pissarides (2001); : (data) target ; : normalization; : (data) implied by ; : symmetry - see text for details; : (data) ; : McLaughlin and Bils (2001) - see text for details; : quadratic search disutility when relevant; : quarterly interest rate at ; , , : MA-OS endogenously implied - see text for details.

Table 2: Properties and Model-Generated Elasticities of Key Variables

V/U Ratio: C1 | V/U Ratio: C2 | V/U Ratio: C3 | Vacancies: C4 | Vacancies: C5 | Vacancies: C6 | Unemp.: C7 | Unemp.: C8 | ∂v/∂u | OPW: | |
---|---|---|---|---|---|---|---|---|---|---|

R1: US | 7.79 | 12 | 60% | 3.88 | 12 | 60% | -3.93 | 15 | -0.86 | - |

R2: st. | 2.12 | 0 | 0% | 1.19 | 0 | 0% | -0.99 | 11 | -1.13 | 1.00 |

R3: MA | 2.19 | 3 | 5% | 0.96 | 0 | 0% | -1.21 | 8 | -0.79 | 1.00 |

R4: st.-OS | 4.25 | 0 | 0% | 1.23 | 0 | 0% | -2.91 | 14 | -0.43 | 1.00 |

R5: MA-OS | 4.58 | 4 | 12% | 1.20 | 0 | 0% | -3.28 | 9 | -0.37 | 1.00 |

R6: MA-OS | 4.99 | 6 | 24% | 1.40 | 0 | 0% | -3.47 | 11 | -0.40 | 1.00 |

R7: MA-OS | 5.44 | 10 | 35% | 1.64 | 0 | 0% | -3.66 | 12 | -0.45 | 0.99 |

R8: MA-OS | 6.30 | 14 | 54% | 2.09 | 0 | 0% | -4.03 | 13 | -0.52 | 0.98 |

R9: MA-OS | 7.15 | 15 | 68% | 2.53 | 10 | 9% | -4.41 | 14 | -0.57 | 0.97 |

R10: MA-OS | 7.95 | 16 | 78% | 2.94 | 12 | 42% | -4.75 | 15 | -0.62 | 0.96 |

R11: MA-OS | 8.49 | 17 | 85% | 3.22 | 15 | 59% | -4.98 | 15 | -0.65 | 0.96 |

Notes (columns and rows; see text for details on US data): : elasticity of V/U ratio with respect to output per worker (OPW); : months after shock over which V/U ratio continues to increase (propagation); : percent of total increase in V/U ratio occurring over period of slow-moving adjustment; : elasticity of aggregate vacancies with respect to OPW; : months after shock over which vacancies continue to increase (propagation); : percent of total increase in vacancies occurring over period of slow-moving adjustment; : elasticity of aggregate unemployment with respect to OPW; : month at which unemployment reaches a trough; : slope of Beveridge curve; : elasticity of output per worker with respect to on-impact size of exogenous shock (0.007). : US data; -: model-specific response given -induced on-impact increase in OPW of 0.7%; -: response in MA-OS model to joint increases in and that (in all cases) induce an on-impact increase in OPW of 0.7%, where size of shock to is half of shock in (), equal to shock in (), and two (), three (), four (), and five () times the shock in . Model-generated elasticities are based off permanent change in productivity.

Figure 1: MA-OS and Standard Models

Figure 2: MA-OS Model

Figure 3: MA-OS Model

Figure 4: MA-OS Model

Figure 5: MA-OS Model

Figure 6: MA-OS Model

Figures 1-6: responses to 1% permanent (unanticipated) increase in economy-wide productivity of V/U ratio (Fig. 1: MA-OS and Standard models), skill-mismatch probability (Fig. 2: MA-OS model), skill-mismatch to unemployment ratio to (Fig. 3: MA-OS model), effective search : U - skill-match, : U - skill-mismatch, : OTJ search (Fig. 4: MA-OS model), rate of skill-mismatch and unemployment rate (Fig. 5: MA-OS model), and fraction of skill-mismatch employment (Fig. 6: MA-OS model).

Figure 7: MA-OS and Standard Models

Figure 8: MA-OS and Standard Models

Figure 9: MA-OS and Standard Models

Figure 10: MA-OS and Standard models

Figure 11: MA-OS and Standard Models

Figure 12: MA-OS and Standard Models

Figures 7-8: responses to 1% permanent (unanticipated) increase in economy-wide productivity of aggregate vacancies (Fig. 7: MA-OS and Standard models) and aggregate unemployment (MA-OS and Standard models).

Figures 9-12: responses in MA-OS model to 1% permanent (unanticipated) increase in relative productivity of V/U ratio (Fig. 9), aggregate vacancies (Fig. 10), aggregate unemployment (Fig. 11), and skill-mismatch probability (Fig. 12).

Figure 13: MA-OS and Standard models

Figure 14: MA-OS and Standard models

Figure 15: MA-OS and Standard Models

Figure 16: MA-OS and Standard Models

Figure 17: MA-OS and Standard Models

Figure 18: MA-OS and Standard Models

Figures 13-15: responses in MA-OS model to 1% permanent (unanticipated) increase in relative productivity of effective search (Fig. 13; legend as in Fig. 4), rate of skill-mismatch and unemployment rate (Fig. 14), and fraction of skill-mismatch employment (Fig. 15).

Figures 16-18 (legend in Fig. 16 applies to all): responses in MA-OS model to permanent (unanticipated) increase in economy-wide productivity , and joint shocks in and relative productivity ; 0*p indicates shock in , only. 0.5*p indicates that size of shock in is half as size as shock to , etc. In all cases, on impact of exogenous shocks OPW increases by 0.7%. Response of V/U ratio (Fig. 16), response of aggregate vacancies (Fig. 17), and response of aggregate unemployment (Fig. 18).

Figure 19: MA-OS and Standard Models

Figure 20: MA-OS and Standard Models

Figures 19-20 (legend in Fig. 16 applies to all): responses in MA-OS model to permanent (unanticipated) increase in economy-wide productivity , and joint shocks in and relative productivity ; 0*p indicates shock in , only. 0.5*p indicates that size of shock in is half as size as shock to , etc.Response of skill-mismatch to unemployment ratio (Fig. 19), and response of fraction of skill-mismatched employment (Fig. 20).

As noted in Shimer (2005) and Mortensen and Nagypal (2007), the cyclical properties of models of the sort developed in this paper are well assessed by considering differences between steady states. Therefore, in what follows, relevant mathematical inferences are based on steady-state to steady-state changes.

**Proposition C.1.1.** *In the
multi-agent model the probability of skill mismatched
employment,*
*, is countercyclical in
economy-wide productivity* *.*

**Proof.** Given symmetry, for
, and
. Then, using
equation
(11) and
rearranging implies that
,
where , and I have used the assumption that
all effective search is fixed at unity. Therefore, if is procyclical, then
is countercyclical. For
symmetry implies that
. Then,

, | (22) |

since is procyclical.

**Proposition C.1.2.** *In the
absence of OTJ search, in the multi-agent model*
*is (always) constant.*

**Proof.** Assuming OTJ search is not possible is equivalent
to setting
, in which case
is equal to the constant
.

**Corollary C.1.1.** *In the
multi-agent model skill-mismatched employment is necessary, but not
sufficient, for propagation in the V/U ratio to occur in response
to changes in economy-wide productivity* *.*

**Remark C.1.1.** Within the
context of the MA-OS model, Proposition 1 can be thought of
carrying over as follows: on impact of an increase in economy-wide
productivity the probability of skill-mismatch can jump, but
relative to its position after this initial jump, the probability
of skill-mismatch will thereafter decrease.

**Proposition C.2.1.** *In
the standard-OS model effective search is procyclical in
economy-wide productivity* *.*

**Proof.**^{25}Consistent with the functional forms
assumed for the MA-OS model, for the standard-OS model let
and
, where . In this case,
, where
. Moreover, denote the
job-finding probability of unemployed individuals by , where
. The relevant first-order
condition for optimal search implies that

, | (23) |

where is the employment surplus (recall that individuals choose effective search taking market conditions as given). Moreover, the job-creation condition is now given by

. | (24) |

Substituting this into equation (23) implies, after total differentiation, that

. | (25) |

Since is procyclical, the equation above implies that so is effective search . Showing formally that is procyclical, to which I now proceed, offers additional insight. Using the definition of surplus, , and substituting in for the relevant value functions implies that

. | (26) |

Combining with equation (24) yields

. | (27) |

Therefore,

. | (28) |

Using the fact that and , this can be stated as

. | (29) |

Of course, and , the latter by assumption: otherwise, it would not be optimal for individuals to seek employment. Moreover, ). Then, given equation (29), to show that it is enough to show that the second term in the coefficient on is greater than the third term. Note that

, | (30) |

which will always hold (in the second line above, I have made use of equation (26). Therefore, , , and .

**Proposition C.2.2.** *If
effective search is endogenized in the standard model, then the
elasticity of the V/U ratio with respect to economy-wide
productivity* *is greater than
otherwise.*

**Proof.**^{26}Consistent with the functional forms
assumed for the MA-OS model, for the standard model let
. In this case,
, where
. Moreover, denote the job-finding
probability of unemployed individuals by , where
. Using the definition of
surplus, , and substituting in for the
relevant value functions implies that

. | (31) |

Combining with the job-creation condition implies that

. | (32) |

Hence,

, | (33) |

and

, | (34) |

Above, , and denotes that effective search is exogenous (and fixed at unity: ). Moreover, for reasonable values for , , , and , .

Note from the proof of Proposition 4 that equation (28) can be written as

, | (35) |

where denotes that effective search is endogenous. From earlier, ; therefore, . Moreover, from the proof of Proposition 4 and . Of course, . Hence, using equation , equation can be stated as

. | (36) |

C.3 Equilibrium in the MA and MA-OS Models

Once searchers are determined, sectoral vacancies can be backed out from . Given knowledge of the masses of unemployed and skill-mismatched individuals it is straightforward to derive , and therefore, , where . Hence, for and knowledge of the key endogenous variables , , , and (eight variables total) is sufficient to derive all of the model's remaining endogenous variables. There are four employment values: , , , and . Using the surplus-sharing rule in equation (12) these can be stated in terms of employment surpluses, and solved for using the four job values implicit in equations (8) and (9). As noted before, and are defined by the two corresponding job creation conditions implied by equation (13). Finally, the remaining six key variables , , , , , and , are defined through the 6 expressions implicit in equations (11), (6), and (7); recall that the environment is symmetric. Thus, the model reduces to 8 equations in 8 unknowns.

Consider the employment surpluses

and

.

Given symmetry, for a type-1 firm the expected gains from posting vacancies satisfy

.

Then, since the skill-mismatch probability is slow moving, what matters for the on-impact effect of a change in (or, for that matter, the on-impact effect of a change in ), is the expression

.

From the surplus definitions above, given a change in , that is, a change that only affects ,

and

Let

,
,

,

and

Then,

,

and

. |

Note that if , then , meaning that (for reasonable parameter values, the same is true when ), in which case the sign of is equal to the sign of and the sign of is opposite to that of . In particular, an increase in induces an increase in , and therefore an increase in and a decrease in . Note, then, that

if and only if

.

If this is true if and only if

, |

which holds for since . Therefore, as long as , an increase in induces an on-impact decrease in the expected gains from posting vacancies.

Petrongolo and Pissarides (2001) document that across a wide range of studies aggregate matching functions are found, empirically, to be approximately Cobb-Douglas in aggregate vacancies and unemployment. They note in their concluding section that the coefficient on unemployment is generally found to be in the range 0.5 to 0.7 when matches are formed with individuals exiting unemployment, only, and in the range 0.3-0.4 when total hires are used as the dependent variable (not only hires from unemployment). The nature of the standard and standard-OS models is such that the former applies, while the latter applies to the MA and MA-OS models.

In the standard and standard-OS models total matches are given by , where denotes effective search; it clearly follows that in both cases the partial elasticities of with respect to and are, respectively, and . In the MA and MA-OS models, given sectoral matching functions and symmetry, it is straightforward to show that total matches in the economy are given by

,

where

. Then, , and
.

As the effective fraction of on-the-job searchers converges to zero, this elasticity converges to ,

as in the standard model. As long as and , then . Furthermore, if and only if

.

Numerical analysis reveals that . Hence, for , the partial elasticities of aggregate matches with respect to and are broadly consistent with the findings in Petrongolo and Pissarides (2001).

Table C.6.1 Model calibrations (weekly frequency)

Difference in | Parameter: b | Parameter: ∝ | Parameter: A | Parameter: T | Parameter: c | Parameter: Ø | Parameter: ε | Parameter: δ | Parameter: ß | Parameter: η |
---|---|---|---|---|---|---|---|---|---|---|

none | 0.72 | 0.60 | 0.20 | 1.00 | 0.72 | 0.12 | 1.00 | 0.0081 | 0.999 | 0.50 |

ε | 0.73 | 0.60 | 0.20 | 1.00 | 0.70 | 0.12 | 1.10 | 0.0081 | 0.999 | 0.50 |

Ø | 0.72 | 0.60 | 0.20 | 1.00 | 0.71 | 0.13 | 1.00 | 0.0081 | 0.999 | 0.50 |

η | 0.72 | 0.60 | 0.19 | 1.00 | 0.60 | 0.12 | 1.00 | 0.0081 | 0.999 | 0.55 |

z | 0.75 | 0.60 | 0.20 | 1.00 | 0.63 | 0.13 | 1.00 | 0.0081 | 0.999 | 0.50 |

Notes for reference, first row shows benchmark calibration; second row refers to calibration in which is 10 percent higher, but all targets remain as in benchmark; third row refers to calibration in which is 10% higher, but all targets remain as in benchmark except that implied ratio of skill-mismatched wage to average sectoral wage is now 0.83; fourth row refers to calibration in which is 10% higher, but all targets remain as in benchmark; fifth row refers to case in which net unemployment flow benefits are targeted to be 10% higher (0.55 instead of 0.5), but all other targets remain as in benchmark.

Table C.6.2: properties of key variables under ε = 1:1

V/U Ratio: C1 | V/U Ratio: C2 | V/U Ratio: C3 | Vacancies: C4 | Vacancies: C5 | Vacancies: C6 | Unemp.: C7 | Unemp.: C8 | ∂v/∂u | OPW: | |
---|---|---|---|---|---|---|---|---|---|---|

R1: US |
7.79 | 12 | 60% | 3.88 | 12 | 60% | -3.93 | 15 | -0.86 | - |

R2: MA-OS | 4.81 | 4 | 11% | 1.22 | 0 | 0% | -3.47 | 11 | -0.35 | 1.00 |

R3: MA-OS | 5.22 | 7 | 22% | 1.43 | 0 | 0% | -3.65 | 12 | -0.39 | 0.99 |

R4: MA-OS | 5.66 | 8 | 32% | 1.68 | 0 | 0% | -3.83 | 12 | -0.44 | 0.99 |

R5: MA-OS | 6.52 | 15 | 50% | 2.15 | 0 | 0% | -4.19 | 12 | -0.51 | 0.97 |

R6: MA-OS | 7.40 | 15 | 63% | 2.62 | 0 | 0% | -4.55 | 13 | -0.58 | 0.96 |

R7: MA-OS | 8.22 | 16 | 73% | 3.10 | 12 | 26% | -4.89 | 13 | -0.63 | 0.95 |

R8: MA-OS | 8.79 | 16 | 78% | 3.37 | 13 | 43% | -5.12 | 14 | -0.66 | 0.94 |

Notes: Alternative calibration that applies to results in - implements value for 10% higher than in benchmark calibration, but all targets remain the same. : elasticity of V/U ratio with respect to output per worker (OPW); : months after shock over which V/U ratio continues to increase (propagation); : percent of total increase in V/U ratio occurring over period of slow-moving adjustment; : elasticity of aggregate vacancies with respect to OPW; : months after shock over which vacancies continue to increase (propagation); : percent of total increase in vacancies occurring over period of slow-moving adjustment; : elasticity of aggregate unemployment with respect to OPW; : month at which unemployment reaches a trough; : slope of Beveridge curve; : elasticity of output per worker with respect to on-impact size of exogenous shock (0.007). : US data; : responses given -induced on-impact increase in OPW of 0.7% under benchmark calibration; -: response in MA-OS model to joint increases in and that (in all cases) induce an on-impact increase in OPW of 0.7%, where size of shock to is half of shock in (), equal to shock in (), and two (), three (), four (), and five () times the shock in .

Table C.6.3: properties of key variables under φ = 0:13

V/U Ratio: C1 | V/U Ratio: C2 | V/U Ratio: C3 | Vacancies: C4 | Vacancies: C5 | Vacancies: C6 | Unemp.: C7 | Unemp.: C8 | ∂v/∂u | OPW: | |
---|---|---|---|---|---|---|---|---|---|---|

R 1: US |
7.79 | 12 | 60% | 3.88 | 12 | 60% | -3.93 | 15 | -0.86 | - |

R2: MA-OS |
4.62 | 4 | 12% | 1.22 | 0 | 0% | -3.29 | 10 | -0.37 | 1.00 |

R3: MA-OS | 5.03 | 5 | 23% | 1.43 | 0 | 0% | -3.47 | 10 | -0.41 | 1.00 |

R4: MA-OS | 5.45 | 9 | 34% | 1.65 | 0 | 0% | -3.66 | 11 | -0.45 | 0.99 |

R5: MA-OS | 6.30 | 11 | 52% | 2.10 | 0 | 0% | -4.03 | 11 | -0.52 | 0.98 |

R6: MA-OS | 7.15 | 14 | 65% | 2.54 | 6 | 2% | -4.40 | 13 | -0.58 | 0.96 |

R7: MA-OS | 7.94 | 15 | 76% | 2.95 | 11 | 34% | -4.75 | 14 | -0.62 | 0.95 |

R8: MA-OS | 8.49 | 15 | 81% | 3.22 | 12 | 51% | -4.98 | 15 | -0.65 | 0.95 |

Notes: Alternative calibration that applies to results in - implements skill-mismatch parameter 10% higher than in benchmark calibration, but all targets remain the same. Row and column references are as in Table C.5.2.

Table C.6.4: properties of key variables under η = 0:55

V/U Ratio: C1 | V/U Ratio: C2 | V/U Ratio: C3 | Vacancies: C4 | Vacancies: C5 | Vacancies: C6 | Unemp.: C7 | Unemp.: C8 | ∂v/∂u | OPW: | |
---|---|---|---|---|---|---|---|---|---|---|

R 1: US |
7.79 | 12 | 60% | 3.88 | 12 | 60% | -3.93 | 15 | -0.86 | - |

R2: MA-OS |
4.61 | 4 | 14% | 1.21 | 0 | 0% | -3.28 | 9 | -0.37 | 1.00 |

R3: MA-OS | 5.06 | 6 | 26% | 1.45 | 0 | 0% | -3.48 | 12 | -0.42 | 1.00 |

R4: MA-OS | 5.52 | 9 | 37% | 1.67 | 0 | 0% | -3.69 | 11 | -0.46 | 0.99 |

R5: MA-OS | 6.43 | 12 | 56% | 2.16 | 0 | 0% | -4.09 | 12 | -0.53 | 0.98 |

R6: MA-OS | 7.35 | 14 | 70% | 2.63 | 13 | 17% | -4.50 | 13 | -0.59 | 0.96 |

R7: MA-OS | 8.20 | 15 | 81% | 3.07 | 13 | 49% | -4.87 | 13 | -0.63 | 0.96 |

R8: MA-OS | 8.79 | 18 | 87% | 3.36 | 18 | 66% | -5.12 | 14 | -0.66 | 0.95 |

Notes: Alternative calibration that applies to results in - implements bargaining power of workers 10% higher than in benchmark calibration, but all targets remain the same. Row and column references are as in Table C.5.2.

Table C.6.5: properties of key variables under *z* = 0:55

V/U Ratio: C1 | V/U Ratio: C2 | V/U Ratio: C3 | Vacancies: C4 | Vacancies: C5 | Vacancies: C6 | Unemp.: C7 | Unemp.: C8 | ∂v/∂u | OPW: | |
---|---|---|---|---|---|---|---|---|---|---|

R 1: US |
7.79 | 12 | 60% | 3.88 | 12 | 60% | -3.93 | 15 | -0.86 | - |

R2: MA-OS |
5.19 | 4 | 13% | 1.38 | 0 | 0% | -3.67 | 9 | -0.38 | 1.00 |

R3: MA-OS | 5.65 | 6 | 24% | 1.62 | 0 | 0% | -3.88 | 10 | -0.42 | 1.00 |

R4: MA-OS | 6.12 | 8 | 34% | 1.87 | 0 | 0% | -4.10 | 11 | -0.46 | 0.99 |

R5: MA-OS | 7.06 | 12 | 50% | 2.36 | 0 | 0% | -4.48 | 14 | -0.53 | 0.97 |

R6: MA-OS | 8.00 | 13 | 63% | 2.86 | 0 | 0% | -4.89 | 15 | -0.59 | 0.96 |

R7: MA-OS | 8.90 | 13 | 73% | 3.32 | 10 | 27% | -5.27 | 15 | -0.63 | 0.95 |

R8: MA-OS | 9.51 | 15 | 78% | 3.64 | 12 | 43% | -5.53 | 16 | -0.66 | 0.94 |

Notes: Alternative calibration that applies to results in - targets net unemployment flow benefits 10% higher than in benchmark calibration, while all other targets remain the same. Row and column references are as in Table C.5.2.

**. International Finance Division, Board of Governors of the Federal Reserve System, Washington, D.C. 20551. E-mail: Brendan.Epstein@frb.gov. My greatest thanks to Mike Elsby, Etienne Gagnon, Miles Kimball, Toshihiko Mukoyama, Jeff Smith, and Brian Jacob for very helpful comments and discussions, and also to seminar participants at the Board of Governors of the Federal Reserve System, Brown University, McMaster University, the University of Michigan, the 2012 Midwest Macro Meetings hosted by the University of Notre Dame, and the 2012 Canadian Economics Association annual conference hosted by the University of Calgary. All errors are my own. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Return to text

1. See, for example, Shimer (2005), Hall (2005), Fujita and Ramey (2007), Mortensen and Nagypal (2007), and Hagedorn and Manovskii (2008) for additional discussion on the V/U ratio's emprical response to changes in productivity, and also with regards to the standard/benchmark model's limitations in fully accounting for the data. In broader reference to the standard model, see, for instance Diamond (1982), Pissarides (1985), Mortensen and Pissarides (1994), and Pissarides (2000). Return to text

2. In particular, given an increase in productivity of about 0.7% the V/U ratio jumps and thereafter continues to slowly rise over approximately 12 months. Return to text

3. As is standard in the literature, by this I mean an exogenous change in productivity. Return to text

4. I assume that it is always optimal for them to do so, as the central focus of this paper is tied to examining the implications of skill-mismatched employment. Return to text

5. I elaborate on the meaning of surplus further below, as well as on the overall determination of wages. Return to text

6. This is based on the theory developed in Pissarides (2000), chapter 5, in which the standard, homogeneous-agent model features an endogenous job-seeking technological component. Later in the paper I endogenize the choice of idiosyncratic search technologies within the present multi-agent framework. Return to text

7. Total matches in sector-1 are given by

. |

8. See the appendix for further details Return to text

9. See, for instance, Pissarides (2000). Return to text

10. See the appendix for further details. Return to text

11. Indeed, numerical analysis implies that is endogenously optimal. Quite simply, this captures differences in outside options between unemployed and OTJ searchers. Return to text

12. In their cross-country investigation, Krueger and Mueller (2008) find that, conditional on searching, the average search time ranges from 40 minutes per week in Slovenia, to slightly less than 4 hours per week in Canada (which is a small amount more than in the U.S.). Return to text

13. Individuals choose effective search taking market conditions as given (in particular, ). Note that it is endogenously optimal to set . Return to text

14. See the appendix, as well as Pissarides (2000), chapter 5, for further details on what I refer to as the "standard-OS model." Return to text

15. With regards to procyclical effective search, see, for instance, Christensen et al. (2005). As noted earlier, the procyclicality of effective search is also a feature of the standard, homogenous-agent model enhanced to account for endogenous effective search (Pissarides (2000), chapter 5). Return to text

16. Numerical analysis reveals that differences in results stemming from temporary and permanent changes in the economy's driving forces are for all purposes negligible. In terms of impulse response functions, in particular, differences are broadly limited to the absence of mean reversion. This is broadly in line with the fact that, as noted in Shimer (2005) and Mortensen and Nagypal (2007), the cyclical properties of models of the sort developed in the present paper are well assessed by considering differences between steady states. Return to text

17. The unemployment rate is at monthly frequency, spans 1951:M1 through 2011:M10, and is taken from the Bureau of Labor Statistics (BLS). Vacancies are at monthly frequency and obtained by using the Job Openings and Labor Turnover Survey (JOLTS) data since 2000:M12 (when first available) combined with the Conference Board's Help-Wanted Index (HWI) from 1951:M1 through 2000:M11 (adjusted to the JOLTS units of measurement). Return to text

18. See the appendix for further details. Return to text

19. The appendix shows that implementing alternative values for , , and does not substantially alter overall results. Return to text

20. To further understand this, consider an extreme example. Suppose the expected gains from skill-mismatched search were zero; then, the first-order conditions for optimal effective search imply that would be zero as well. Given a positive productivity shock, assume these expected gains increase by an arbitrarily small amount. Then, so will . However, because was originally zero, the percent change in would technically be infinity. However, recall from earlier that by comparative advantage in all states of the economy. Return to text

21. See, for instance, Pissarides (2009). Return to text

22. Empirical elasticities are based off the logarithm of quarterly data spanning 1951:Q1 through 2011:Q3 that, following Shimer (2005), is detrended using an HP-filter with smoothing parameter . Data on unemployment and output per worker is taken from Bureau of Labor Statistics. Vacancies are obtained by using the Job Openings and Labor Turnover Survey (JOLTS) data since 2000:M12 (when first available) combined with the Conference Board's Help-Wanted Index (HWI) from 1951:M1 through 2000:M11 (adjusted to the JOLTS units of measurement). Both unemployment and vacancy data are originally at monthly frequency (statistics are based off quarterly averages); the series of output per worker is only available at quarterly frequency. Return to text

23. In line with the overall preceding analysis, note that in all cases referred to henceforth, model-generated elasticities are based off permanent changes in productivity, and hence are implied by changes between steady states. Return to text

24. In particular, net unemployment flow benefits must be approximately 96% of OPW in order for the standard model to match the data's amplification properties. Recall that the present calibration is such that net unemployment flow benefits are approximately 50% of OPW. Intuitively, the fact that employment surpluses are a function of the difference between the productivity of a match and net unemployment flow benefits means that the smaller this difference is, the more percentage-wise sensitive it is to changes in productivity, and therefore, the greater the elasticity of with respect to OPW. Return to text

25. For simplicity, this proof assumes functional forms consistent with the overall development of the model. Return to text

26. For simplicity, this proof assumes functional forms consistent with the overall development of the model. Return to text

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