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07 20
Diagnosing Labor Market Search Models:
A Multiple Shock Approach
by Kenneth Beauchemin and Murat Tasci
FEDERAL RESERVE BANK OF CLEVELAND
Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to
stimulate discussion and critical comment on research in progress. They may not have been subject to the
formal editorial review accorded official Federal Reserve Bank of Cleveland publications. The views stated
herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of
the Board of Governors of the Federal Reserve System.
Working papers are now available electronically through the Cleveland Fed’s site on the World Wide Web:
www.clevelandfed.org/research.
Working Paper 07-20
December 2007
Diagnosing Labor Market Search Models: A Multiple-Shock Approach
by Kenneth Beauchemin and Murat Tasci
We construct a multiple-shock version of the Mortensen-Pissarides labor market
search model to investigate the basic model’s well-known tendency to under
predict the volatility of key labor market variables. Data on U.S. job finding
and job separation probabilities are used to help estimate the parameters of a
three-dimensional shock process comprising labor productivity, job separation,
and matching or ‘allocative’ efficiency. We show that the Mortensen-Pissarides
labor market search model requires significantly procyclical and volatile job
separations to simultaneously account for high procyclical variations in jobfinding probabilities as well as relatively small net employment changes. Hence, the
model is more fundamentally flawed than its inability to amplify shocks would
suggest. This leads us to conclude that the model lacks mechanisms to generate
procyclical matching efficiency and labor force reallocation. As for the latter, we
conjecture that nontrivial labor force participation and job-to-job transitions are
promising avenues of research.
Key words: Labor Market Search; Mismatch; Business Cycles; Unemployment;
Job Vacancies.
JEL code: E24; E32; J40
Kenneth Beauchemin is at the University of Albany, SUNY and can be reached at
k.beauchemin@albany.edu. Murat Tasci is an economist at the Federal Reserve
Bank of Cleveland. He can be reached at murat.tasci@clev.frb.org.
The authors thank Russell W. Cooper, P. Dean Corbae and John Jones for insightful comments and suggestions. They also wish to thank participants of the
Midwest Macroeconomics Meetings (2005), North America Econometric Society
Meetings (2006), Society for Economic Dynamics (2006) and Macro Tea seminar
at the University Texas at Austin for helpful comments on an earlier version of
this paper titled as “On the Cyclicality of Labor Market Mismatch and Aggregate
Employment Flows.” The views expressed herein are those of the authors and not
necessarily those of the Federal Reserve Bank of Cleveland or the Federal Reserve System.
1
Introduction
There is now a fairly rich literature using the Mortensen-Pissarides labor market search model
to understand business cycle movements in the labor markets1 . Shimer (2005a) has recently
criticized this model, arguing that it requires implausibly large shocks to labor productivity to
generate substantial variation in key variables: unemployment, vacancies and the vacancy to
unemployment ratio2 .
We explore whether other reasonable sources of exogenous variation,
including job separation and job-matching shocks, can satisfactorily resolve this puzzle. In
particular, we identify the realizations of a multiple-shock process required for the model to
fit the data perfectly. Our results are striking. The perfect-fit experiment strongly indicates
that the standard labor market search model is more fundamentally flawed than its inability to
amplify shocks would suggest.
Our multiple shock approach allows for exogenous shocks to job separation, labor productivity and matching efficiency that are mutually correlated over the business cycle. In keeping
with the most basic Mortensen-Pissarides model, the rate of job separation is exogenous and
simply gives the fraction of employed persons that will separate from their jobs, for whatever
reason, during a particular period. The shock to the matching function captures the efficiency
with which existing labor market institutions pair searching workers with available jobs. We
call this the ‘allocative efficiency’ shock as in Andolfatto (1996). We apply data on monthly
separation and job finding probabilities as well as unemployment and job vacancies to estimate
the process that governs these shocks. The estimation strategy provides us with empirically
plausible variations in labor market transition probabilities of the average U.S. worker.
Although realistic variation in the transition probabilities substantially increase the volatil1
For a textbook treatment of this class of models see Pissarides (2000). Broadly speaking, we can identify
two separate but closely related strands in this literature. The first group, including works by Andolfatto
(1996), den Haan, Ramey and Watson (2000) and Merz (1995, 1999), incorporates labor market search into
otherwise standard real business cycle environments to improve upon their cyclical implications for labor market
variables such as employment. A second group of papers, such as Cole and Rogerson (1999) and Mortensen
and Pissarides (1994), focuses on the implications of the standard labor market search model in relation to the
empirical evidence on job creation and job destruction provided by Davis and Haltiwanger (1992) and Davis,
Haltiwanger and Schuh (1996).
2
Earlier studies either failed to address the magnitude of the exogenous forcing process (Mortensen and
Pissarides (1994), Cole and Rogerson (1999)) or implied counterfactually positive relationship between unemployment and vacancies (Andolfatto (1996), Merz (1995)). Merz (1995) provides two versions of the model, one
with constant and one with variable search effort. To be precise, her version with variable search effort gives
this counterfactual finding.
2
ity of key variables in the model, it does so at an enormous descriptive cost. The simulated
cyclical behavior of unemployment and vacancies is entirely counterfactual, displaying procyclical unemployment and countercyclical job vacancies. We conduct the ‘perfect-fit’ experiment
to better understand the counterfactual finding. That is, we posit the model as the actual data
generating process and subsequently infer the shocks required to match the data perfectly. Our
findings are startling. To be consistent with the observed fluctuations in unemployment and
job vacancies, the multiple-shock model requires volatile and procyclical job separations to simultaneously account for the pronounced procyclical variation in job finding and the relatively
small net employment changes.
These counterfactual findings are due to two reasons. First, the substantial fluctuations
observed in market tightness require significantly procyclical and volatile allocative efficiency
shocks. Given this, however, observed net employment changes could only be reconciled with
significantly procyclical and substantially volatile job separations. This is due to the fact
that all of the worker reallocation required to accommodate cyclical variations in employment
must involve an unemployment spell in the standard environment. However, it is impossible to
reconcile procyclical job separation/destruction with the existing empirical evidence (Blanchard
and Diamond (1990), Davis and Haltiwanger (1992), Davis, Haltiwanger and Schuh (1996) and
Shimer (2005a)).
Therefore, the defects in the labor market search model are even more
fundamental than Shimer (2005a) argued. We show that these findings are robust to different
calibrations.
We conclude that the basic model lacks sufficiently strong mechanisms to reallocate workers
over the course of the business cycle. Our results point to potentially productive extensions
of the basic model. If not acceptable, a priori, one could search for mechanisms that underly
procyclical allocational efficiency and modify the model accordingly. The results also indicate
that any such modification be accompanied by a theoretical expansion of the searching-workers
pool or the allowance of job-to-job transitions.
This paper is related to various other studies in the literature.
Our investigation into
the mechanics of the standard labor market search model echoes Shimer’s (2005a) diagnostic
exploration of Mortensen-Pissarides framework. We argue that the model, even when it has
substantial degrees of freedom with multiple shocks and empirically plausible transition proba3
bilities, has counterfactual implications. We emphasize this point further with an experiment
that requires our model to be the data generating process and gives us the unique realization of
the shocks for perfect fit. Although our objective for this experiment is diagnosis rather than
measurement, it is similar to accounting exercises employed in Chari, Keheo and McGrattan
(Forthcoming) and Ingram, Kocherlakota and Savin (1994). Our model is identical to Merz
(1995) except that we abstract from capital stock. Finally, we discuss our findings and several
avenues for future research in conjunction with the literature that tries to resolve the puzzle
presented in Shimer (2005a).
The remainder of the paper is organized as follows. Section 2 outlines our version of the
Mortensen-Pissarides model. In Section 3, we briefly describe the data and its basic statistical
properties. Section 4 discusses our calibration and presents the simulation results. Section 5
analyzes the simulation results and presents our diagnostic procedure. We also interpret our
findings in the context of recent literature. Section 6 discusses some robustness issues. We
briefly outline our conclusions and set a direction for future research in Section 7.
2
The Model
The economy is inhabited by a continuum of infinitely-lived worker/households distributed
uniformly along the unit interval; there is also a continuum of firms. At the beginning of
each period, a worker is considered either employed or unemployed. The measure of employed
workers is denoted Nt ; the measure of unemployed workers is the complement Ut ≡ 1 − Nt . The
representative household has preferences over state-contingent consumption and employment
given by
E0
∞
X
β t U (Ct , Nt ) ,
0 < β < 1,
(1)
t=0
where β is the subjective discount factor. Following Merz (1995), the period utility function is
separable in consumption and employment, with
1+ γ1
N
U (Ct , Nt ) = log Ct − t
1+
4
1
γ
,
γ > 0,
where γ defines the wage elasticity of labor supply at a constant marginal utility of wealth (the
“Frisch elasticity” of labor supply).
Both workers and firms must undergo a costly search process before jobs are created and
output is produced. At the beginning of each period, each unemployed worker searches for a
job expending φ consumption units in the process. Aggregate period t search costs incurred
therefore equal φ (1 − Nt ) consumption units. Firms create job vacancies, but only by expending
κ units of output per vacancy per period, generating aggregate “recruiting” costs equal to κVt .
Here, as in the traditional Mortensen-Pissarides framework, all jobs must be posted as vacancies
before they can be filled. Once a job is filled, it produces output equal to Zt generating aggregate
output
Yt = Zt Nt
(2)
where Zt > 0 is the exogenously determined productivity of labor.
The matching function captures the labor market search frictions. The typical formulation
determines the number of job matches formed in a given period, M (Vt , Ut ), as an increasing
function M of job vacancies, Vt , and the number of job seekers, Ut , where M exhibits constant
returns to scale. With search costs ultimately arising from heterogeneity-induced information
problems, we interpret the matching function as a mapping from not only from the quantities
of vacancies and searching workers, but also from the degree of mismatch between those two
populations. To allow for fluctuations in mismatch, we generalize the matching function to
include a multiplicative shock term, χt . Hence, the number of matches formed in period t is
given by
Mt = χt M (Vt , Ut ) = χt Vtα (1 − Nt )1−α
(3)
where 0 < α < 1 and χt is the period t realization of an unobserved shock process. Increases
in χt raise the number matches formed given the numbers of searching workers and available
positions. From a searching worker’s perspective, an increase in χt raises the probability of being
matched with a vacant position; from the perspective of a single firm, it improves its chances
of filling a vacancy. Consequently, fluctuations in χt signify improvements or deteriorations in
the allocative efficiency of the labor market.
While job matches are being formed, others are dissolved. We assume that the fraction
5
of existing matches dissolved during period t, σ t , is also determined as the realization of an
exogenous stochastic process. The period t change in aggregate employment, i.e. the net
employment flow, is defined as the difference between the period gross employment inflow and
gross employment outflow:
Nt+1 − Nt = Mt − σ t Nt .
(4)
Note that each flow is directly impacted by unobserved shocks: the flow into employment by
the allocative efficiency term, χt , and the outflow by the rate at which workers separate from
jobs, σ t .
The state of the economy in a given period, or (Nt , et ), consists of the beginning-of-period
employment level Nt , and values of the unobserved and exogenous state vector et = (Zt , χt , σ t ).
We make the standard Markovian assumption which allows agents to form expectations of
future-period quantities using knowledge of the current state only. Given the current state, the
socially efficient allocation of employment, vacancies, and consumption, {Nt+1 , Vt , Ct }, solves
the following recursively-defined social planner’s problem:
υ (Nt , et ) =
max
Nt+1 ,Vt ,Ct
{U (Ct , Nt ) + βEt υ (Nt+1 , et+1 )}
(5)
subject to
Ct + φ (1 − Nt ) + κVt ≤ Zt Nt .
(6)
Nt+1 = (1 − σ t ) Nt + χt M (Vt , 1 − Nt ) .
(7)
where υ (Nt , et ) is the future discounted social value of employment level Nt and the exogenous
state et . Equation (6) represents the period t resource constraint prohibiting the sum of current
expenditures on consumption, job search, and vacancy creation to exceed current output, and
equation (7) describes the trajectory of employment (4) with the matching function (3) determining the current-period flow into employment. Finally, we assume that a VAR(1) process
governs the exogenous state et :
et+1 = Aet + εt+1 , E(εε0 ) = Ω.
6
(8)
The autoregressive process for exogenous shocks plays a key role in this paper.
We will
devise a way to estimate this joint process and try to understand its implications for business
cycles in this model.
The corresponding first-order and envelope conditions imply an Euler equation describing
an intertemporally efficient vacancy-posting scheme for the economy. Suppressing arguments
and letting primes denote one-period-ahead quantities, we write
κ
UC
= βEt UC0
χMV
½
¾
0
¢
¤
UN
κ £¡
0
0
0
0
Z + φ + 0 + 0 0 1 − σ − χ MU
UC
χ MV
(9)
equating the loss in welfare due to vacancy creation with its expected future social benefit. In
this expression,
1
V
= α−1
χMV
χM
gives the average duration of vacancies multiplied by the elasticity of vacancies in matching,
α =
V MV .
M .
The left-hand side of (9), therefore, represents the utility loss associated with
a marginal increase in vacancies. The expected gain of the marginal vacancy, given by the
right-hand side of (9), derives from many sources. The expression Z 0 + φ +
0
UN
0
UC
gives the one-
period-ahead net social benefit of an additional match formed in the current period. The term
Z 0 equals the output flowing from the match; φ represents the (constant) search costs foregone
by the worker in the match. The final term in the sum,
0
UN
0 ,
UC
represents the consumption value
of the leisure foregone by the newly matched worker. In the basic Mortensen-Pissarides setup
this quantity is a constant, whereas we allow it to vary over the business cycle.
The final term in braces represents the net future social benefit arising from the expected
persistence of a job match. Given that any single current-period match survives with probability
1 − σ 0 , future social welfare will increase simply by reducing expected future recruiting costs
by the quantity
κ(1−σ0 )
χ0 MV0 .
The second term in this sum, −χ0 MU0 , represents the future reduction
in the future job-finding rate
χM
U
due to the current depletion of the unemployment stock; the
expected recruiting cost in future consumption units equals
0
κMU
MV0 .
As a system, equations (6)—(9) characterize the socially-optimal allocation of employment,
vacancies, and consumption given a joint distribution for the exogenous forcing variables or
7
shocks: Zt , χt and σ t . The traditional Mortensen-Pissarides approach determines these quantities in a market equilibrium with a real wage emerging as the outcome of Nash bargaining
between firms and households. The socially optimal allocation characterized above is supported
by a similar market allocation mechanism provided that: 1) asset markets are rich enough for
households to diversify away employment risk, and 2) the relative bargaining power between
households and firms is such that the positive and negative search externalities net out to
zero.3 Although we do not take a position on the precise nature of the allocation mechanism,
we maintain that existing market and institutional arrangements direct the realized allocation
sufficiently close to the social optimum to establish equations (6)—(9) as a useful instrument of
measure.
3
The Data
Before proceeding to shock measurement, we briefly review the salient facts regarding the
observed aggregate U.S. labor market measures that bear on our analysis. Given that the
model presented in the previous section does not require a labor market participation decision
for worker/households, we must choose whether to express our employment and unemployment
variables, Nt and Ut ≡ 1 − Nt , relative to the labor force or the age 16-and-over population.
Although there are valid arguments in favor of both normalizations, we find that the choice
little affects our results, and choose the labor force (employment plus unemployment) as our
reference population4
In the absence of a long time series on actual job vacancies, we follow standard practice and
construct vacancies from the Conference Board’s help-wanted advertising index. The resulting
vacancy series, Vt , is also expressed per member of the labor force5 . Also, since our model
3
Hosios (1990) determines the conditions under which the Pareto-optimum is supported as a decentralized
market equilibrium in a static environment; Merz (1995) and Andalfatto (1996) do the same in dynamic general
equilibrium settings. The market equilibrium in the current work closely follows those of Merz and Andalfatto.
4
Specifically we use the unemployment rate (unemployed persons per member of the labor force) constructed
as a quarterly average of the seasonally adjusted monthly series from the Current Population Survey (CPS) of
the Bureau of Labor Statistics (BLS). The civilian labor force measure is also provided by BLS as part of the
CPS. Both series can be downloaded from the CPS home page http://www.bls.gov/cps.
5
We construct a vacancy series by multiplying two seasonally adjusted monthly series — the ratio of helpwanted advertising to unemployed compiled by the Conference Board (downloaded as variable LHELX from the
DRI Basic database), and the unemployment rate U (defined above) — and averaging the monthly values to
obtain the quarterly series. The commonly reported help-wanted advertising index is a scalar transformation of
this series.
8
abstracts from the capital accumulation decision, we must choose between aggregate output
and aggregate consumption — a choice that reflects our desire to preserve a consistent and
well-understood labor productivity measure and one that can be more readily compared to
those in other studies. Since the aggregate labor input Nt produces all goods and services,
including private investment goods and those purchased by government, real GDP provides
the appropriate output measure.
member of the labor force6 .
Therefore, consumption, C, is proxied by real GDP per
We divide this series by the seasonally-adjusted civilian labor
force (averaged from monthly to quarterly), appropriately scaled, to express the variable in
year 2000 chained dollars per person. Time series data on U , V and C are constructed at the
quarterly frequency and run from 1951:1 to 2003:4.
We use real output per person in the non-farm business sector as our productivity measure.
This particular series is chosen to ensure comparability with the recent body of literature. It
is also a natural way to think about productivity in the standard labor market search model.
This series is part of BLS’s Major Sector Productivity and Costs program. It is normalized to
100 for 1992.
We also use U.S. labor market transition probabilities for our shock measurement process.
These probabilities were constructed by Shimer (2005a), but our discussion follows that of
Shimer (2005b). In accord with Shimer (2005b), the relevant labor market states are unemployment (u) and employment (e) with the job finding probability governing the rate at which
a worker switches between unemployment and employment and the separation probability determining the rate at which a worker switches between employment and unemployment.
Shimer’s (2005b) definitions of job finding and separation probabilities are as follows:
ft = 1 −
st =
s
Ut+1 − Ut+1
Ut
s
Ut+1
Et (1 − ft /2)
(10)
(11)
where in a given month t, Ut is the number of unemployed, Uts is the number of workers
6
Real GDP (billions of chained 2000 dollars, seasonally adjusted annual rate) was downloaded from the
Federal Reserve Bank of St. Louis FRED II database at http://research.stlouisfed.org/fred2/series/GDPC1
9
unemployed less than one month, and Et is the number of workers employed. These definitions
correct for time aggregation bias in the job separation probability by allowing for the possibility
of short unemployment spells within a given month.
To aggregate monthly transition probabilities, we account for all possible histories of employment states within a quarter. Temporal aggregation of labor market transition probabilities
might imply different cyclical features at various frequencies. The idea behind this argument
is simple. An unemployed worker at the beginning of a quarter potentially switches between
employment and unemployment before being counted as employed at the end of the quarter.
Therefore, the quarterly job finding probability of an average worker will not only reflect the
cyclical features of monthly job finding probabilities, but also of monthly separation probabilities. Since we are interested in the cyclical properties of labor market variables, it is vital for
us to be precise in aggregating Shimer’s monthly transition probabilities. We use the following
aggregation:
Ft = (1 − f(3∗(t−1)+1) ) ∗ (1 − f(3∗(t−1)+2) ) ∗ f(3∗(t−1)+3) +
¡
¢
(1 − f(3∗(t−1)+1) ) ∗ f(3∗(t−1)+2) ∗ 1 − s(3∗(t−1)+3) +
(12)
St = (1 − s(3∗(t−1)+1) ) ∗ (1 − s(3∗(t−1)+2) ) ∗ s(3∗(t−1)+3) +
¡
¢
(1 − s(3∗(t−1)+1) ) ∗ s(3∗(t−1)+2) ∗ 1 − f(3∗(t−1)+3) +
(13)
f(3∗(t−1)+1) ∗ s(3∗(t−1)+2) ∗ f(3∗(t−1)+3) +
¡
¢
f(3∗(t−1)+1) ∗ (1 − s(3∗(t−1)+2) ) ∗ 1 − s(3∗(t−1)+3) . ∀t
s(3∗(t−1)+1) ∗ f(3∗(t−1)+2) ∗ s(3∗(t−1)+3) +
¢
¡
s(3∗(t−1)+1) ∗ (1 − f(3∗(t−1)+2) ) ∗ 1 − f(3∗(t−1)+3) . ∀t
Note that in this aggregation four possible histories arise.
probabilities ignores these different experiences.
10
A simple averaging of monthly
We summarize the key business cycle features of the data in Table 1. In addition to the
labor market variables, we include the properties of the official BLS labor productivity measure
for the non-farm business sector7 .
With respect to transition probabilities, we report both
averages of f ’s and s’s and F ’s and S’s to facilitate comparison with Shimer (2005a).
To
describe the business-cycle variation in these quantities, we follow Shimer (2005a) and remove
the low-frequency trend in all variables implied by the Hodrick-Prescott filter under a smoothing
parameter of 105 . We apply this procedure to remove movements in the aggregates induced
by institutional and technological changes associated with job-matching, so that they are not
spuriously assigned to matching function instability arising from cyclical movements in labor
market mismatch. Key business cycle features of the U.S. data is summarized in Table 1.
Table 1: U.S. DATA
(Quarterly, 1951Q1-2003Q4)
u
v
v/u
u→e
u→e*
e→u
e→u*
z**
Standard Dev.
0.190
0.202
0.381
0.117
0.059
0.075
0.118
0.020
Autocorrelation
0.938
0.947
0.946
0.910
0.916
0.731
0.870
0.889
−0.894
−0.971
−0.949
−0.938
0.712
0.889
−0.417
0.974
0.898
0.908
−0.689
−0.852
0.369
0.948
0.948
−0.718
−0.893
0.402
0.990
−0.578
−0.841
0.406
−0.590
−0.840
0.414
0.910
−0.518
Cross Correlations
u
v
v/u
u→e
u→e*
e→u
e→u*
−0.546
* Quarterly transition probabilities aggregated as in (12) and (13).
** Official BLS labor productivity measure for the non-farm business sector.
From Table 1, we observe that employment, vacancies, and the vacancies-unemployment
ratio are all strongly procyclical and persistent; unemployment is strongly countercyclical and
7
This series is part of BLS’s Major Sector Productivity and Costs program and is downloaded from the
Federal Reserve Bank of St. Louis FRED II database at http://research.stlouisfed.org/fred2/series/GDPC It is
normalized to 100 for 1992.
11
persistent. These data also affirm the Beveridge curve with a strong contemporaneous correlation between vacancies and unemployment of −0.894.
Note that these cyclical properties
are all in accord with the qualitative predictions of the standard labor market search model.
However, Table 1 also show unemployment and vacancies to be almost 10 times more volatile
than labor productivity.
Note as well, the extreme volatility of the vacancy-unemployment
ratio (market tightness) with a standard deviation of 38 percent around its trend.
The ex-
treme volatility observed in unemployment, vacancy and market tightness provides the main
motivation of Shimer (2005a) and the subsequent literature on the business cycle implications
of standard labor market search models. We will contrast these findings with the implications
of a search model in the following section.
Although different ways of aggregation for transition probabilities give strongly correlated
series (note that the correlation between u→e and u→e* is 0.99 and e→u and e→u* is 0.91), the
relative variation changes significantly. With aggregations according to (12) and(13), quarterly
separations become relatively more volatile than quarterly job findings. It is crucial, however,
to keep in mind that this reversal of relative variation is a by-product of the aggregation. It
does not imply that the fluctuations in unemployment at a higher frequency is dominated by
separations.
4
Results
In this section, we explore the cyclical properties of the search model presented in section 2
with two sets of simulation results. First, we subject the model only to a labor productivity
shock holding allocational efficiency and job destruction constant. This experiment provides a
direct comparison to the standard labor market search model as in Shimer (2005a). The second
simulation incorporates the allocational efficiency and job destruction shocks with all three
governed by the VAR(1) process. Before presenting our results, we briefly describe calibration
of our model.
12
4.1
Calibration
With a large empirical literature to draw upon and stationary labor market variables at hand,
we combine micro-evidence with long-run data averages to calibrate the steady state values
of the exogenous shocks and the technology/preference parameters. We begin by setting the
steady state values of the labor market variables, Nt , Vt , and Ut , equal to the corresponding
data first moments: N = 0.943, V = 0.048, and U = .057. Given these values, we observe that
the steady-state version of the equation-of-motion for employment (7),
σN = χV α U 1−α ,
(14)
sharply restricts the steady state values of the shocks, χt and σ t , and matching technology
parameter, α. We set α equal to 0.28, which is the value used by Shimer (2005a)8 .
The
steady state rate of job separation is chosen to be 6.9 percent of total employment per quarter,
or σ = 0.069, which is the implied quarterly average of job separation probability discussed
in the previous section.
Under these settings, the steady state employment condition (14)
subsequently pins down steady state allocative efficiency level: χ = 1.056. These parameters
imply an average vacancy duration, (M/V )−1 , of 0.85 quarters or about 76 days, which is a
bit higher than the value reported by van Ours and Ridder (1992) using data from the Dutch
economy. The implied unemployment duration is 0.98 quarters, or about 12.7 weeks, which is
consistent with U.S.data.
Without loss of generality, we normalize the steady state of inferred aggregate output to
equal one, ZN = 1, yielding steady-state labor productivity Z = 1/N = 1.06. Under this
assumption, the steady state resource constraint becomes
C + φU + κV = 1.
Note that in the absence of search and recruiting costs, i.e. φ = κ = 0, labor productivity
reduces to the traditional average product of labor definition. Steady state labor productivity
equals C −1 in that case. (Recall that we must proxy consumption with aggregate output, or
8
This is at the low end of the estimates surveyed by Petrongolo and Pissarides (2001).
13
real GDP.) In the presence of search and recruiting costs, our imputed output measure deviates
from measured real GDP somewhat, but we anticipate the magnitude of the difference to be
small, with the settings of parameters φ and κ largely determining the gap. Unlike the model’s
other parameters, independent evidence regarding these two parameters is scarce. We follow
Andalfatto (1996) in assuming steady state recruiting expenditures to be one percent of output,
or κV = .01, implying κ = .206.
We assume that steady state search costs for workers are
also one percent of aggregate output, φU = .01, yielding φ = .176. The steady state value of
consumption is therefore C = .98, or 98 percent of output.
Table 2: Calibrated Parameters
Parameter
Value
Source
β
0.99
4% interest
α
0.28
Shimer (2005a)
γ
1.25
Merz (1995)
φ
0.1762
1% of Output
κ
0.2056
1% of Output
χss
1.0561
uss and vss
z ss
1.0602
Avg. Output = 1
σ ss
0.0609
Shimer (2005b)
Next, we consider the two preference parameters, β and γ, the subjective discount factor and
the Frisch elasticity of the labor supply, respectively. We choose β = .99 to be consistent with
a steady-state risk-free real interest rate of 4 percent. We follow Merz’s (1995) interpretation
of the empirical literature and choose γ = 1.5 for the Frisch elasticity. Table 2 summarizes our
calibration. In section 6, we consider different values of γ and α.
To calibrate the shock process, we first define the data series that comprise the VAR(1)
specification. Recall that the job separation probability series is taken from Shimer (2005b)
corrected for time aggregation effects. Next, observe that resource constraint along with the
data on V , U and C defines the period t productivity shock as follows:
Zt =
Ct + φ (1 − Nt ) + κVt .
Nt
14
(15)
Although we have reported cyclical properties of U.S. labor market data relative to the official
BLS measure of non-farm business sector labor productivity, it is comforting to note that the
correlation between this series and the inferred series is nearly perfect — 0.998, specifically.9
To measure the allocational efficiency shocks, we make use of job finding probabilities and
the matching function. Recall that total flow into employment in a given period is dictated by
the matching function, (3). We can rewrite this flow in the following way:
χt Vtα (1 − Nt )1−α = χt Vtα (1 − Nt )−α (1 − Nt ) = Pr(u → e) ∗ (1 − Nt )
(16)
Hence, total matches equals the job finding probability multiplied by the number of unemployed
workers. This decomposition along with data on V , U and Pr(u → e) identifies a time series
for χt .
We depict our series on Z, χ, and σ in Figure 5. With the knowledge that productivity
is strongly procyclical, Figure 5 shows allocative efficiency and job separation to be strongly
countercyclical. The contemporaneous correlations with Z are −0.55 and −0.69 for χ and σ,
respectively. As we have confirmed that the cyclical features of Z are nearly identical to labor
productivity as measured by the BLS, we also want to see whether the inferred series of χ
and σ imply reasonable fluctuations in unemployment when compared to the BLS measure.
Figure 6 compares the actual unemployment rate with the unemployment rate implied by the
shocks that are pictured in Figure 5 and inferred using the equation of motion (7). Figure 6
reveals that the shocks we use to estimate the VAR(1) lead to unemployment dynamics that are
virtually identical to those implied by the BLS series. Since this paper focuses on fluctuations
in unemployment rather than its level, the difference is inconsequential.
Finally, with these data series, we estimate the coefficients of A and Ω using the usual
equation-by-equation OLS procedure. Estimates of A and Ω are as follows.
⎤
⎢ 0.9081 0.1212 −0.0360⎥
⎥
⎢
⎥
A=⎢
⎢−0.0214 0.3078 0.2915 ⎥
⎦
⎣
−1.0305 0.0438 0.7558
⎡
9
⎡
⎤
⎢ 0.00007 −0.00003 −0.00014⎥
⎥
⎢
Ω=⎢
0.00047 ⎥
⎥
⎢−0.00003 0.00056
⎦
⎣
−0.00014 0.00047
0.00318
The correlation between H-P detrended measures is 0.869.
15
(17)
Next we turn to the simulation details of two main experiments.
4.2
Simulating the Benchmark Economy
We start with the analysis of the model presented in Section 2 without stochastic σ and χ.
Here we ask whether our benchmark model is consistent with Shimer (2005b), even though we
work with a decentralized economy version of the standard labor market search model. Since
σ and χ are assumed to be parameters in this benchmark model, we only need to define a
stochastic process for Z. Parameters σ and χ are set to their steady state values defined in
Table 2. We assume that Z follows a first-order autoregressive process such that the standard
deviation and the first-order autocorrelation match the corresponding moments in the data.
Our general solution algorithm is based on Christiano (2002) and relies on the linearized first
order condition (9). We posit linear decision rules for log deviations of the endogenous variables
Vt , Nt+1 and Ct around their respective steady states as a function of Nt and et = (Zt , χt , σ t ).
In the benchmark model, exogenous state only consists of Zt .
Table 3 presents sample moments from 100 simulations of the model economy where each
simulation is 500 periods in length.
To facilitate comparison with Table 1, each variable
is H-P filtered with smoothing parameter 105 and only deviations from trend are reported.
We can summarize this table in three broad findings.
First, vacancies and market tightness
(v/u) are significantly procyclical and unemployment is countercyclical. Second, the Beveridge
curve relationship is consistent with the benchmark model as shown by the negative correlation
between unemployment and vacancies of −0.846.
Finally, variations in the labor market
variables are much less than the underlying variation in productivity.
16
Table 3:
Simulations of Benchmark Economy
u
v
v/u
u→e
e→u
z
Standard Dev.
0.005
0.016
0.020
0.006
0.000
0.020
Autocorrelation
0.900
0.845
0.910
0.910
1.000
0.880
−0.846
−0.913
0.085
0.000
−0.890
0.989
0.989
0.000
0.996
1.000
0.000
1.000
0.000
1.000
Cross Correlations
u
v
v/u
u→e
e→u
0.000
This last observation regarding the relative standard deviation of vacancies, unemployment
and market tightness is the emphasis of Shimer (2005b). He argues that there is virtually no
amplification in the standard labor market search model. The third rows of Table 1 and Table
3 confirm this point.
Therefore, we conclude that the model with only productivity shocks
behaves similarly to the criticized search model, even though we focus on the social planner’s
problem. In what follows, we refer to this discrepency between the model and the data as the
amplification puzzle.
4.3
Simulating the Multiple Shock Economy
We now focus on the model presented in section 2 where the exogenous state space consists
of (Zt , χt , σ t ). Having introduced two additional shocks to the model, we expect to resolve
the amplification puzzle to some extent.
In other words, the allocational efficiency and job
destruction shocks are expected to improve the model’s performance. To show the consequence
of introducing the additional shocks to the model, we again report the moments similar to those
in Tables 1 and 3. Table 4 presents sample averages of moments from 100 simulations of the
model economy where each simulation is 500 periods at length.
percentage deviations from trend.
17
Once again, we report the
Table 4: Simulations of Multiple Shock Economy
u
v
v/u
u→e
e→u ( σ)
χ
z
Standard Dev.
0.048
0.386
0.420
0.159
0.109
0.051
0.020
Autocorrelation
0.883
0.732
0.779
0.860
0.840
0.806
0.890
−0.690
−0.745
−0.839
−0.800
−0.890
0.270
0.990
0.957
0.974
0.670
−0.539
0.974
0.986
0.716
−0.526
0.990
0.855
−0.500
0.810
−0.530
Cross Correlations
u
v
v/u
u→e
e→u ( σ)
χ
−0.340
Simulation results of this economy are striking. First, we observe significantly more volatility in all key variables relative to productivity, especially in V and V /U .
This is even ac-
companied by a seemingly correct Beveridge curve relationship, i.e. −0.69 correlation between
vacancies and unemployment. However, the model predicts completely counterfactual cyclicallity for vacancies, unemployment and market tightness. As cross correlations in Table 4 show,
this model implies mildly procyclical unemployment and significantly countercyclical vacancies
and market tightness. Adding two possible channels for possible fluctuations partly resolves
the amplification puzzle but it entails seriously counterfactual cyclical features with respect to
its key endogenous variables.
5
Discussion
It is no surprise the multiple shock approach produces better results in terms of the volatility of
endogenous variables. Beyond that, however, our results provide more questions than answers.
When we augment the standard labor market search model with shocks to allocative efficiency
and job separation, unemployment and vacancies present counterfactual cyclical properties.
Our objective here is to diagnose the reasons behind this erratic behavior to help reconcile the
predictions of search models with the cyclical labor market facts. To this end, we present an
insightful experiment. In what follows, we compute the innovations of the shock processes that
18
would obtain if the actual data generating process is indeed the multiple shock model10 . We
then analyze the characteristics of the realized shock processes to form economically meaningful
conjectures for the counterfactual behavior implied by the simulation in the preceding section.
This leads us to productive avenues for future research.
5.1
Required Shocks for Perfect Fit
Given that we apply a linearization-based algorithm to solve for the economy’s decision rules,
solving for the exogenous shock series that gives the multiple shock model a perfect fit in
expectation requires a straightforward inversion of the log-linearized model.
The solution procedure generates log deviations of endogenous variables around the steadystate as a function of Nt and et = (Zt , χt , σ t ). Dropping the time subscript to denote steadystate values and using lower-case letters to represent the corresponding log-deviation from
¡ t¢
¡ Vt ¢
steady-state, we define the endogenous variables as follows: nt ≡ ln N
N , vt ≡ ln V , and
¡ ¢
¡ ¢
ct ≡ ln CCt . The log-deviations of exogenous variables are similarly defined: z̃t ≡ ln ZZt ,
³ ´
¡ ¢
χ
e t ≡ ln χχt , and σ
et ≡ ln σσt . Similar transformation can be applied to the VAR(1) shock
process:
eet+1 = Ae
et + ε̃t+1
(18)
where eet = (z̃t , χ
et, σ
et )0 , A is a 3 × 3 matrix of constants, and ε̃t is trivariate normal with Eε̃t = 0
and E[εt ε0t ] = Ω.
Given values for the parameters comprising the VAR(1) matrix of coefficients A, the decision
rules mapping the period t state (nt , eet ) into values for the endogenous variables (nt+1 , vt , ct )
are required to be log-linear:
⎡
⎢ nt+1
⎢
⎢ v
⎢ t
⎣
ct
⎤
⎡
⎤
n
⎢ t ⎥
⎢
⎥
⎥
⎢ zt ⎥
⎥
⎢
⎥
⎥ = Π⎢
⎥,
⎥
⎢
⎥
⎦
et ⎥
⎢ χ
⎣
⎦
σ
et
⎡
⎢ π nn π nz π nhχ π nhσ
⎢
Π=⎢
⎢ π vn π vz π vhχ π vhσ
⎣
π cn π cz π chχ π chσ
⎤
⎥
⎥
⎥
⎥
⎦
(19)
where the π parameters comprise expressions of technology and preference parameters. Easy
10
This exercise is partly in the spirit of Chari, Kehoe and McGrattan (Forthcoming) and Ingram, Kocherlakota
and Savin (1994).
19
manipulation segregates the observed variables from the unobserved exogenous variables:
⎡
⎤
⎤
z
n
−
π
n
nn t ⎥
⎢ t ⎥
⎢ t+1
⎢
⎢
⎥
⎥
b⎢ χ
⎢ v −π n ⎥=Π
⎥
t
vn t ⎥
⎢ et ⎥ ,
⎢
⎣
⎣
⎦
⎦
ct − π cn nt
σ
et
⎡
⎡
⎢ π nz π nhχ π nhσ
⎢
b =⎢ π
Π
⎢ vz π vhχ π vhσ
⎣
π cz π chχ π chσ
⎤
⎥
⎥
⎥.
⎥
⎦
(20)
Given data series for employment, vacancies, and consumption, the left-hand side of this expression is a vector of constants in any given period. With values of all model parameters in
b is easily inverted to yield the period t realization of the forcing process:
hand, the matrix Π
(z̃t , χ
et , σ
et ). Our estimates in (17) and the mapping in (20) yield a unique set of realizations
for (z̃t , χ
et , σ
et ). Figure 7 plots the implied time series.
It is clear from the figure that both job destruction and allocative efficiency shocks move
in the same direction over the cycle. They are both significantly procyclical and much more
volatile than the productivity shock. One interesting observation is that, job destruction shock
clearly identifies NBER recession dates. Table 5 presents statistics on the implied shocks11 .
We see that standard deviation of allocative efficiency is required to be almost 8 times that of
labor productivity. The relative difference is even larger for the job destruction shock, which
displays 30 times the variation observed in labor productivity. All three exogenous shocks are
required to be quite persistent.
Table 5: Required Shocks
z
χ
σ
Standard Deviation
0.016
0.120
0.464
Autocorrelation
0.894
0.928
0.922
0.589
0.693
Cross Correlations
z
χ
0.859
σ
There is obviously nothing surprising with the implied series on labor productivity, z. Since
we are able to identify χ only indirectly when we use restrictions from the model, we can
11
Once again, all variables are log-deviations from their H-P trend, with smoothing parameter, 105 .
20
be agnostic about the true nature of the allocative efficiency shock.
destruction series poses a significant challenge.
However, implied job
It is impossible to reconcile a procyclical
job destruction shock with the existing evidence (Blanchard and Diamond (1990), Davis and
Haltiwanger (1992), Davis, Haltiwanger and Schuh (1996) and Shimer (2005a)). Since we cast
this experiment as a diagnostic procedure, we need an answer to the following question: What
are the properties of the multiple-shock search model that require it to produce a procyclical
and volatile job separation rate to account for U.S. employment fluctuations?
A successful
answer to this puzzle requires a deeper understanding of the mechanics of the standard labor
market search model.
5.2
Diagnosing the Search Model
We begin our analysis by identifying the model mechanisms and the cyclical properties of the
observed data that produce the results highlighted above. Motivated by the persistent and
procyclical movements of labor productivity Zt , we first trace out the dynamics generated by
the search model in response to a sudden and persistent increase in labor productivity, holding
constant allocative efficiency χt and the rate of job separation σ t . In doing so, we make use of
the equations (6), (7), and (9).
Consider first, the effects of an innovation to labor productivity. By signaling greater future
productivity [as captured by the term Z 0 in the intertemporal efficiency condition (9)], it encourages an immediate spike in vacancies as firms respond to the higher anticipated productivity
benefits of filled positions. Consequently, additional job matches form in the period of impact
thereby increasing employment and reducing unemployment. These effects are summarized by
an increasing vacancies-unemployment ratio.
The productivity innovation also sets in motion forces that oppose the increasing vacancyunemployment ratio. To see this, one first notes that the resource constraint (6) translates
the anticipated increase in future productivity and employment into higher future consumption through a more rapid output flow.12
The increases in employment and consumption
subsequently reduces the representative worker’s marginal willingness to substitute non-market
12
The sum of search and vacancy-creation costs, φ (1 − Nt ) + κVt , small and the increase in vacancy-creation
costs κVt counteract the reduction in search costs φ (1 − Nt ).
21
activities for consumption, i.e. decreases
0
UN
0
UC
in equation (9). This offsets, to some extent, an
individual firm’s vacancy-creation motive and the subsequent increase in employment. Furthermore, the draining of the unemployment pool persists and offsets some of the future benefits
of currently high productivity by frustrating future hiring efforts through the term −
0
κMU
MV0 ;
this
term represents the additional future recruiting costs exacted by the depleted stock of searching workers on the right hand side of (9). Recall that this last quantity (or more precisely,
its absolute value) is directly proportional to the vacancy-unemployment ratio—a proxy for the
‘tightness’ of the labor market. The data, as we have seen, displays extremely large procyclical
variation in this ratio, and casts doubt on the model’s ability to produce the required cyclical
variation in response to realistically sized shocks to labor productivity.13
By allowing both matching efficiency and the job separation rate to vary over the business
cycle, the preceding diagnostic procedure responds to this tension by, in effect, equating the
observed vacancy-unemployment ratio with the socially optimal one in each period. The highly
variable and procyclical allocative efficiency shock χt implied by this exercise (Table 5 and
Figure 7) effectively increases the expected gains of vacancy creation in response to exogenous
increases in labor productivity, thus generating additional vacancies while also increasing the
rate at which unemployed workers meet up with them. As a result, the flow of workers from
unemployment to employment increases, reducing the unemployment pool. The increase in
vacancies coupled with falling unemployment, thus gives an additional upward push to the
vacancy-unemployment ratio moving the economy along the Beveridge curve in accord with
the data. Although the vacancy-employment ratio moves decidedly in the proper direction, it
cannot do so with a sizeable increase in net employment, all else constant. However, given that
the aggregate employment data reveal relatively small period-to-period changes, a complete
picture of the labor market dynamics implied by this model requires a much larger employment
outflow to restock the unemployment pool depleted by the enhanced matching efficiency. This
element, of course, could only be provided by the required procyclical rate of job separation σ t
(Table 5 and Figure 7). The picture that emerges from this exercise is a more dynamic labor
13
This point has already been convincingly demonstrated by Shimer (2005a) using a more conventional
Mortensen-Pissarides model with a structurally stable matching function. We are indebted to his work for
articulating the opposing forces on the theoretical vacancy-unemployment ratio restraining its response to labor
productivity shocks.
22
market than the standard, single shock search model allows with workers cycling more quickly
through labor market states during expansions than in periods of recession.
5.3
A Resolution: Procyclical Mismatch and Reallocation
At this point we could simply accept the results of our experiment with a claim that matching
efficiency and job separation are indeed both strongly procyclical. To our knowledge, there
is no direct evidence on the efficiency of labor market matching over the business cycle. We
could (and do) simply argue that procyclical matching efficiency is a more reasonable outcome
than the alternatives. As we have already stated however, sharply procyclical job separation is
strongly at odds with existing data. To jointly accept both outcomes would be the equivalent of
believing in an extraordinarily unlikely draw from the distribution of shocks. Instead we look for
economic meaning in the results to conjecture a plausible solution to the puzzle, and by doing
so, propose potentially productive modifications to standard labor market search framework.
First, however, we discuss the current state of the literature regarding amplification puzzle.
Recent studies have attributed the amplification puzzle to different characteristics of the
standard labor market search model with only productivity shock. Shimer (2005a) and Hall
(2005) take a particular stand suggesting that the underlying wage determination mechanism
is the reason for the lack of amplification in these models. Hall (2004, 2005), Shimer (2004)
and Kennan (2006) build on this presumption and introduce wage rigidity either exogenously
or through an endogenous mechanism, such as asymmetric information.
As argued extensively in a recent paper by Mortensen and Nagypal (2006), however, wage
rigidity per se is not the reason for amplification. For instance, assuming no bargaining strength
for workers leads to constant wages that are equal to the reservation wage ( i.e. the value of
leisure).
Even in this case, the variability of labor market variables relative to productivity
are an order of magnitude smaller (Mortensen and Nagypal (2006), p.15).
Therefore, we
conclude that our formulation of the problem as a social planner’s problem, thereby ignoring
wage determination, is not crucial for understanding the amplification puzzle.
Several recent studies also aim to provide a mechanism to amplify the effects of business
cycles on unemployment and vacancies (Hagedorn and Manovskii (2006), Krause and Lubik
(2006), Nagypal (2006) and Silva and Toledo (2006)). Hagedorn and Manovskii (2006) use an
23
unrealistically high value of leisure to generate amplification, which also implies an excessive
unemployment response to a slight increase in the value of leisure (Costain and Reiter (2005)
and Hornstein, Krusell and Violante (2005)). Silva and Toledo (2006)’s result depends on a
particular constellation of parameter values for separation, hiring and training costs, that is
hard to quantify empirically.
In light of our discussion above, we first consider the possibility that the technological efficiency with which labor markets operate to match vacant positions and searching workers varies
systematically over the business cycle. In terms of the standard matching model, we question
whether the typical constant-returns matching function is structurally stable. In motivating
this approach, we take as axiomatic the notion that the matching function owes its existence
to concept of mismatch, “an empirical concept that measures the degree of heterogeneity in
the labor market across a number of dimensions, usually restricted to skills, industrial sector,
and location” (Petrongolo and Pissarides, 2001). In the absence of mismatch, jobs and workers
would match instantly. Accordingly, an exogenous reduction in labor market mismatch, given
the matching inputs of vacancies and unemployment, increases the number of matches formed,
or equivalently, raises the allocative efficiency of labor markets.
To complete the story, procyclical matching efficiency must be paired with a job separation
rate that an emerging literature has shown to be only mildly procyclical or nearly acyclical
in the aggregate—not pronouncedly procyclical as the multiple-shock model and the U.S. data
require.14 As written, the standard model does not allow for a labor force participation decision
or job-to-job transitions. In such a model, all labor force reallocation must be directed through
only two labor market states in sequence, employment and unemployment, with no possibility
of labor-force dropout and re-engagement or moving between jobs without an interceding unemployment spell. Thus, jobs must be destroyed to release workers into unemployment before
being redeployed in a recently created vacancy.
The model creates a vicious circle between
employment and unemployment.
Introducing a labor force participation decision or the possibility of job-to-job transitions
breaks the tight link between job matching and job dissolution in the standard setup by creating
an additional pool of workers to draw upon to fill newly-created vacancies. As to whether these
14
See Shimer (2005b), for instance.
24
flows are significant is an empirical question. The distinction between unemployment and notin-the labor force is relatively vague. However, existing evidence suggests significant job-to-job
movements. Nagypal (2004) and Shimer (2005b) argue that job-to-job transitions are crucial
for the cyclical worker reallocation.
Exploiting dependent interviewing methods introduced
in the CPS in 1994, Fallick and Fleischman (2004) find that these flows are large: on average
2.6% of employed workers change employers each month. Moreover, job-to-job transitions turn
out to be significantly procyclical. This particular flow cannot be analyzed by standard search
models. Thus, on-the-job search seems to be a natural avenue to pursue. Krause and Lubik
(2006), Nagypal (2006) and Tasci (2006) are examples of this approach.
Extending the pool of searching workers to include already employed workers work as an
incentive for firms to post vacancies. Following the same story provided above, consider what
happens in the standard model following a positive productivity shock.
As expected, this
increases labor demand, hence the number of vacancies posted. Higher number of vacancies
imply higher job finding rate and lower unemployment. If all new matches form through the
pool of unemployed workers, an increasingly small pool of unemployed workers dampens the
incentive for firms to post vacancies. In Nagypal (2006), information frictions generate a bias
for firms to hire employed workers, which reduces the counter effect. In Tasci (2006), however,
the underlying match heterogeneity in the form of symmetric incomplete information about the
quality of the job-worker match implies a measure of workers employed in relatively low quality
matches during expansions. These workers have the incentive to accept better quality matches
thereby, providing the additional incentive for firms to post vacancies.
6
Robustness
This section explores the robustness of the multiple-shock model experiments.
We check
whether our results vary with the elasticity of the matching function and the elasticity of labor
supply, α and γ respectively.
Recall that we have calibrated γ to be 1.25 based on Merz
(1995). Since this parameter will determine the response of household labor supply to changes
in productivity, it is important to know whether our results are dependent on a particular
choice.
Similarly we also change the parameter value of the elasticity of matching function
25
and check whether it fundamentally alters our conclusions. We present the relevant tables and
figures for this section in the Appendix.
Our alternatives for γ are 1 and 0.5.
Simulating the multiple-shock economy with these
parameter values changes virtually nothing. The model continues to generate more volatility in
labor market variables than the benchmark single-shock economy. Moreover, the counterfactual
cyclical implications remain in place with procyclical unemployment and countercyclical job
vacancies and labor market tightness. When we repeat our "perfect-fit" experiment, the model
continues to require procyclical job separation and allocative efficiency shocks.
Since a = 0.28 lies at the lower end of the matching function estimates that Petrongolo and
Pissarides (2001) provide, we consider higher alternative values. Increasing the value of α from
0.28 to 0.4 and 0.5, slightly alters our results, generating even less volatility in job vacancies
and market tightness. This is expected, given that α also determines the share of the match
surplus extracted by workers get in the decentralized manifestation of the model. As the firm’s
share falls, vacancy-creation becomes less sensitive to the underlying changes in the value of
a match. However, the shocks required for a perfect fit conntinue to exhibit procyclical job
separation and allocative efficiency. Hence, we conclude that our results remain in place for
reasonably different values of Frisch elasticity of labor supply and the elasticity of matching
function.
7
Conclusion
We have extended a basic discrete-time version of the Mortensen-Pissarides model of labor
market search to include multiple and mutually-correlated sources of exogenous variation. The
shock process comprises labor productivity (as in the basic model), job separation, and matching
efficiency. The process that governs these shocks is partly estimated using data on job finding and separation probabilities for the U.S. economy.
Although the multiple shock process
allows the model to generate additional cyclical variation in unemployment, job vacancies, and
the vacancies-unemployment ratio (as one would expect), the model does so while producing
significant counterfactual implications with mildly procyclical unemployment and significantly
countercyclical vacancies and labor market tightness. Having empirically plausible labor mar-
26
ket transition probabilities increases variability in endogenous variables, but only at the expense
of counterfactual cyclicality in unemployment and job vacancies. Our results, therefore, complement those of Shimer (2005a) and demonstrate that the model is more fundamentally flawed
than its inability to amplify shocks would suggest.
We exploit the relative degrees of freedom we have in the model allowed by the multipleshock structure, to understand the mechanics of the model that generated the empirically
implausible implications. We argue that a convincing model of labor markets should simultaneously account for high procyclical variations in job finding probabilities as well as relatively
small net employment changes. In the basic model, this is only possible through procyclical
job separation. This result leads us to conclude that the Mortensen-Pissarides model of labor
market search lacks a mechanism that would give rise to procyclical labor reallocation and
procyclical matching efficiency.
We have been silent about the true nature of the allocative efficiency and job separation
shocks. Our hope is to stimulate further research into the nature of our findings and to generate
even richer theoretical structures which will eventually give us a more thorough picture of the
fluctuations in the aggregate labor markets.
27
Appendix
Frisch Elasticity of Labor Supply
Table A-1: Multiple Shock Model ( γ = 1)
u
v
v/u
u→e
e→u ( σ)
χ
z
Standard Dev.
0.048
0.389
0.424
0.160
0.109
0.051
0.020
Autocorrelation
0.883
0.734
0.781
0.861
0.840
0.806
0.890
−0.689
−0.746
−0.839
−0.806
−0.886
0.280
0.996
0.957
0.974
0.676
−0.548
0.974
0.986
0.721
−0.535
0.997
0.858
−0.511
0.822
−0.540
Cross Correlations
u
v
v/u
u→e
e→u ( σ)
χ
−0.353
Implied Shocks from the Model
Productivity
Allocative Efficiency
Job Destruction
1
0.5
Percentage Deviations
A
0
−0.5
−1
−1.5
1950
1955
1960
1965
1970
1975
1980
Years
Figure 1:
28
1985
1990
1995
2000
2005
Table A-2: Multiple Shock Model ( γ = 0.5)
u
v
v/u
u→e
e→u ( σ)
χ
z
Standard Dev.
0.047
0.386
0.421
0.158
0.109
0.051
0.020
Autocorrelation
0.882
0.735
0.781
0.861
0.840
0.806
0.890
−0.688
−0.745
−0.838
−0.804
−0.881
0.287
0.996
0.957
0.974
0.667
−0.550
0.974
0.986
0.712
−0.538
0.997
0.852
−0.514
0.815
−0.543
Cross Correlations
u
v
v/u
u→e
e→u ( σ)
χ
−0.352
Implied Shocks from the Model
Productivity
Allocative Efficiency
Job Destruction
1
Percentage Deviations
0.5
0
−0.5
−1
−1.5
1950
1955
1960
1965
1970
1975
1980
Years
Figure 2:
29
1985
1990
1995
2000
2005
Elasticity of Matching Function
Table B-1 : Multiple Shock Model ( α = 0.4)
u
v
v/u
u→e
e→u ( σ)
χ
z
Standard Dev.
0.048
0.262
0.295
0.159
0.109
0.051
0.020
Autocorrelation
0.883
0.707
0.778
0.861
0.840
0.809
0.890
−0.650
−0.739
−0.833
−0.795
−0.883
0.24
0.992
0.946
0.966
0.649
−0.539
0.974
0.986
0.719
−0.513
0.996
0.857
−0.490
0.819
−0.531
Cross Correlations
u
v
v/u
u→e
e→u ( σ)
χ
−0.341
Implied Shocks from the Model
Productivity
Allocative Efficiency
Job Destruction
1
0.5
Percentage Deviations
B
0
−0.5
−1
−1.5
1950
1955
1960
1965
1970
1975
1980
Years
Figure 3:
30
1985
1990
1995
2000
2005
Table B-2 : Multiple Shock Model ( α = 0.5)
u
v
v/u
u→e
e→u ( σ)
χ
z
Standard Dev.
0.048
0.204
0.238
0.160
0.110
0.051
0.020
Autocorrelation
0.880
0.689
0.780
0.861
0.842
0.809
0.890
−0.616
−0.734
−0.827
−0.786
−0.878
0.225
0.987
0.936
0.959
0.633
−0.543
0.975
0.986
0.724
−0.514
0.996
0.859
−0.494
0.820
−0.542
Cross Correlations
u
v
v/u
u→e
e→u ( σ)
χ
−0.350
Implied Shocks from the Model
Productivity
Allocative Efficiency
Job Destruction
1
Percentage Deviations
0.5
0
−0.5
−1
−1.5
1950
1955
1960
1965
1970
1975
1980
Years
Figure 4:
31
1985
1990
1995
2000
2005
References
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Review, 86(1): 112—132.
[2] Blanchard, Olivier Jean, and Peter Diamond. 1990. “The Cyclical Behavior of the Gross
Flows of U.S. Workers.” Brookings Papers on Economic Activity, 1990(2): 85—143.
[3] Chari, V.V., Patrick Kehoe, and Ellen R. McGratten. Forthcoming. "Business Cycle Accounting.” Econometrica.
[4] Christiano, Lawrence. 2002. "Solving Dynamic Equilibrium Models by a Method of Undertermined Coefficients." Computational Economics, 20(1-2): 21-55.
[5] Cole, Harold L., and Richard Rogerson. 1999. “Can the Mortensen-Pissarides Matching
model Match the Business Cycle Facts?” International Economic Review, 40(4): 933-959.
[6] Costain, J. S., and M. Reiter. 2005. "Business cycles, Unemployment Insurance and the
Calibration of the Matching Models." Universitat Pompeu Fabra Economics Working Paper #875.
[7] Den Haan, W., Garey Ramey, and Joel Watson. 2000. “Job Destruction and Propagation
of Shocks.” American Economic Review, 90(3): 482—498.
[8] Davis, Steven, and John Haltiwanger. 1992. “Gross Job Creation, Gross Job Destruction,
and Employment Reallocation.” Quarterly Journal of Economics, 107(3): 819—863.
[9] Davis, Steven, John Haltiwanger, and Scott Schuh. 1996. Job Creation and Destruction.
Cambridge: MIT Press.
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Market: The Complete Picture of Gross Worker Flows." Board of Governors of the Federal
Reserve System Finance and Economics Discussion Series 2004-34.
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[13] Hall, Robert E. 2005. “Employment Fluctuations with Equilibrium Wage Stickiness.”
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Models: A Quantitative Assessment." Federal Reserve Bank of Richmond Working Paper.
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[16] Ingram, B. F., Narayana Kocherlakota and N. Eugene Savin. 1994. “Explaining Business
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[18] Krause, M. U. and T. A. Lubik. 2006. "On-the-Job Search and the Cyclical Dynamics of
the Labor Market." Federal Reserve Bank of Richmond Working Paper.
[19] Merz, Monika. 1995. “Search in the Labor Market and the Real Business Cycle.” Journal
of Monetary Economics, 36(2): 269—300.
[20] Merz, Monika. 1999. "Heterogeneous Job-Matches and the Cyclical Behavior of Labor
Turnover." Journal of Monetary Economics, 43(1): 91-124.
[21] Mortensen, Dale and Christopher A. Pissarides. 1994. “Job Creation and Job Destruction
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[22] Mortensen, Dale and Eva Nagypal. 2006. “More on Unemployment and Vacancy Fluctuations.” Northwestern University Working Paper.
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Job-to-Job Transitions" Northwestern University Working Paper.
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Hire the Unemployed?" In Structural Models of Wage and Employment Dynamics, ed. H.
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33
Shocks with JFP and SP
0.4
Productivity
Allocative Efficiency
Job Destruction
0.3
Percentage Deviations
0.2
0.1
0
−0.1
−0.2
−0.3
1950
1960
1970
1980
Years
1990
2000
2010
Figure 5: Estimated shocks using job finding (JFP) and separation probabilities (SP).
34
Actual vs Implied Unemployment
0.14
Actual Unemployment
Implied Unemployment
0.12
Unemployment
0.1
0.08
0.06
0.04
0.02
1950
1960
1970
1980
Years
1990
2000
Figure 6: Actual Unemployment versus Implied Unemployment
35
2010
Implied Shocks from the Model
Productivity
Allocative Efficiency
Job Destruction
1
Percentage Deviations
0.5
0
−0.5
−1
−1.5
1950
1955
1960
1965
1970
1975
1980
Years
1985
1990
Figure 7: Shocks required for perfect fit.
36
1995
2000
2005