Discussion Paper No. 734

TECHNOLOGICAL PROGRESS,
ON-THE-JOB SEARCH,
AND UNEMPLOYMENT
Hiroaki Miyamoto
andYuya Takahashi
The 11th ISER-Moriguchi Prize (2009)
Awarded Paper

March 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Technological Progress, On-the-Job Search, and Unemployment
Hiroaki Miyamotoy

Yuya Takahashi

University of Wisconsin-Madison

University of Wisconsin-Madison

First Draft: May 2008
This Version: March 2009

Abstract
This paper studies the impact of long-run productivity growth on job …nding and separation rates, and thus the unemployment rate, using a search and matching model. We
incorporate disembodied technological progress and on-the-job search into the endogenous
job separation model of Mortensen and Pissarides (1994). The incorporation of on-the-job
search allows faster growth to reduce unemployment by decreasing the separation rate andinducing job creation. We demonstrate that introducing on-the-job search substantially improves the ability of the Mortensen and Pissarides model to explain the impact of growth onunemployment. Our quantitative analysis shows that our model increases the magnitude of
the negative impact of growth on unemployment compared to the standard matching modelwith disembodied technological progress.
Keywords: Growth; Unemployment; Search-matching model; On-the-job search
JEL Classi…cation: E24;J64;O40

We are grateful to Steven Durlauf, John Kennan, Rasmus Lentz, Yoshimasa Shirai, and Shintaro Yamaguchi
for their invaluable comments and suggestions. We also thank Rebecca Lessem, John Morrow, Michael Rapp,and Jiao Shi and seminar participants at ISER at Osaka University for their helpful comments. This paper wasawarded the 11th ISER-Moriguchi Prize. All remaining errors are ours.
yCorresponding author. Mailing Address: University of Wisconsin-Madison, Department of Economics, 1180
Observatory Dr, Madison, WI 53706-1393, USA. E-mail: hmiyamoto@wisc.edu.

11Introduction

Over the years, economists have been interested in the relationship between productivity growth
and the unemployment rate. The simultaneous slowdown of productivity growth and the risein unemployment in OECD countries in the latter half of the 1970s has led many economists
to believe that there is a close connection between these trends. Several empirical studies …nd
a strong negative impact of productivity growth on unemployment.1 The search and matchingmodel with disembodied technological progress (DTP henceforth) explains the negative impactof growth on unemployment by focusing on job creation. However, data from the US shows thatwhile the job …nding rate is positively correlated with the productivity growth rate, there is anegative correlation between the match separation rate and the productivity growth rate. Thus,
productivity growth a¤ects unemployment through not only job creation but also job separation.
Therefore, when we examine the impact of growth on unemployment, it is necessary to use amodel that endogenously determines both the match creation and separation margins. However,the standard search and matching model with DTP has downplayed the role of job separation.
In most models, the match separation rate is exogenous and assumed to be constant over time.
Furthermore, Prat (2007) qualitatively demonstrates that the impact of DTP on unemployment
is ambiguous because of the so called outside option e¤ect when job separation is endogenously
determined in a matching model. He also shows that under standard parameter values, fasterproductivity growth increases unemployment because of the outside option e¤ect. Thus, the predictions of the endogenous job separation model with DTP are not consistent with the empiricalevidence.
This paper demonstrates that the incorporation of on-the-job search reduces the size of theoutside option e¤ect. In addition, we develop an endogenous job separation model with on-thejob search that can explain the above-mentioned empirical facts. Empirical studies document
the pervasive job-to-job ‡ows in the U.S. economy, suggesting that this particular ‡ow is an
important element in understanding labor market dynamics (Fallick and Fleischman (2004) andNagypál (2005)). Thus, the incorporation of on-the-job search into the standard search model isa natural extension, especially in light of empirical …ndings.
1

See Bruno and Sachs (1985), Ball and Mo¢ tt (2001), Muscatelli and Tirelli (2001), Staiger, Stock and Watson(2001), Tripier (2006) and Pissarides and Vallanti (2007).

2

Incorporating on-the-job search into the Mortensen-Pissarides matching model with DTP
reduces the size of the outside option e¤ect, and the augmented model can generate the empiricallyconsistent negative correlation between productivity growth and the match separation rate. In an
endogenous job separation model, currently employed workers have better outside opportunities
and so ask for higher wages when productivity growth increases. Since this outside option e¤ect
reduces the expected lifetime of jobs, faster productivity growth is more likely to increase thematch separation rate. However, in our model with on-the-job search, bene…ts from on-the-job
search are shared between the …rm and the worker through the wage. In other words, the worker
gets a lower wage than he would be paid if he did not search on the job. This makes otherwise
unproductive jobs to be productive enough to survive, leading to decreased separation when therate of productivity growth rises.
Furthermore, on-the-job search generates more vacancies by accelerating the reallocation ofworkers when the productivity growth rate rises. When growth accelerates, the search e¤ort of
employed workers rises, increasing the value of newly created jobs through recruitment. This
induces more job creation and leads to a lower unemployment rate. Thus, the incorporation of
on-the-job search gives rise to a new channel through which faster productivity growth reduces
unemployment.
On-the-job search therefore improves the model’s performance through the two above-mentioned
channels. Qualitatively, in our model, faster productivity growth reduces the separation rate andincreases the job …nding rate, leading to lower unemployment. Furthermore, introducing on-thejob search substantially improves the ability of the Mortensen and Pissarides model to accountfor the size of the impact of growth on unemployment. Recently, Pissarides and Vallanti (2007)
demonstrate that the matching model with DTP fails to explain the magnitude of the impact ofgrowth on unemployment. Speci…cally, in the standard matching model with DTP and exogenous job separation, a 1% decrease in the growth rate increases unemployment by about 0.01%,
assuimg Nash bargaining over wages. This is far from the estimated magnitude in the empiricalliterature. Blanchard and Wolfers (2000) estimate that a 1% decline in the growth rate leads to
0.25%-0.7% increase in the unemployment rate. Pissarides and Vallanti (2007) …nd the e¤ect tobe 1.3% to 1.5%. After calibrating our model, we …nd that a one percentage point decline inproductivity growth increases the unemployment rate by 0.24%. Although the magnitude of the3e¤ect is still lower than in the literature, the sign of the e¤ect is empirically consistent, and we…nd a larger magnitude than in standard models.
Related Literature:

The search and matching theory predicts that the impact of productivity

growth on unemployment depends on the extent to which new technology is embodied in newjobs (Mortensen and Pissarides (1998), Pissarides and Vallanti (2007)). The standard search
and matching model with disembodied technological progress predicts that a faster rate of productivity growth reduces unemployment through the so called capitalization e¤ect (Pissarides(2000, chapter 3)). On the other hand, in the model with embodied technological progress, fasterproductivity growth increases unemployment through creative destruction (Aghion and Howitt(1994, 1998), Postel-Vinay(2002)). Motivated by the empirical evidence that productivity growthdecreases the unemployment rate, Pissarides and Vallanti (2007) demonstrate that totally disembodied technology is necessary for the model to match empirical evidence. In this paper, wefollow Pissarides and Vallanti (2007), and study the e¤ect of DTP on labor market dynamics.
Prat (2007) studies the impact of DTP on unemployment in a matching model with endogenous job separation. He demonstrates that under plausible parameter values, faster productivity
growth increases the unemployment rate due to the outside option e¤ect. By introducing on-thejob search, we …nd that productivity growth decreases unemployment in a search and matchingmodel with endogenous job separation.
Our paper focuses on the impact of productivity growth on the labor market using on-the-job
search in the search and matching model. A number of studies develop a matching model with onthe-job search. Pissarides (1994) and Krause and Lubik (2006) consider a matching model with
two types of jobs (good and bad) to study the search activity of employed workers in bad jobs. In
this paper, we allow for a continum of job quality instead of just two discrete types. Mortensen(1994) introduces on-the-job search into the endogenous job separation model of Mortensen and
Pissarides (1994). Mortensen (1994) demonstrates that on-the-job search helps to explain thenegative correlation between job creation and destruction over the business cycle. In his model,
workers employed by lower productivity …rms search for better jobs. Thus, employed searchvaries through changes in the number of job seekers, not through the intensity of search. The
set up of Nagypál (2007) is the most similar to this paper. In her model, heterogeneity in job
amenity motivates search by employed workers, while in our model, heterogeneity in productivity
4plays this role. Furthermore, none of these papers discusses the impact of productivity growth
on labor market dynamics or considers the long-run properties of worker reallocation, which is
the main focus of this study.
Following Miyamoto (2008), the paper provides empirical evidence on the long-run relationships between productivity growth, the unemployment rate, and rates of worker ‡ows in the U.S.
labor market. Recently, several studies investigate the contributions of in‡ows and out‡ows tounemployment dynamics. Shimer (2007) and Fujita and Ramey (2007) study the cyclical behavior of job …nding and separation rates. They also measure the contributions of ‡uctuations in
job …nding and separation to the variability of unemployment in the U.S. economy. Similarly,
Petrongolo and Pissarides (2008) study the contribution of in‡ows and out‡ows to unemployment
in European countries. While they focus on labor market dynamics over the business cycle, inthis paper we investigate its long-run properties.
The remainder of the paper is organized as follows. Section 2 presents salient features of the
U.S. aggregate labor market in the long-run, and discusses the relationship between productivity
growth and the labor market. Section 3 describes the theoretical model. We develop a generalized
Mortensen and Pissarides model with on-the-job search. In Section 4 we characterize the steadystate equilibrium of the model. In Section 5, we calibrate the model parameters and present
the results of quantitative comparative statics exercises. Section 6 discusses the sensitivity of
the numerical results to our choice of parameter values. We also assess whether or not theresults of our model crucially depends on the incorporation of on-the-job search. Conclusions
and suggestions for future research are presented in Section 7.

2

U.S. labor market facts

In this section, we present some of the salient features of the U.S. aggregate labor market in
the long-run. We use this to discuss the relationship between productivity growth and labormarket variables. We focus on productivity growth, g; and three labor market variables: theunemployment rate u, the job …nding rate f , and the separation rate s.2
Labor productivity growth is measured by the …rst di¤erence of logged labor productivity.
We use real output per person in the non-farm business sector as labor productivity. The Bureau
2This section is based on Miyamoto (2008).

5

of Labor Statistics (BLS) constructs this quarterly time series as part of its Major Sector Productivity and Costs program. Using output per hour or total factor productivity as a measureof labor productivity yields similar results, but we use this series because it is a natural way toconsider productivity in the standard search and matching model. Unemployment rate is thequarterly average of seasonally adjusted monthly data constructed by the BLS using the Current
Population Survey (CPS) data.
Dynamics of the unemployment rate are determined by the underlying ‡ows into and outof unemployment, particularly by the rates at which workers match with and separate fromjobs. In this paper, we de…ne the job …nding rate as the rate of transition from unemployment toemployment, and the separation rate as the rate of transition from employment to unemployment.
Shimer (2007) uses short-term unemployment data and total unemployment data to pin downthese rates. Following Shimer’s (2007) time aggregation correction, we measure job …nding andseparation rates from the CPS over the 1948Q1-2005Q1 period.
Since our focus is the long-run relationship between productivity growth, unemployment rate,
and rates of worker ‡ows, we use band-pass …ltering to smooth the data. The band-pass …lter
is a linear …lter which retains the cyclical components of each series within a speci…c band of
frequency and removes other components. By using the band-pass …lter, we can isolate the longterm components of productivity growth and labor market variables. Let yt be a quarterly timeseries, and let yt denote its trend. Following Staiger et al. (2001), yt is estimated by passing
yt through a two-sided low pass …lter, with a cuto¤ frequency of 15 years.3 Essentially, thisestimates yt as a long two-sided weighted moving average of yt with weights that sum to one.
Estimates of the trend at the beginning and end of the sample are obtained by extending theseries with autoregressive forecasts and backcasts of yt , constructed from an estimated AR(4)model for the …rst di¤erence of yt .
Figure 1 presents quarterly time series data and their estimated trends for (a) labor productivity growth, (b) the unemployment rate, (c) the job …nding rate, and (d) the separation rate.
Table 1 summarizes the relationship among smoothed series of these four variables.4
3

We also adopt the de…nition of the business cycle as the cyclical components between 1.5 years and 8 years,

following Baxter and King (1999) and Stock and Watson (1999). When we use these limits as the de…nition of thebusiness cycle, we get similar results.
4
For comparison purpose, we also use the HP …ltering to smooth the data and get similar results. However, using

6

Table 1: Summary statistics: Correlation matrix
g

u

f

s

g

1

-0.6590.310-0.842

u-1-0.841

0.854

f

-

-

1

-0.447

s

-

-

-

1

Note: Correlation between the productivity growth rate (g), the job …nding rate(f ), the separation rate (s), and unemployment rate (u). All series are smoothed withband-pass …lter with a cuto¤ frequency of 15 years. Sample covers 1948Q1-2005Q1.
Figure 1 and Table 1 show that productivity growth and the unemployment rate have anegative correlation. In Figure 2, we reproduce the smoothed series for these two variables.
The smoothed series move closer together until the early 1960s, then apart during the 1970s,
and then come slowly back to near their starting levels. The correlation between these twoseries during the sample period is -0.659. Thus, there is a strong negative relationship between
productivity growth and the unemployment rate at low frequencies. This …nding con…rms theempirical evidences from previous studies5 , showing a negative relationship between productivity
growth and unemployment, as predicted by Pissarides (2000) and Pissarides and Vallanti (2007).
Figures 1-(c) and 1-(d) show smoothed series for job …nding and separation rates. The job…nding rate is large, averaging 27 percent during the sample period. In contrast, the separation
rate is low and averaged 1.5 percent. The separation rate moved upward in the 1960’s and 1970’s
and downwards afterwards. It is important to note that the separation rate moves positively
with the unemployment rate and negatively with productivity growth. The correlation between
the HP …lter is not suitable for the analysis of the long run components of an economic series. The HP …lter is bestinterpreted as a high-pass …lter isolating frequencies of 8 years and higher in economic data and is not intended forother frequencies. Moreover, although the HP …lter produces cyclical components that are covariance stationary
for raw series that are integrated up to order four, the trend component of it re‡ects the non-stationarirty of theraw data.
5
Staiger et al. (2001) …nd a strong negative relationship between productivity growth and the unemployment
rate in the U.S. economy. Muscatelli and Tirelli (2002) …nd a negative correlation between these two variables
for Japan, Germany, Italy, France, and Canada. Pissarides and Vallanti (2007) also …nd a negative relationshipbetween total factor productivity (TFP) and unemployment in the US, Japan and European countries.

7the separation rate and unemployment rate is 0.854. The correlation between the separation
rate and productivity growth is -0.842. On the other hand, the job …nding rate is negatively
correlated with the unemployment rate and the correlation is -0.841. We can also see the positive
relationship between the job …nding rate and productivity growth. Their correlation is 0.310.
We have seen that job …nding rates co-move negatively with the unemployment rate and theseparation rate co-move positively with the unemployment rate in the long-run. Now we quantify
the contributions of separation and job …nding rates to overall unemployment variability in thelong-run following Shimer (2007) and Fujita and Ramey (2007). To analyze how hazard ratesa¤ect unemployment variability, Shimer (2007) and Fujita and Ramey (2007) approximate theunemployment rate using the theoretical steady-state value associated with the contemporaneousjob …nding and separation rates. Thus,ut 'stst + ftu~t ;where ut is the unemployment rate, st is the rate of transition from employment to unemployment(i.e., the separation rate), and ft is the rate of transition from unemployment and employment(i.e., the job …nding rate). Let f and s denote the average value of ft and st during the sample
period. Then we compute the hypothetical unemployment ratesuFtsand uSts + ftst
st + fas measures of the contributions of ‡uctuations in the job …nding and employment exit rates tooverall ‡uctuations in the unemployment rate.
Figure 3 plots hypothetical unemployment rates u~t , uFt and uSt together with the unemployment rate. Figure 3 shows that both the job …nding rate and the separation rate tend to movewith the unemployment rate.
To quantify the contribution of separation and job …nding rates to overall unemployment
variability, we look at the co-movement of long-run components of the data. Over the sample
period, the covariance of the cyclical component of ut and uFt accounts for about half of thevariance of the cyclical component of ut . Similarly, cyclical ‡uctuations in the separation rate
explain 47 percent of the ‡uctuation in the unemployment rate. This suggests that the job …nding
rate and the separation rate account for roughly similar proportions of overall unemployment
variability in the long-run.
8

Since both job …nding and separation rates contribute to variability in unemployment in thelong-run, the fact that the job …nding rate is positively correlated with productivity growth
seems to support the capitalization e¤ect theory of Pissarides (2000). Furthermore, the negative
correlation between the separation rate and productivity growth suggests a new channel for
productivity growth to a¤ect unemployment, which has not been pointed out in the literature.

3

The model

We consider a search and matching model with endogenous job separation, in the spirit ofMortensen and Pissarides (1994). We allow for search by employed workers. Jobs di¤er by idiosyncratic productivity level, and the heterogeneity in productivity motivates on-the-job search.
Furthermore, in order to study the impact of productivity growth on labor market dynamics, we
introduce disembodied technological progress, as in Pissarides (2000) and Pissarides and Vallanti(2007).
There is a large measure of ex-ante identical …rms and a unit measure of ex-ante identicalworkers. Both workers and …rms are in…nitely lived and maximize the present discounted valueof income with discount rate r. Time is continuous.
Firms are free to enter the market and post a vacancy at ‡ow costt in order to recruita worker. If a …rm succeeds in recruiting a worker, the …rm starts to produce output and theworker earns an endogenous wage wt . An employed worker is allowed to search for other jobs.
The …rm-worker pair continues to produce output until the match is destroyed by a negative
productivity shock or if the worker …nds a better job. An unemployed worker receives a ‡owutility zt and searches for a job.
Both unemployed and employed workers can choose to search for a new job at a ‡ow cost ct (e),where e is the search e¤ort and ct (e) is a strictly increasing, strictly convex, twice continuouslydi¤erentiable function with ct (0) = c0t (0) = 0. Workers exerting search e¤ort e encounter new
job opportunities at the Poisson rate q( )e, where q( ) is the matching rate per unit of searchintensity that will be de…ned later.
Production technology Production takes place when one frim is matched with one worker.
Let the output of each …rm at time t be given by pt x, where pt is a general productivity parameter9

which is common to all producing jobs, and x is an idiosyncratic productivity speci…c to eachjob. Suppose that the leading technology in the economy is driven by an exogenous invention
process that grows at the rate g < r, so
pt = p0 exp(gt)where p0 > 0 is some initial productivity level which is normalized to be one.
The initial value of idiosyncratic productivity is drawn from a distribution F : [xF ; xF ] !
[0; 1]: Subsequently, let fxt g be a jump process characterized by arrival rate

and a distributionof new realizaitons G : [xG ; xG ] ! [0; 1]:
Matching technology

There is a single matching function determines the number of meetings

between workers and …rms, as a function of the total amount of search e¤ort of workers, e, andthe number of vacancies posted, v:
mt = m(vt ; et )The matching function m(v; e) is continuous, twice di¤erentiable, increasing in its arguments andhas constant returns to scale. The meeting rate per unit of vacant jobs isq( t ) =1m(vt ; et )= m 1;
vt
t

where
vt
et

t

is labor market tightness at time t. Then, the meeting rate per unit of search e¤ort for workers
ist q( t ) =m(vt ; et )=m
etRendering the growth model stationary

vt
;1 :
et

We focus on the steady state. This corresponds

to a balanced growth path where the economy grows at the rate of disembodied technologicalprogress g. To make the model stationary, we assume that all exogenous variables grow at therate of disembodied technological progress g.6 Thus, we de…ne two positive exogenous parameters
6

In the literature, in order to ensure the existence of a balanced growth path, usually all the exogenous para-

meters are assumed to follow the pace of productivity growth. See, for example, Mortensen and Pissarides (1999)
and Pissarides and Vallanti (2007).

10

z andsuch that zt = pt z andt = pt. Furthermore, the search cost function can be rewritten

as ct (e) = pt c(e).

3.1Value functions

The values of workers and …rms at time t are described by a series of Bellman equations. We
start with the worker’s side. Let the value for an employed worker in a job with idiosyncratic
productivity x be Wt (x) and the value of an unemployed worker at time t be Ut .
The value of an employed worker in a job with productivity x is characterized by the following
Bellman equation:Z

rWt (x) = max wt (x) pt c(e) +max Wt (x0 ); Ut
Wt (x) dG(x0 )e 0Z x_ t (x) :+e t q( t )IfW (x0 ) > W (x)g Wt (x0 ) Wt (x) dF (x0 ) + W(1)x

In equation (1), If g is an indicator function that equals one if its expression is true and equals
zero otherwise. The value of an employed worker in a job with productivity x is determined byseveral factors. The worker receives wage w(x): The match draws a new value of idiosyncratic
productivity at rate , in which case the worker loses the current asset value W (x) and gains
the asset value associated with working at the new productivity level x0 or being unemployed,
whichever is greater. Moreover, the worker optimally chooses his search intensity at cost pc(e)and obtains the bene…t of meeting new job opportunities at rate e q( ). If the worker encounters
a new …rm, he accepts any job that has a higher asset value W (x0 ) than the current asset value
W (x). Finally, the asset value of a match is expected to change over time due to the exogenous
technological progress.
The value of an unemployed is
rUt = max pt ze

pt c(e) + e t q( t )Z(max [Wt (x); Ut ]Ut ) dF (x) + U_ t :An unemployed worker also chooses his search e¤ort e at cost pt c(e).
We now turn to the side of a …rm. Let

t (x) denote the asset value for a …rm with a …lled job

with idiosyncratic productivity x. Denote the optimal search intensity of an employed worker ina job with productivity x by e(x). The value of a …lled job with an idiosyncratic productivity x

11satis…esrZ

wt (x) +max t (x0 ); Vt
Z x
IfW (x0 ) > W (x)g [Vt+e(x) t q( t )t (x)= pt xt (x)dG(x0 )

0_t (x)] dF (x ) + t (x);(2)

xwhere Vt is the value of posting a vacancy. A …rm with a …lled job receives ‡ow revenues px w(x),which is the productive output of the match minus the wage paid to the worker. The match draws
a new value of idiosyncratic productivity at rate . Facing the changed productivity, the …rm(x0 ) is larger than the value of a vacant job. The job may be
Rxdestroyed by the quit of the worker at rate e(x) q( ) x IfW (x0 ) > W (x)gdF (x0 ), in which casedecides to continue producing if

the …rm loses its asset value. Finally, the asset value of a match is expected to change over time
due to exogenous technological progress.
The …rm that has a job with valuegt (x) at time t expects to make a capital gain of dt =dt =

on it. The same holds for the value of an employed worker Wt (x) and an unemployed worker

Ut , where the capital gain is gW (x) and gU , respectively. But the value of a vacant job Vt ,because it is zero by the free entry condition, does not change. Replacing the capital gain by its
steady-state value, the above Bellman equations can be rewritten as follows:(rZg) W (x) = max w(x) pc(e) +max W (x0 ); U
W (x) dG(x0 )e 0Z xIfW (x0 ) > W (x)g W (x0 ) W (x) dF (x0 ) ;+e q( )

(3)

x

(r

g) U = max pz
epc(e) + e q( )Z(max [W (x); U ]U ) dF (x)

(4)

and
(r

Z

w(x) +max (x0 ); V(x) dG(x0 )Z x+e(x) q( )IfW (x0 ) > W (x)g [V(x)] dF (x0 ):g) (x) = px

(5)

x

3.2

Wage determination

Wages are determined by the sharing of the surplus from the match, where the worker and the…rm receive shares

and 1, respectively. It is assumed that wages can be revised continuously

12at no cost, so the long-run contracts are ruled out.7 Furthermore, we assume that matches cannot
be recalled. Note that the outside option of the worker is unemployment. At each instant, …rst
the worker decides the level of search intensity, in anticipation of the wage outcome, and thenthe surplus sharing takes place.
The key object for characterizing the model is the match surplus function. Let S(x) be thejoint gross return from a match with job-speci…c productivity x. Then the surplus function isS(x) =(x) + W (x)

U

V:

(x)

V

= (1

)S(x)

W (x)

U

=

(6)

Surplus sharing implies

S(x)Because of the timing at the model and the nature of bargaining, the wage cannot reduce thelikelihood that a worker quits. Therefore, the non-convexity of the Pareto frontier discussed in
Shimer (2006) does not arise in our model. In this model, at each instance a worker decides how
much to search, in anticipation of the wage outcome, and then the surplus-sharing takes place.
Thus, the surplus-sharing rule does not allow for the wage to in‡uence the search behavior of
the worker. This allows us to determine the wage as an outcome of Nash bargaining, since thefeasible payo¤ set is convex.8
Using of equations (3),(5), and (6), we obtain the equation characterizing S(x); as follows:(r +g) S(x) = px(rg) [U + V ]+e(x) q( )

Z x
x

7

pc(e(x)) +Z xmax[S(x0 ); 0]dG(x0 )xIfW (x0 ) > W (x)gS(x0 )

S(x) dF (x0 ):(7)Thus, even if an employed worker could start negotiations with a new employer before resigning from thecurrent job, this would not a¤ect the equilibrium outcome. The new employer would immediately renegotiate thewage once the worker breaks the relationship with the previous employer.
8
It may be helpful to consider a discrete time version of the model to understand this argument. In the discrete
time model, the search behavior of a worker is governed by returns to search that accrue in the next period. Thus,to reduce the probability that the worker quits, the …rm should commit to higher wages in the future. However,with continuous renegotiation, such a commitment cannot be made. Therefore, under these assumptions, the payo¤
set is convex, and surplus sharing is equivalent to the Nash bargaining solution.

13

Lemma 1 S(x) is strictly increasing in x. As a result, W (x) and J(x) are strictly increasing inx:
Since W ( ) is strictly increasing, the acceptance decisions of an unemployed worker has thereservation property. Thus, an unemployed worker accepts any job with productivity xR,where R is de…ned by
W (R) = U , S(R) = 0:
Since the surplus function S( ) is strictly increasing, the …rm and the worker will choose to
adopt a reservation policy, i.e., they will continue their match if S(x)0 but stop if S(x) < 0:Thus, a separation takes place at x = R. Note that the reservation productivity at the time thematch is formed is the same as the one at match dissolution, even though the initial distributionsof productivity di¤er.

3.3

The search choice

Since W (x) is strictly increasing in x, an employed worker accepts all new jobs with a higher
initial productivity than his current job. Thus,
Z xx

IfW (x0 ) > W (x)g W (x0 )W (x) dF (x0 ) = 0 for x

x;

and
Z x
x

0

0

0

IfW (x ) > W (x)g W (x )W (x) dF (x ) =Z x

W (x0 )

W (x) dF (x0 ) for x < x:xThen, equation (3) can be rewritten as(rZg) W (x) = max w(x) pc(e) +max W (x0 ); Ue 0Z x+e q( )
W (x0 ) W (x) dF (x0 ) :W (x) dG(x0 )(8)xThe optimal search intensity of an employed worker is found by using the equation above. The…rst-order condition of (8) is0pc (e(x)) = q( )Z x

0W (x )

0

W (x) dF (x ) = q( )

x

Z x
x

14

S(x0 )

S(x) dF (x0 ):(9)Because the search cost function is strictly convex and S(x) is strictly increasing, the optimal
search e¤ort by employed workers is strictly decreasing in x for x < x. Furthermore, by theconvexity of c( ) and c0 (0) = 0, e(x) = 0 for xx.

The value function of an unemployed worker can be rewritten as
Z x[W (x) U ] dF (x) :(r g) U = max pz pc(e) + e q( )
e

(10)

R

Denote the optimal search intensity of an unemployed worker by eu . Then, the …rst-order condition of (10) yields
0

pc (eu ) + q( )Z x

[W (x)

R

0

U ] dF (x) = 0 , pc (eu ) =

q( )

Z x

S(x)dF (x):

(11)

R

Under the assumption that the cost of search e¤ort is the same whether employed or not, acomparison of equations (9) and (11) implies that the optimal search e¤ort when unemployedequals the search e¤ort when employed at x = R, i.e.,
eu = e(R):

3.4

Vacancy creation

The value of posting a vacancy isrV =p + q( )

Z x

[ (x)

V ] (x)dF (x);

x

where

(x) is the probability that a searching worker accepts a job with productivity x: This is

the ratio of search e¤ort by workers who are willing to accept a match with initial productivity
x to the total amount of search e¤ort e exerted by all workers.
The measure of employed workers in jobs with an idiosyncratic productivity less then or equal
to x is denoted by H(x). Thus, H(x) is the distribution of employed workers.
The total amount of search e¤ort e exerted by all workers isZ x
e = ueu + (1 u)e(x0 )dH(x0 ):R

Then, the acceptance probability will be8< e 1 ueu + (1
(x) =
:

Rx
u) R e(x0 )dH(x0 )
0

15

if x

R

if x < R

Free entry implies that the value of a vacancy is zero in equilibrium. Thus,

V = 0:

3.5

Labor market dynamics

The steady-state unemployment rate is determined by equating the ‡ow into unemployment withthe ‡ow out of it. The ‡ow into unemployment is equal to the number of employed workers who
work in jobs that get hit by an idiosyncratic shock below the reservation value. Therefore the‡ow into unemployment is given by G(R)(1u). The ‡ow out of unemployment is equal to thenumber of unemployed workers who …nd jobs. This is q( )eu [1F (R)] u. Then, the evolutionof unemployment is given by
u_ = G(R)(1u)

q( )eu [1

F (R)] u;

and its steady-state value is

u=

G(R)
G(R) + q( )eu [1F (R)]:To close the model, we need to derive the stationary distribution of employed workers acrossproductivity. Denote the measure of employed workers in jobs with an idiosyncratic productivity less than or equal to x by H(x). The ‡ow into this subset consists of unemployed andemployed workers. The ‡ow of unemployed workers who obtain a job with productivity x or lessis q( )eu [F (x)F (R)] u. Because of the idiosyncratic productivity shocks, employed workers

who originally work in jobs with a productivity higher than x may move to this subset, whichis equal to[G(x)G(R)] [H(x)H(x)]. Thus, the sum of them is the total ‡ow into the setof employed workers in jobs with productivity x or less. The ‡ow out of this subset consists ofthree ‡ows. First is those who lose their jobs after a productivity shock, equal to G(R)H(x).
Second is the ‡ow of those whose jobs’ productivity becomes higher than x after the arrival ofa productivity shock; this is equal to H(x)[1G(x)]. Third is the ‡ow of those who …nd jobs

that have a higher productivity than their current jobs. Since the rate at which a worker searches
depends on this current productivity, the ‡ow that …nds a job with productivity higher than x is
Z x
q( ) [1 F (x)]e(x0 )dH(x0 ):R

16Then, the stationary distribution of employed workers across idiosyncratic productivity is givenbyq( )eu [F (x)=F (R)] u + [G(x)G(R)H(x) + H(x)[1G(R)] [H(x)G(x)] + q( ) [1

H(x)]
Z x
e(x0 )dH(x0 ):F (x)](12)R

The measure of job-to-job transition is given by
Z x
q( )e(x)[1JJ = (1 u)

F (x)]h(x)dx;

Rso the quit rate isQ=Z xq( )e(x)[1F (x)]h(x)dx:

R

Since the ‡ow of employment-to-unemployment transition is G(R)(1u), the separation rate,

de…ned as the ‡ow rate from employment to unemployment, is G(R). Then, the job destruction
rate is also G(R).
The ratio of job-to-job to employment-to-unemployment transition is given by
Rx
F (x)]h(x)dx
R q( )e(x)[1:G(R)
Note that not all job separation implies layo¤s because of on-the-job search. Matches are destroyed by not only productivity shocks but also when employed workers …nd another job.
The stationary equilibrium is characterized by a constant unemployment rate, vacancy rate,
and a constant distribution of employed workers across productivities.
De…nition 2 A stationary (stationarized balanced growth) equilibrium is a list of unemploymentu, vacancy rate v, total search intensity e, asset values f (x); W (x); V; U g, wage equation w(x),search intensity of unemployed workers eu , search intensity of employed workers e(x), and a
distribution of employed workers H(x) such that
W ( ) and U are the value of employed workers and unemployed workers making optimal
searching and matching decisions, given u; v; e; w( ) and H( ). eu and e( ) are the corresponding optimal search e¤ ort policies.
( ) and V are the value of a …lled job and a vacancy for …rms making optimal vacancy
creation decisions, given u; v; e; w( ),e( ) and H( ):17Wages are determined by Nash bargaining, where workers and …rms get sharesand 1of the surplus, resepectively, given e(x):There is free entry of vacancies; thus V = 0.
The distribution H( ), the unemployment rate u, the vacancy rate v, and the total searchintensity e are consistent with decisions of agents in the economy.

4

Characterization

Because S(x) and W (x) are strictly increasing and the free entry condition V = 0, equation (7)can be rewritten as(r +g) S(x) = px

(r

g) U
Z x
+e(x) q( )pc(e(x)) +Z x

S(x0 )dG(x0 )R

S(x0 )

S(x) dF (x0 )

(13)

x

By substituting the value of an unemployed worker into equation (13), we haveZ x(r +g) S(x) = p (x z) + pc(eu ) eu q( )S(x0 )dF (x0 ) pc(e(x))
R
Z x
Z x
0
0
+
S(x )dG(x ) + e(x) q( )S(x0 ) S(x) dF (x0 ):
R

(14)

x

Evaluating (14) at x = R and using the optimal separation rule S(R) = 0, we haveZ x
0 = p(R z) +S(x0 )dG(x0 )

(15)

R

This equation implies that the di¤erence between R and z comes from the option value of changing
idiosyncratic productivity.
From (14) and (15), we obtain(r +g) S(x) = p (xR)

pc(e(x)) + pc(e(R)) + e(x) q( )Z xe(R) q( )S(x0 )dF (x0 )Z xS(x0 )

S(x) dF (x0 )xRDi¤erentiating with respect to x on both sides of the surplus equation and by the envelope
theorem gives(r ++ e(x) q( ) [1

F (x)]g) S 0 (x) = p + (1) q( )S(x) e(x)f (x)e0 (x) [1

F (x)] :(16)

18

Given a search intensity function e(x), this is a …rst-order linear di¤erential equation in S(x).
Note that since the di¤erential equation has variable coe¢ cients, it does not generally have aclosed-form solution.
Since e(x) = 0 for xx, equation (16) yields

S 0 (x) =

p
r+

g

:

Totally di¤erentiating the optimal search intensity condition (9) yields
e0 (x) =q( ) S 0 (x) [1 F (x)]:pc00 (e(x))

This is an ordinary di¤erential equation for e(x) with the boundary condition e(x) = 0: Thus,
now we have a system of equations for S(x) and e(x).
Di¤erentiating equation (12) with respect to x and rearranging yields
H 0 (x) =q( )eu F 0 (x)u + G0 (x) [10u] + 1F F(x)(x) f q( )eu [F (x)+ ( ) [1F (R)] u + [G(x)G(R)] (1F (x)] e(x)with boundary condition
H(R) = 0:This di¤erential equation characterizes the distribution of employed workers.

5

Quantitative analysis

In this section, we calculate the equilibrium of the above model using numerical methods, since it
is not possible to solve analytically. First, we calibrate the model to match the several dimensionsof the data. Then, we perform quantitative comparative statics exercises by calculating thesteady-state response to an increase in the rate of disembodied technological growth.

5.1

Basic calibration

In order to investigate the quantitative impact of disembodied technological progress on unemployment, we calibrate the model to match U.S. labor market facts. The following 11 parameters
have to be determined: the discount rate r, the level of productivity p, the value of leisure z,
the worker’s bargaining power , two matching function parameters m0 and
19, two search cost

u)

H(x)g

function parameters c0 and , the technological growth rate g, the arrival rate of idiosyncratic
productivity shocks , and the vacancy cost .
We choose the model period to be one-year and set the discount rate r = 0:05 because theannual real interest rate has been around 5%. Since the level of productivity does not in‡uence
the steady state, we normalize p = 1 without loss of generality. For the benchmark case, we set
g to 2%, the average productivity growth rate in the US from 1948 to 2007.
We assume that the matching function is Cobb-Douglas,m(v; e) = m0 v 1
where m0 is the matching constant ande ;is the matching elasticity with respect to the total

search e¤ort of workers. Then, the job …nding rate is q( ) = m0 1
rate is q( ) = m0

and the worker …nding. Following Mortensen and Naygpál (2007), we choose the elasticity of thematching function to equal 0:5.9 We also set the workers’bargaining power

to 0:5.

In order to pin down the scale parameter m0 , we combine the monthly job …nding rate,
f = 0:45, estimated in Shimer (2005) with the monthly vacancy …lling rate, q = 0:71, proposed
by Hagedorn and Manovskii (2008). The equilibrium labor market tightness

must be equal totheir ratio f =q, namely 0.634. The matching scale parameter m = 6:845 is chosen to match thevacancy …lling rate.
The search cost function is speci…ed by c(e) = c0 e1+ , where c0 is a scale parameter and> 0: Since the value of c0 can be eliminated from the equilibrium condition, we set it to equal
to 1. In the benchmark case, we assume= 1. Thus, the search cost function is quadratic.

We now determine the unemployment ‡ow utility parameter z. In calibrations of search andmatching models, this parameter has been the subject of some discussion.10 For the benchmark
case, we set z equal to the e¤ective replacement rate 0.71, estimated by Hall and Milgrom (2008).
9

Shimer (2005) estimated that the elasticity with respect to a vacancy equals 0:28 by calculating the elasticityof the job …nding rate with respect to the vacancy-unemployment ratio:However, with on-the-job search, market
tightness is no longer equal to the vacancy-unemployment ratio, so this is not the appropriate value. In the model,
employed workers contact vacancies at the same rate as unemployed workers, so market tightness is proportional
to the number of vacancies.
10
Shimer (2005) sets z=y equal to 0.4 in order to capture unemployment bene…ts. Hagedorn and Manovskii(2008) argue that Shimer’s choice of the value of opportunity cost of employment is too low because it does notallow for the value of leisure, home production, disutility of work forgone when employed, as well as unemployment
bene…ts. They calibrate the opportunity cost of employment and the worker’s bargaining power to match the20

It includes both unemployment insurance and the value of time. 11 Since the aggregate level ofproductivity in the economy is
y=p

Z x

xdH(x);

R

the value of z is chosen to satisfy z=y = 0:71.
Since there is no obvious empirical counterpart to which the distribution of idiosyncratic
productivity should be matched, we assume that both F and G are uniform on support [0; 1] as
it is commonly used in the literature. Moreover, we can assume G = F:
It remains to selectthe vacancy costand : The average unemployment rate in the US is 6%. We choose

and the rate of arrival of the idiosyncratic shocks

to match the rate of job…nding and the unemployment rate, respectively. The parameter values are summarized in Table
2.
Table 2: Parameter values
Value

Parameter

Discriptionr

Discount rate

0:05

Data

p

A general productivity parameter

1:0

Normalization

m0

Scale parameter of Matching function

6:78

Vacancy …lling rate

Elasticity of Matching function

0:5

Mortensen-Nagypál (2007)

Workers’s bargaining power

0:5

See text

Flow value of unemployment

0:598

Hall-Milgrom (2008)

Cost of posting a vacancy

1:049

Job …nding rate

Scale parameter in search cost function

1:0

Normalization

Parameter in search cost function

1:0

Quadratic cost function

Arrival rate of idiosyncratic shock

0:103

Unemployment rate

The rate of productivity growth

0:02

Data

z

c0

g

Source / Target

observed cyclical response of wages and average pro…t rate. Their results are z = 0:955 and= 0:052. Mortensen

and Nagypál (2007) criticize Hagedorn and Manovskii (2008) for using these parameters because these parameters
yield workers a gain of 2.8% in ‡ow utility by going from unemployment to employment.
11
Hall and Milgrom (2008) use utility parameter values based on the empirical literature on household consumption and labor supply and reports of the e¤ective replacement ratio to estimate the value of z.

21

5.2

Results

Some of the model solutions under the chosen parameter values are reported in Table 3.
Table 3: Model solutions
Variable

description

solution

Variable

description

solutionlabor market tightness

0.634

q( )

worker …nding rate

8.52

R

reservation productivity

0.568

G(R)

separation rate

0.058

u

unemployment rate

0.060

JJ

job-to-job transtions

0.080

v

vacancy

0.060

EUEU ‡ows

0.055

e

total amount of search e¤ort

0.095

y

Average productivity

0.843

eu

search e¤or of an unemployed

0.392

Job destruction rate

0.058

q( )

job …nding rate

5.40

Job creation rate

0.058

The labor market tightness, the job-…nding rate of workers, the worker …nding rate of avacant job and the unemployment rate are equal to their calibrated values.12 The average levelof productivity in the economy is 0.843, and together with the value of z, the e¤ective replacement(z=y) is 0.71. The reservation productivity R is 0.568. This gives an endogenous separation rateof 5.8% per year, which is slightly lower than what is observed in U.S. data. The value of thereservation productivity is signi…cantly lower than the value of unemployment ‡ow utility. Thus,
unemployed workers are willing to match with a job with a low productivity. This result comesfrom the option value of working.
The calibrated job-to-job transitions and ‡ows of workers from employment to unemployment are 0:080 and 0:055, respectively. Using data from the Current Population Survey, Fallick and Fleischman (2004) and Nagypál (2005) report that job-to-job transitions are twice aslarge as employment-to-unemployment transitions. In our model, the ratio of job-to-job ‡ows to
employment-unemployment ‡ows is 0:080=0:055 = 1:45. Thus, the model predictions are slightly
lower than in the data.
In Figure 4, we plot the initial density of the distribution of initial idiosyncratic productivity
draws F and the endogenous equilibrium distribution of employed workers across productivity H.
Since employed workers move from low to high productivity jobs, the equilibrium distribution of
12

The resulting job …nding rate is 5.4, so that the monthly rate is 0.45%, which is exactly equal to the target

value.

22

employed workers H …rst-order stochastically dominates the distribution of initial productivity
F.
The equilibrium search intensity as a function of idiosyncratic productivity is plotted in
Figure 5. The search intensity of workers is decreasing with productivity. Thus, the search e¤ortof unemployed workers is higher than that of employed workers. In the literature, it is reported
that only a small fraction of employed workers searches for jobs intensively, and these actively
searching employed workers account for a small portion of job-to-job transitions. In our model,
the optimal search intensity of an unemployed worker is 0.392. Employed workers in a job withproductivity higher than 0.687 search for jobs with an intensity level that equals 80% of theintensity level of unemployed workers. In equilibrium, about 96% of employed workers are above
this productivity. Thus, the model can match the fact that the few employed workers actively
search for a job.

5.3

Productivity growth and the labor market

We now calculate steady-state responses to an increase in the growth of disembodied technologicalprogress. Figure 6 reports the results.
A faster rate of productivity growth reduces the separation rate, which matches the data. Thisimplies that the introduction of on-the-job search enables the endogenous job separation modelwith DTP to explain the negative relationship between productivity growth and the separation
rate, and thus the negative relationship between growth and unemployment.
The mechanism of our model can be understood by comparing to the standard endogenous
job separation model with DTP. In the standard model, the impact of disembodied technologicalprogress on the separation rate is ambiguous because there are two counteracting e¤ects: thelabor hoarding e¤ect and the outside option e¤ect. The labor hoarding e¤ect reduces the valueof the reservation productivity R because the option value of a match is an increasing functionof g. On the other hand, the outside option e¤ect, identi…ed by Prat (2007), raises the valueof the reservation productivity R. Since faster technological progress increases the job …nding
rate through the capitalization e¤ect, workers have better outside opportunities and so ask forhigher wages. This reduces the value of the match, and the …rm raises the reservation value of
productivity R. When the labor hoarding e¤ect dominates the outside option e¤ect, productivity

23

growth reduces the reservation value and thus the separation rate. As seen in the next section,
under plausible parameter values, the model without on-the-job search fails to generate a negative
impact of productivity growth on the separation rate. Furthermore, the model generates a positive
impact of productivity growth on unemployment rate.
Faster productivity growth reduces the separation rate in our model because the outside
option e¤ect is weakened due to on-the-job search. In the model, workers in …rms with low
productivity jobs search on the job. Some of the bene…t from on-the-job search is shared with
the …rm through the wage. Thus, the worker gets a lower wage than he would be paid if he did
not search on the job. This makes otherwise unproductive jobs productive enough to survive,
leading to decreased separation when the rate of productivity growth increases.
The job …nding rate and the vacancy rate increase with disembodied technological progress.
Since a higher rate of productivity growth increases the return from creating a job, …rms have agreater incentive to enter the market and the job …nding rate of workers rises. This is because
the cost of creating a vacancy is paid at the start but the pro…ts accrue in the future. Whenthe rate of productivity growth increases, all future income ‡ows are discounted at lower rate,
so …rms are encouraged to create more vacancies. This e¤ect is well-known as the capitalizatione¤ect (Aghion and Howitt (1994,1998), Pissarides (2000)).
In addition to the capitalization e¤ect, the incorporation of on-the-job search generates more
job creation as DTP increases. In a model without on-the-job search, all job creation has to be
fed from the pool of unemployed, which is quickly exhausted in a high growth economy. Instead,in our model, when productivity growth is high, increasing search activity by employed workers
expands the pool of potential hires for …rms. This induces more job creation. This mechanism issimilar to what we see in the recent literature of the search and matching model in the businesscycle, which …nds on-the-job search into the search and matching model ampli…es the e¤ect of
productivity shocks on unemployment (Nagypál (2007), Krause and Lubik (2006)). Thus, ourresults suggest that on-the-job search is not only relevant to the business cycle properties of thesearch and matching model but also to its long-run predictions.
To summarize, the incorporation of on-the-job search gives rise to two new channels throughwhich a faster productivity growth may reduce unemployment: an increased job …nding rate
and a reduced separation rate. Both factors leads to lower unemployment when productivity

24growth increases. Furthermore, incorporating on-the-job search substantially improves the performance of the search and matching model in accounting for the size of the impact of growthon unemployment. In the standard exogenous job separation matching model with DTP, a 1%
decrease in growth rate increases the unemployment rate by less than 0.01%. This is far below
the estimated magnitude of the impact of growth on unemployment in the literature. Blanchard
and Wolfers (2000) estimate that a 1% decline in the growth rate leads to a 0.25-0.7% increasein the unemployment rate. Pissarides and Vallanti (2007) …nd the e¤ect to be 1.3% to 1.5%.
Our result shows that a one percentage point decline in the rate of productivity growth increases
the unemployment rate by 0.24%. Thus, our model generates not only an empirically consistentsign of the e¤ect, but also a larger size of impact of growth on unemployment than the standard
model.
The model predicts an increase in job-to-job transitions as productivity growth increases.
Technological progress a¤ects job-to-job transitions through three channels. First, the capitalization e¤ect increases the job …nding rate of employed workers. Second, a faster rate of productivity
growth increases the bene…ts of on-the-job search and reduces its cost. Third, a faster rate ofproductivity growth increases the number of employed job seekers.
It is also important to understand how the optimal search intensity of workers changes in
response to productivity growth. In order to illustrate this, in Figure 7 we plot the equilibrium
search intensity as a function of the idiosyncratic productivity. Figure 7 shows that the optimal
search intensity is higher when productivity growth is high. This is because a faster rate ofproductivity growth increases the bene…ts of on-the-job search and reduces its cost.
Figure 8 plots the e¤ect of productivity growth on the equilibrium distribution of employedworkers. A change in productivity growth a¤ects the characteristics of the distribution in two
ways. Since faster productivity growth reduces the reservation value of productivity, the support of the endogenous distribution of employed workers becomes larger. Also, the compositionof employed worker in jobs with high productivity increases. Figure 8 shows that with fasterproductivity growth, more employed workers are in jobs with high productivity.

25

6

Discussion

This section evaluates the robustness of our results in the previous section. First, we discuss thesensitivity of our results to our choice of parameter values. Then, we assess whether or not theresults of our model crucially depends on the incorporation of on-the-job search.

6.1

Sensitivity analysis

Our main result is that an increase in the rate of productivity growth decreases the separation
rate and the unemployment rate. We now study how our results vary with the value of theworker’s bargaining power , the ‡ow value of unemployment z, and the curvature parameter ofthe search cost function . When we change these parameters, the value ofthat the model keeps matching the labor market tightness

andare reset soof 0.634 and the unemployment rateof 6%.
First, we consider the impact of the value of the workers’ bargaining power. Figure 9

reports the relationship between the rate of productivity growth and the unemployment rate
and the relationship between the growth rate and the separation rate for di¤erent values ofthe bargaining power of workers. Although the sign of the relationship between growth andunemployment does not change, the size of the impact is magni…ed as
because a decrease in

decreases. This is

reduces the size of the outside option e¤ect. It is also clear that allowing

for workers’ bargaining power to vary does not have a signi…cant impact on the relationshipbetween the productivity growth rate and the separation rate.
Next, we discuss the sensitivity of the results to our choice of parameter value unemployment‡ow utility z. The choice of the parameter value of z has been the subject of some discussion
in business cycle properties of the search and matching model. In our benchmark calibration,
we use the parameter value z = 0:598, which corresponds the value chosen by Hall and Milgrom(2008). Now we consider two di¤erent values of z, 0.4 (z=y = 0:55) and 0.87 (z=y = 0:955).
Former one is the value chosen by Mortensen and Nagypál (2008) and latter one is the value ofHagedorn and Manovskii (2008). Figure 10 reports the relationship between the growth rate andthe unemployment rate and the relationship between the growth rate and the unemployment rate
for di¤erent values of z. Varying the value of unemployment ‡ow utility substantially a¤ects themagnitude of the impact of productivity growth on unemployment. In the benchmark case, a one
26

percentage point decline in productivity growth leads to an increase in the unemployment rate of
0.24%. When we use z = 0:4 and z = 0:87, a one percentage point decline in productivity growthleads to an increase in the unemployment rate of 0.13% and 1.1%, respectively. This analysis
suggests that the calibration of the ‡ow unemployment utility is not only relevant to the businesscycles properties of the search-matching model but also to its long-run predictions.
Finally, we consider the impact of the value of the curvature parameter of the search costfunction. Note that a larger value ofincreases the convexity of the search cost function

and implies a higher search intensity of employed workers. This leads to more on-the-job search.
Since the extent of on-the-job search increases, …rms are more likely to open vacancies. Figure
11 reports the relationship between these two variables for di¤erent values of the curvatureparameter of the search cost function . Recall that in the bench mark case, i.e.,= 1, a one

percentage point decline in productivity growth leads to an increase in the unemployment rateof 0.24%. With

= 0:5 and= 1:5, the magnitude of the negative impact is 0.36% and 0.18%,

respectively. Thus, this sensitivity analysis suggests that a larger value ofyields a smaller e¤ectof productivity growth on unemployment.

6.2

The role of on-the-job search

In our model with on-the-job search, a faster productivity growth reduces the separation rate,
leading to a fall in the unemployment rate. Now we assess the contribution of on-the-job search
to this key result by examining a model without on-the-job search. This experiment allows us
to determine whether or not the results of our model crucially depend on the incorporation of
on-the-job search.
Now we consider an endogenous job separation model with productivity growth. The basic
structure of the model is same to that of Mortensen and Pissarides (1994). To analyze the impact
of productivity growth on unemployment, we incorporate disembodied technological progress into
the model of Mortensen and Pissarides (1994). Furthermore, to facilitate comparison between
our original model and this model, we assume that initial value of idiosyncratic productivity x is
drawn from a distribution F , and subsequently let fxg be a jump process characterized by arrivalrate

and a distribution of new realizations G. Note that in Mortensen and Pissarides (1994),

all new jobs are created at the highest productivity. The details of the model can be found in27

Appendix.
Without on the job search, the following job creation and job destruction conditions determine
endogenous variables

and R:
(1
=
q( )
r+

)
g

Z 1

x0

Given the equilibrium

z

+

1

(17)

R

and
0=R

R dF (x0 )

r+

g

Z 1

x0

R dG(x0 ):

(18)

R

and R, the unemployment rate is determined by
u=

G(R)
:
G(R) + q( )F (R)Note that in the model without on-the-job search, the labor market tightness

is equal to thevacancy-unemployment ratio. The job …nding rate is q( )F (R), and the separation rate isG(R).
Now we examine the impact of disembodied technological progress on labor market variables.
Total di¤erentiating (17) and (18) shows that labor market tightness is an increasing function ofg. The intuition is that faster growth raises the returns to job creation, so …rms are encouraged
to post more vacancies, resulting in higher labor market tightness. When the match separation
is exogenously determined, faster productivity growth raises the job …nding rate, lowering unemployment. However, once endogenous job separation is incorporated, the impact of DTP on
unemployment rate is ambiguous. We can see this by looking at Figure 12. The locus of the pairsofand R that satis…es the job creation condition (17) is a downward sloping curve JC overR-space. On the other hand, the locus of the pairs ofand R that satis…es the job destruction

condition (18) is an upward sloping curve JD on the same plane. The intersection of the JC and
JD curves determines the equilibrium

and R. When the productivity growth rate g increases,

the JC curve shifts upward and the JD curve shifts downward. Thus, the labor market tightness
increases and the reservation productivity R may increase or decrease. Hence, job …nding andseparation rates may increase or decrease.13
The ambiguity is due to a higher workers’outside option from a higher labor market tightness
Since faster productivity growth increases market tightness, workers have better outside oppor13

Notice that the separation rate is increasing function of the reservation productivity R, and the job …nding

rate is increasing in the market tightness

but decreasing in R.

28

tunities and so ask for higher wages. This reduces the value of the match, and the …rm raises thereservation value of productivity R. This e¤ect is identi…ed by Prat (2007) as the outside optione¤ect. On the other hand, a faster productivity growth increases the option value of the match,leading to a fall in the reservation productivity. This e¤ect is called the labor hoarding e¤ect.
When the outside option e¤ect dominates the labor hoarding e¤ect, a faster productivity growthincreases the reservation value of productivity, and thus the separation rate.
The impact of DTP on the separation rate can be understood by examining the job destruction
condition. Totally di¤erentiation of (18) yields
dR
=
dg

r+
g
r + F (R) g

1

d
dg
|{z}

R1

0
R (x

[r + F (R)R) dG(x0 ):g] (r +g)(+)

The …rst term and the second term of the RHS capture the outside option e¤ect and the laborhoarding e¤ect, respectively. We can see that the sign of the impact of productivity growth on theseparation rate depends on which e¤ect dominates. When the labor hoarding e¤ect dominates,
faster productivity growth reduces the value of the reservation productivity. Then, the separation
rate falls and the job …nding rate rises, leading to a lower unemployment rate. On the other hand,when the outside option e¤ect dominates, faster productivity growth increases the value of thereservation productivity, and thus separation rate. The job …nding rate may increase or decrease.
When the job …nding rate falls, the unemployment rate rises. When the job …nding rate increases,
the impact of a faster growth on unemployment is ambiguous.
We quantitatively analyze this model to help clarify these ambiguities. Basic parameter values
are the same as those used before. We normalize to the time period to be one year and choose adiscount rate r of 5%. Again, we normalize the average productivity to be one. We set g to 0:02.
The matching function is assumed to be Cobb-Douglas, and the matching constant m0 is chosen
to match the vacancy …lling rate given the elasticity with respect to unemployment . Similar to
before, the cost of posting a vacancy

and the arrival rate of idiosyncratic shocks

are chosen

to match the job …nding rate and the average unemployment rate.
It remains to select the elasticity of matching function, workers’bargaining power , andthe value of non-market activity z. First, we follow the calibration strategy proposed by Shimer(2005), and set== 0:72 and z = 0:4. The parameter values under this calibration strategy

are reported in column (1) of Table 4. In the second calibration strategy, we use the value of29

non-market activity chosen by Hall and Milgrom (2008) and set z to be 0:71, and the elasticity
parameter

is set to 0:5, as suggested by the matching function estimates in Petrongolo andPissarides (2001). The workers’bargaining power

is set to 0:5 to internalize the search exter-

nalities. The parameter values under this calibration strategy are reported in column (2) of Table
4. Lastly, we use the calibration strategy proposed by Hagedorn and Manovskii (2008), and set= 0:052 and z = 0:955. Also, we use their parameter value for the elasticity of the matching
function,= 0:455. Parameter values chosen by this calibration strategy are reported in column(3) of Table 4.
Table 4: Parameter values(1)(2)

Parameter

Interpretationr

Discount rate

0:05

m0

Scale parameter of Matching function

z

g

(3)

(4)

(5)

(6)

0:05

0:05

0:05

0:05

0:05

55:24

92:87

131:5

6:13

6:78

6:92

Elasticity of Matching function

0:72

0:5

0:455

0:72

0:5

0:455

Workers’s bargaining power

0:72

0:5

0:052

0:72

0:5

0:052

Flow value of unemployment

0:4

0:71

0:955

0:4

0:71

0:955

Cost of posting a vacancy

0:317

0:387

0:540

0:335

0:551

0:401

Arrival rate of idiosyncratic shock

0:388

0:372

0:364

0:365

0:354

0:358

The rate of productivity growth

0:02

0:02

0:02

0:02

0:02

0:02

Again we perform comparative static exercises by calculating the steady-state responses toan increase in the productivity growth rate. Figure 13 reports the results of this experiment.
We see a positive relationship between the productivity growth rate and the separation rate.
This is because the outside option e¤ect prevails over the labor hoarding e¤ect. This result is
not consistent with the empirical negative relation between long-run productivity growth and theseparation rate. We use various parameter values and …nd that this is more likely to be the rulethan an exception. Thus, an endogenous job separation model with productivity growth tends
to generate a positive impact of productivity growth on the separation rate.
It is worth noting that although faster productivity growth increases the labor market tightness, it reduces the job …nding rate. This is because a faster growth increases the reservation
value R.
The overall e¤ect of DTP on the unemployment rate is determined by interactions between the30

job …nding rate and the separation rate. Since faster productivity growth reduces the job …nding
rate and increases the separation rate, the unemployment rate rises. Calibrating the model showsthat a one percentage point rise in the productivity growth rate leads to a 0:013

0:07% increasein the unemployment rate. Thus, the model fails to generate the empirically consistent sign ofthe impact of productivity growth on unemployment.
Our result is similar to that of Prat (2007). Prat (2007) considers the endogenous job separation model in which the job’s idiosyncratic productivity follows a Geometric Brownian Motion.
He demonstrates that the e¤ect of productivity growth rate on unemployment depends on theworker’s value of non-working z and the workers’ bargaining powerz = 0:4 and. He shows that when

= 0:72; as used by Shimer (2005) and others, higher growth rate increases theunemployment rate. However, when z is large andis very small, as is used by Hagedorn andManovskii (2008), higher growth decreases the unemployment rate. Our result is stronger than
this. In other words, for various parameter values of z and , the model generates the positive
relationship between productivity growth rate and the unemployment rate.
This di¤erence between result of our paper without on-the-job search and one of Prat (2007)
comes from the assumption about the job’s initial productivity. While the original Mortensen
and Pissarides (1994) model assumes that the job’s initial productivity starts at the highest
productivity, our model assumes that the job’s initial productivity is drawn from the distribution
F . Now we consider the case in which F (x) is degenerate at x. Again, we assess the quantitative
property of the model by using three di¤erent calibration strategies. The calibrated parametervalues are summarized in columns (4)-(6) of Table 4, and the result of quantitative comparative
statics exercises is reported in Figure 14.
The impact of productivity growth on the unemployment rate depends on the parametervalues of z andas seen in Prat (2007). When z is large andis very small, higher growth

reduces the unemployment rate, which is consistent with the data. This result comes from anincreased job …nding rate due to the capitalization e¤ect. When the job’s initial productivity startat the highest value, the job …nding rate q( ) does not depend on the reservation productivity.
Hence, the higher labor market tightness due to the capitalization e¤ect unambiguously increases
the job …nding rate. When z = 0:955 and= 0:052, since the capitalization e¤ect dominates

the outside option e¤ect, a higher growth rate reduces the unemployment rate. However, the31

impact of productivity growth on the separation rate is not consistent with empirical …ndings.
In all calibration strategies, a higher growth rate increases the separation rate. This implies
that the size of the outside option e¤ect is not small, and endogenous job separation models
with productivity growth cannot account for the empirical relationship between the long-run
productivity growth and the separation rate.

7

Conclusion

This paper studies the impact of long-run productivity growth on the job …nding rate, the separation rate, and unemployment rates in a search and matching model. By examining the long-run
relationship between the productivity growth rate and labor markets variables in the U.S., we…nd that, while the job …nding rate is positively correlated with the growth rate, there is astrong negative correlation between the separation rate and growth. Furthermore, we …nd thatboth job …nding and separation rates contribute to overall unemployment variability in the longrun. These empirical …ndings suggest that productivity growth reduces the unemployment rate
through not only increased job …nding but also decreased separation.
In order to explain these empirical facts, we incorporate on-the-job search and endogenous
search intensity into the endogenous job separation model of Mortensen and Pissarides (1994)
with disembodied technological progress. The incorporation of on-the-job search gives rise to newchannels through which faster growth may reduce unemployment, by reducing the separation rateand inducing more job creation. Our model demonstrates that faster productivity growth reduces
the separation rate and increases the job …nding rate, leading to lower unemployment. This result
is consistent with empirical …ndings. Furthermore, the model not only generates an empiricallyconsistent sign of the impact of growth on unemployment, but also generates a larger magnitude
than the standard matching models with productivity growth.
A number of important issues remain for future research. One issue to be considered is themagnitude of the impact of productivity growth on unemployment. Although our model generates
a larger magnitude than the standard models, it is still smaller than the estimated one in thedata. Also, considering another wage determination mechanisms is an important issue. In the
typical on-the-job search model, surplus sharing is not equivalent to the Nash bargaining solution
because of the non-convexity of the Pareto set. A number of studies considers alternative wage
32

determination mechanism in a matching model with on-the-job search. However, most of thesestudies assume exogenous job destruction. To consider an alternative wage setting mechanism inan endogenous job destruction model with on-the-job search is a fruitful avenue for research.

33

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37

8

Appendix

8.1

Generalized Mortensen and Pissarides (1994) ModelIn this appendix, we develop a generalized endogenous job separation model of Mortensen and
Pissarides (1994) with disembodied technological progress. The basic structure of the model isthe same as that of Mortensen and Pissarides (1994). While all new jobs are created at the highest
productivity in Mortensen and Pissarides (1994), we assume that initial value of idiosyncratic
productivity x is drawn from a distribution F: We let fxg be a jump process characterized byarrival rate

and a distribution of new realization G. Furthermore, in order to study the impactof long-run productivity growth on unemployment, we incorporate disembodied technologicalprogress into the model. We use the same notations in our original model with on-the-job search
to describe the generalized Mortensen and Pissarides (1994) model.
The value functions of …rms and workers areZ(r g) (x) = px w(x) +max[ (x0 ); V dG(x0 )
(x)
Z
rV =
p + q( )max[ (x0 ); V dF (x0 ) V (x)Z(r g) W (x) = w(x) +max[ W (x0 ); U dG(x0 ) W (x)Z(r g) U = pz + q( )max[ W (x0 ); U dF (x0 ) U
Given the free entry condition V = 0, the surplus function S(x) is characterized by
Z
Z
0
0
(r +
g) S(x) = px pz +max[ S(x ); 0 dG(x )q( )max S(x0 ); 0 dF (x0 ): (A1)This implies
S 0 (x) =

p
r+

g

> 0:

Since the surplus function is increasing in x, the …rm and the worker will choose to form andcontinue any match that has am idiosyncratic productivity x

R: The reservation productivity

is determined by S(R) = 0. Using integration by parts and the free entry condition, we …nd thefollowing job creation condition

q( )

=

(1
r+

)
g

Z 1
R

38

x0

R dF (x0 ):Evaluating (A1) at x = R and using the above job creation condition, we have the following job
destruction condition
0=R

z

1

+

r+

These two equations determine equilibrium value of

g

Z 1

x0

and R. Given these values, the unemploy-ment rate is determined by
u=

G(R)
:
G(R) + q( )F (R)39

R dG(x0 ):R(a) Productivity Growth(b) Unemployment Rate

20

11
Trend
Raw data

15

Trend
Raw data

1098Percent

Percent

10

5

0

7
6
5
4

-5
3
-10

2

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Year(c) Job Finding Rate(d) Separation Rate

50

2.4
Trend
Raw data

40

2

35

1.8

30

1.6

25

1.4

20

1.2

15

Trend
Raw data

2.2

Percent

Percent

45

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Year

1

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Year

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Year

Figure 1: Macro series and their trend values. Note: The rate of labor productivity growth ismeasured by the …rst di¤erence of logged labor productivity. Labor productivity is real output
per person in the non-farm business sector, constructed by the BLS Major Sector Productivity
and Costs program. The unemployment rate is a quarterly average of the seasonally adjusted
monthly series constructed by the BLS from the CPS. The job …nding and the separation rates
are constructed by Shimer (2007). See Shimer (2007) for data construction details. The trends
are estimated by passing raw time series data through a two-sided low pass …lter, with a cuto¤
frequency corresponding to 15 years. Sample covers 1948Q1-2005Q1.

40

8
Productivity Growth
Unemployment Rate
7
6

Percent

5
4
3
2
1
0

1950

1955

1960

1965

1970

1975 1980
Year

1985

1990

1995

2000

2005

Figure 2: Productivity growth and unemployment rate. Note: The dashed line indicates the trendof the productivity growth rate. The solid line indicates the trend of the unemployment rate. The
rate of labor productivity growth is measured by the …rst di¤erence of logged labor productivity.
Labor productivity is real output per person in the non-farm business sector, constructed by
the BLS Major Sector Productivity and Costs program. The unemployment rate is a quarterly
average of the seasonally adjusted monthly series constructed by the BLS from the CPS. The
trends are estimated by passing raw time series data through a two-sided low pass …lter, with acuto¤ frequency corresponding to 15 years. Sample covers 1948Q1-2005Q1.

41

8
7.5
7

Percent

6.5
6
5.5
5
Actual unemployment rate
Hypothetical unemployment rate

4.5

uFt

4
3.5

uSt

1950

1955

1960

1965

1970

1975 1980
Year

1985

1990

1995

2000

2005

Figure 3: Contribution of Unemployment Rate Variability. Note: The solid line indicates theactual unemployment rate. The line with circle indicates hypothetical unemployment rate u~.
The dashed line indicates the hypothetical unemployment rate if there were only ‡uctuationsin the job …nding rate uF . The dash-dotted line indicates the hypothetical unemployment ratewith only ‡uctuations in the separation rate uS . See text for de…nitions of u~, uF and uS . The
unemployment rate is a quarterly average of the seasonally adjusted monthly series constructedby the BLS from the CPS. Sample covers 1948Q1-2005Q1.

42

7
Initial productivity distribution
Equilibrium distribution of employed workers
6

5

Density

4

3

2

1

0

0.6

0.65

0.7

0.75
0.8
Match productivity

0.85

0.9

Figure 4: Distribution of employed workers

43

0.95

1

0.4

0.35

0.3

Search effort

0.25

0.2

0.15

0.1

0.05

0

0.6

0.65

0.7

0.75
0.8
Match productivity

0.85

Figure 5: Optimal Search Intensity

44

0.9

0.95

1

Vacancy rate

Unemployment rate
0.075

0.066
0.064

0.07
0.062
0.065

0.06
0.058

0.06

0.056
0.055
0.054
0.052

0

0.01

0.02
0.03
Growth rate

0.04

0.05

0.05

0

0.01

Job finding rate

0.02
0.03
Growth rate

0.04

0.05

0.04

0.05

Separation rate

5.7

0.05855.65

0.0585

5.6
0.05845.55
5.5

0.05845.45

0.0583

5.4

0.05835.35
0.0582

5.3
5.25

0

0.01

0.02
0.03
Growth rate

0.04

0.05

0.0582

0

0.01

0.02
0.03
Growth rate

Figure 6: Comparative statics for disembodied technical change g:

45

0.45
At growth rate g=0.02
At growth rate g=0.04
0.4

0.35

Search effot

0.3

0.25

0.2

0.15

0.1

0.05

0

0.6

0.65

0.7

0.75
0.8
Match productivity

0.85

0.9

0.95

Figure 7: The e¤ect of productivity growth on optimal search intensity

46

1

7

6

At growth rate g=0.02
At growth rate g=0.04

5Density

4

3

2

1

0

0.6

0.65

0.7

0.75
0.8
Match productivity

0.85

0.9

0.95

1

Figure 8: The e¤ect of productivity growth on the distribution of employed workers

47

Separation rate

Unemployment rate
0.062

0.066

β =0.4β =0.5β =0.6

0.064

0.062

0.06

0.06

0.059

0.058

0.058

0.056

0.057

0.054

0.056

0.052

0.055

0.05

0

0.01

0.02
0.03
Growth rate

0.04

β =0.4β =0.5β =0.6

0.061

0.054

0.05

0

0.01

0.02
0.03
Growth rate

Figure 9: Sensitivity analysis with respect to

48

0.04

0.05

Unemployment rate

Separation rate

0.09

0.12
z/y=0.55
z/y=0.71
z/y=0.955

z/y=0.55
z/y=0.71
z/y=0.955

0.08

0.1

0.07

0.08

0.06

0.06

0.05

0.04

0.04

0.02

0.03

0

0.01

0.02
0.03
Growth rate

0.04

0

0.05

0

0.01

0.02
0.03
Growth rate

Figure 10: Sensitivity analysis with respect to z

49

0.04

0.05

Separation rate

Unemployment rate
0.0625

0.07
µ=0.5
µ=1.0
µ=1.5

µ=0.5
µ=1.0
µ=1.5

0.062

0.0615

0.065

0.061

0.06

0.06050.06

0.0595

0.055

0.059

0.05

0.0585

0.058

0.045

0

0.01

0.02
0.03
Growth rate

0.04

0.0575

0.05

0

0.01

0.02
0.03
Growth rate

Figure 11: Sensitivity analysis with respect to

50

0.04

0.05

Figure 12: Job creation and job destruction curves in R-space.

51

Unemployment rate

Labor market tightness

0.06250.67
z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

0.062
0.0615z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

0.66
0.65

0.061
0.0605

0.64

0.06

0.63

0.05950.62

0.059
0.0585

0

0.01

0.02
0.03
Growth rate

0.04

0.05

0.61

0

0.01

Job finding rate

0.04

0.05

0.04

0.05

Separation rate

5.55

0.347
z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

5.5

0.346

5.4

0.3455

5.35

0.345

5.3

0.3445

5.25

0.344
0

0.01

0.02
0.03
Growth rate

0.04

0.05

z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

0.3465

5.45

5.2

0.02
0.03
Growth rate

0.3435

0

0.01

0.02
0.03
Growth rate

Figure 13: Comparative statics for disembodied technical change g in the generalized MortensenPissarides model with disembodied technological progress.

52

Unemployment rate

Labor market tightness

0.062

0.68
z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

0.06150.061

0.66

0.0605

0.65

0.06

0.64

0.0595

0.63

0.059

0.62

0.0585

0.61

0.058

0

0.01

0.02
0.03
Growth rate

0.04

z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

0.67

0.05

0.6

0

0.01

Job finding rate

0.04

0.05

0.04

0.05

Separation rate

5.6

0.347
z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

5.55

0.346

5.45

0.3455

5.4

0.345

5.35

0.3445

5.3

0.344
0

0.01

0.02
0.03
Growth rate

z=0.4 and β =0.72
z=0.71 and β =0.5
z=0.955 and β =0.052

0.3465

5.5

5.25

0.02
0.03
Growth rate

0.04

0.05

0.3435

0

0.01

0.02
0.03
Growth rate

Figure 14: Comparative statics for disembodied technical change g in the Mortensen-Pissarides
model with disembodied technological progress.

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