1. society
2. work
3. unemployment

Large Firms and Within Firm Occupational
Reallocation
Theodore Papageorgiou
McGill University, Department of Economics y
April 2015
Abstract
This paper considers the equilibrium interaction between within …rm and across
…rm reallocation in the presence of labor market frictions. While a sizable literature has investigated frictional labor markets, it has ignored within …rm mobility.
Nonetheless, every year a sizable fraction of workers switch occupations without
changing …rms. Employees in large …rms can sample from a larger selection and
across …rm mobility is replaced by within …rm mobility. Bringing together within
and across …rm reallocation along with labor market frictions, naturally accounts
for the observed di¤erences in worker ‡ows and wages across …rms of di¤erent sizes.
Keywords: Frictional Labor Markets, Occupational Mobility, Within Firm Occupational Reallocation, Firm Size-Wage Premium, Separation Rates
JEL Classi…cation: E24, J24, J31
I thank Joe Altonji, Dimitris Cheliotis, Jed DeVaro, Mike Elsby, Manolis Galenianos, Ed Green,
William Hawkins, Christian Hellwig, Erik Hurst, Philipp Kircher, Fabian Lange, Ed Lazear, Alex Monge,
Giuseppe Moscarini, Elena Pastorino, Andrés Rodríguez-Clare, Richard Rogerson, Rob Shimer, Russell
Toth, Neil Wallace, as well as numerous seminar participants for useful comments. I am grateful to
Anders Frederiksen for generating results from the IDA dataset.
y
855 Sherbrooke St West, Room 519, Montreal, QC H3A 2T7, Canada.
E-mail:
[email protected]. Web: http://www.theodorepapageorgiou.com
1
Introduction
A key role of labor markets is to assign workers to the task they perform best. A large
literature investigates this assignment process in the presence of labor market frictions.
This literature focuses on moves of workers across …rms,1 but ignores an important empirical phenomenon: every year 8% of workers switch occupations within their …rm.2 These
moves within the …rm allow workers to allocate without being subject to labor market
frictions and thus facilitate the assignment of workers to occupations. This is the …rst
paper to consider how between and within …rm mobility interact in the presence of labor
market frictions.
In the …rst part of the paper, I examine empirically worker reallocation within and
across …rms. Large …rm workers switch occupations more often within their …rm. In
addition, they separate less often, even controlling for wages. Workers with high wages
switch occupations less often within the …rm and also separate from their …rm less often.
I also …nd that within a …rm, occupational ‡ows are largely o¤setting, so that for every
worker going from one occupation to another, there is another worker switching in the
opposite direction.
I also con…rm the well-known size-wage premium, whereby workers in …rms with more
employees earn higher wages. This wage premium is higher for longer tenured workers.
In addition, using matched employer-employee data from Denmark, I examine whether
the wage premium is due to the number of …rm employees per se or the increased number
of occupations of large …rms. I …nd that, controlling for the number of …rm employees,
workers in …rms with more occupations have substantially higher earnings than workers
in …rms with fewer occupations.
In the second part of the paper, motivated by these …ndings, I introduce an equilibrium
model of within …rm and across …rm reallocation. The setup is tractable and I am able
to show analytically that it can replicate the abovementioned facts. It is also amenable
to aggregation and general equilibrium analysis and therefore I am able to characterize
analytically the wage distributions both within the …rm, as well as the entire economy.
In my baseline setup, a …rm constitutes a labor market without search frictions and
workers can costlessly switch occupations within a …rm. In contrast, reallocation across
…rms is subject to frictions. The quality of a match between a worker and an occupation
is uncertain and revealed gradually. Workers and …rms observe output realizations and
1
Examples include Mortensen and Pissarides (1994), Burdett and Mortentsen (1998), Postel-Vinay
and Robin (2002) and Moscarini (2005).
2
1996 Panel of the Survey of Income and Program Participation.
1
update their beliefs in Bayesian fashion. At any point a worker has the option of leaving
his current occupation and going to another one within his …rm or to unemployment.
The optimal behavior of the worker is one of a reservation strategy: if his belief regarding the quality of his match with his current occupation falls below a certain threshold,
the worker moves on to another one; this threshold is increasing in the number of occupations within the …rm that the worker has not tried. Workers in large …rms have more
occupations remaining and are therefore willing to abandon unpromising matches more
easily. In equilibrium, workers in large …rms are better matched.
The model correctly predicts that workers in larger …rms are less likely to separate
from their …rm both because they have more occupations available and also because they
are better matched. Moreover, the …rm size remains important even when conditioning on
wages. In my model, as well as the data, workers in large …rms replace across …rm mobility
with within …rm mobility: conditional on wage, workers in larger …rms are more likely to
switch occupations within their …rm, consistent with the data. In addition, workers with
higher wages are less likely to switch occupations, both within as well as across …rms.
Workers who switch occupations within a …rm see their wages rise. I also prove that the
setup replicates the observed wage premium.
Furthermore, I am able to derive analytically the within-…rm and overall wage distributions. In particular, the shape of the within …rm wage distribution is the same as
that of the overall wage distribution of my economy. This is consistent with empirical
evidence from several countries presented in Lazear and Shaw (2009). Moreover, both
theoretical wage distributions have an empirically accurate shape and can replicate the
fat Pareto-type tail that we observe in the data.
In addition, I discuss a number of extensions, including human capital accumulation,
on-the-job search, search frictions inside the …rm and wage posting. Finally, I calibrate
the model, by targeting moments related to within and across …rm mobility and examine
its predictions regarding the wage premium, as well as other non-targeted moments.
A key assumption of the above setup is that frictions are lower within …rms compared
to across. From the beginning, the search literature has emphasized that search frictions
are a shortcut for the costly acquisition of information regarding both the location of
vacancies and job-seekers, as well as the characteristics of a particular job and worker.3
It is reasonable to expect that both types of information disseminate faster within …rms
than in the market: for instance, a …rm is better informed about the characteristics of its
current workers, rather than other applicants. Similarly, an employed job-seeker is more
3
See for instance Mortensen (1978) or the survey by Pissarides (2000).
2
likely to learn about a job opening in his own …rm rather than another one.
The contribution of the present paper is that it considers the equilibrium interaction
between within …rm and across …rm reallocation in the presence of labor market frictions
and how it a¤ects worker outcomes. Idson (1993) investigates empirically labor turnover
inside the …rm and argues that large …rms provide more …rm-speci…c training to their
workers. Novos (1992) investigates worker turnover and …rm scope in the presence of
learning-by-doing and asymmetric information. Moscarini and Thomsson (2007), using
data from the Current Population Survey, document that a signi…cant fraction of occupational switches do not involve an employer switch. The idea of match uncertainty between
a worker and a …rm dates back to Jovanovic (1979).4;5 The importance of occupations in
labor market outcomes has been emphasized by a sizable literature.6
In a related paper, Pastorino (2014) estimates a model of learning, job assignment
and human capital accumulation within the …rm. She …nds that the impact of learning
is particularly important, especially when taking into account its role in job assignment
within the …rm. See also Kahn and Lange (2014) and Pastorino (2015).
Since this paper …rst appeared, some new papers have followed the approach introduced in the present paper and also started investigating the equilibrium interaction
between within …rm and across …rm reallocation in the presence of labor market frictions:
In ongoing work, Kramarz, Postel-Vinay and Robin (2014) use French data to study the
importance of within-…rm reallocation in the assignment process of workers to occupations in the presence of frictions. Their approach is in many ways complementary to mine:
My setup is very tractable and I am able to obtain results analytically, whereas their approach relies on solving the model numerically, while also allowing for ‡exible worker and
…rm heterogeneity and focusing on the quantitative exercise.7 Their approach builds on
Postel-Vinay and Robin (2002) with commitment on the part of the …rm and Bertrand
4
The present paper is also related to Neal (1999) who argues that workers follow a two-stage search
strategy: …rst they search for a career (occupation), and then shop for jobs within the chosen career. In
his setup however workers cannot search for another occupation/career within an employer.
5
In addition, there’s an extensive empirical literature that documents the observed size-wage premium.
Brown and Medo¤ (1989), Idson and Feaster (1990) and Schmidt and Zimmermann (1991) …nd that the
positive relationship between …rm size and wage holds even conditional on observed and unobserved
labor quality and therefore cannot be explained away by worker selection. Brown and Medo¤ (1989)
also demonstrate that it remains even when considering …rms that o¤er a piece-rate system, therefore
indicating that it is not just the result of wage backloading.
6
Examples include Kambourov and Manovskii (2009), Alvarez and Shimer (2009) and (2011), Eeckhout and Weng, (2010), Antonovics and Golan (2012), Hsieh et al. (2013), Carrillo-Tudela and Visschers
(2014), Gervais, Jaimovich, Siu and Yedid-Levi (2014), Groes, Kircher and Manovskii (2014), Papageorgiou (2014).
7
It is possible to extend the setup introduced here to allow for some ex-ante heterogeneity on the
worker and the …rm side (e.g. assume a certain number of types).
3
competition between employers; my setup instead builds on the matching theory of Jovanovic (1979) and the equilibrium unemployment theory of Mortensen and Pissarides
(1994) with a lack of commitment on the part of the …rm (see also Moscarini, 2005).
While their work is still ongoing, it is ambitious and consistent with the idea put forward
in the present paper that within and across …rm reallocation should no longer be viewed
in isolation. Similarly, Li and Tian (2013) modify the setup introduced in the present
paper to allow for wage posting by the …rms and directed, rather than random search on
the part of the workers.
The next section investigates empirically worker mobility within and across …rms, as
well as how it’s a¤ected by the size of the …rm and wages. I also examine the wage premium
and how it’s a¤ected by the number of occupations of the …rm. Section 3 introduces a
model of within …rm and across …rm reallocation that is consistent with the above facts.
I solve the model, consider several extensions and calibrate it. Section 4 concludes.
2
Empirical Evidence
In this section I empirically examine within …rm and across …rm mobility and how they
depend on …rm size and wages. I focus on occupational matching and use data from the
1996 panel of the Survey of Income and Program Participation (SIPP). The SIPP is a
nationally representative sample of households in the civilian non-institutionalized U.S.
population. Interviews were conducted every four months for four years and included
approximately 36,000 households. It contains information on the worker’s wage, 3-digit
occupation, current employer and employer size. The SIPP reports employer size in three
size bins: fewer than 25 employees (22% of the sample), between 25 and 99 employees
(13% of the sample) and more than 100 employees (65% of the sample). Because the
SIPP reports both an employer identi…er as well as an occupation identi…er, I am able
to distinguish between within and across …rm mobility. The 1996 SIPP uses dependent
interviewing, which is found to reduce occupational coding error (Hill, 1994). Hourly
wages are de‡ated to real 1996 dollars using the Consumer Price Index.
2.1
Within Firm Occupational Reallocation
I begin by investigating whether there is a typical career progression or career-ladder
within …rms when considering occupations. For instance, in the case of 3 occupations, A,
B and C, a typical ladder might involve workers switching from occupation A to B and
4
Occupation i
A
A
C
A
A
I
A
J
I
D
C
C
D
C
Occupation j
D
C
D
G
I
J
B
L
L
G
G
L
I
I
F low(i; j) + F low(j; i)
968
541
330
300
274
262
219
167
164
161
136
133
119
103
jF low(i;j) F low(j;i)j
F low(i;j)+F low(j;i)
0.141
0.117
0.055
0.193
0.183
0.107
0.26
0.054
0.098
0.044
0.132
0.173
0.025
0.01
Table 1: Within Firm Net Reallocation. 1996 Panel of Survey of Income and Program
Participation. 4-month transitions. Occupational groups de…ned as A. Managerial and
Professional Specialty Occupations (003-199); B. Technical Support (203-235); C. Sales
(243-285); D. Administrative Support (303-389); E. Private Household Occupations (403407); F. Protective Service Occupations (413-427); G. Service (433-469); H. Farming
(473-499); I. Precision Production (503-699); J. Machine Operators (703-799); K. Transportation (803-859); L. Handlers (864-889)
then from B to C (from the bottom to the top). We would not expect to see ‡ows from B
to A or from C to B or to A. Put di¤erently gross ‡ows across these occupations should
equal net ‡ows.
To test this prediction of the career-ladder model, I focus on 12 major occupational
groups and examine the ‡ows of workers who switch occupations within their …rms. It’s
worth noting that 64% of 3-digit switches within …rms involve a switch among major
occupations as well. Table 1 presents the pairs of occupations with the most ‡ows. I’m
interested in whether these ‡ows are ‡ows in one direction only, or whether they are
bidirectional. With that in mind, I compute the ratio of the absolute di¤erence between
the two ‡ows over the sum of the two ‡ows, in other words the ratio of net ‡ows over
gross ‡ows. The career-ladder model would predict that this ratio should be close to 1.
Instead, for all occupational pairs shown in Table 1 the ratio is far from 1 and for several
pairs it is close to zero. This pattern is not driven by the pairs of occupations with the
most ‡ows presented in the table. When I compute the weighted average of the above
ratio for all pairs (5743 observations), I …nd that it’s equal to 0.137 (the ratios for all
5
Occ. Switching
Occ. Switch
Occ. Switch
Occ. Switch
Occ. Switch
Prob. (Probit) Prob. (Probit) Prob. (Probit) Prob. (Probit) Prob. (Probit)
Same Employer
Same Empl
Same Empl
Same Empl
Same Empl
All
Services
Manufact
High School
College
100 employees
0.0122
0.0129
0.0093
0.0125
0.0158
(0:0014)
(0:0021)
(0:0047)
(0:0018)
(0:0049)
ln(wage)
-0.0046
(0:0013)
-0.0056
(0:002)
-0.0068
(0:0035)
-0.0048
(0:0017)
Table 2: Wage and Size Impact on Within Firm Reallocation. 1996 Panel of Survey of
Income and Program Participation. 4-month probabilities. 3-digit occupations. Controls
include gender, race, education, medium size …rm dummy, quartic in age, marital status,
population density, 11 industry dummies, 13 occupation dummies. Services includes
industry codes 700 through 893. Manufacturing includes industry codes 100 through
392. Standard errors clustered by individual. Coe¢ cients represent marginal e¤ects
evaluated at the average value of the 4-month probability, which equals 0.028 (all), 0.026
(services), 0.033 (manufacturing), 0.029 (high school), 0.03 (college) respectively. 130,839
(all), 47,345 (services), 26,983 (manufacturing), 93,669 (high school), 13,439 (college)
observations respectively.
‡ows are presented in the Online Appendix). Put di¤erently, gross ‡ows are on average
almost an order of magnitude greater than net ‡ows and therefore I conclude that most
‡ows within …rms are bidirectional.8
I next examine how within …rm mobility is a¤ected by …rm size and wage. Table
2 shows the impact of wages and …rm size on 3-digit occupational switching within a
…rm. Among workers who remain with the same employer for two consecutive waves
(4 month period), workers earning higher wages are less likely to switch occupations.
Furthermore, conditional on wages, workers in large …rms are signi…cantly more likely to
switch occupations within a …rm. The size e¤ect is quantitatively large, as working in a
large …rm increases the four-month probability of an occupational switch within a …rm by
almost 45%.9 The results remain quantitatively large when focusing on manufacturing,
services, high school graduates or college graduates.
8
I reach the same conclusion when looking at within-…rm ‡ows between 3-digit occupations. While
there are a lot fewer ‡ows for each pair, when I focus on pairs that have at least 30 ‡ows between them,
the weighted ratio of net over grows ‡ows is equal to 0.191.
9
If workers in small …rms perform multiple tasks, whereas in large they specialize (Lazear, 2004), this
might arti…cially generate more mobility in small …rms. Then the true di¤erence between large and small
…rms would be larger.
6
-0.0053
(0:0031)
Occ Switch Dummy
ln(wage)t
1
ln(wage)t
Same
Employer
All
0.0242
(0:0029)
0.8914
(0:005)
ln(wage)t
Same
Employer
Services
0.0244
(0:0052)
ln(wage)t
Same
Employer
Manufact
0.0189
(0:0051)
ln(wage)t
ln(wage)t
Same
Same
Employer
Employer
High School
College
0.0219
0.0559
(0:0033)
(0:0106)
0.8868
(0:0091)
0.9009
(0:0088)
0.8942
(0:0054)
0.8943
(0:0215)
Table 3: Impact of Occupation Switch Within a Firm on Wage. 1996 Panel of Survey
of Income and Program Participation. 4-month intervals. 3-digit occupations. Controls
include gender, race, education, quartic in age, marital status, population density, 11
industry dummies, 13 occupation dummies. Services includes industry codes 700 through
893. Manufacturing includes industry codes 100 through 392. Standard errors clustered by
individual. 122,536 (all), 43,714 (services), 25,797 (manufacturing), 88,135 (high school)
and 11,645 (college) observations respectively.
It’s also worth noting that 88% of within …rm ‡ows are switches to occupations that
the worker has not worked in, during the past 3 years.
Finally, I study the impact of within …rm mobility on wages. As shown in Table 3,
workers who switch occupations within their …rm experience a wage increase of 2.42% on
average. Again, this result holds when the two industries and two education groups are
considered separately.
2.2
Across Firm Occupational Reallocation
I now turn to the across …rm reallocation and consider how …rm separation probability is
a¤ected by wages and …rm size. The …rst column of the top panel of Table 4 shows that
workers in large …rms are signi…cantly less likely to separate from the …rm. In particular,
the …rm separation probability is 26% lower in large …rms. In the second column of
the top panel of Table 4 it’s worth noting that workers in large …rms are less likely to
separate, even conditional on the worker’s wage. Even though the size coe¢ cient falls in
magnitude as expected, it continues to be economically large and statistically signi…cant:
indeed, the impact of size on the separation probability is as large as that of a 28% wage
increase. Again these facts hold when considering workers in manufacturing or services,
7
100
empl
Separation
Probability
All
-0.0283
(0:0023)
ln(wage)
Separation
Probability
All
-0.0202
(0:0024)
Separation Separation
Probability Probability
Services
Services
-0.0197
-0.015
(0:0038)
(0:0038)
-0.0721
(0:0029)
Separation
Probability
Manufact
-0.0437
(0:005)
-0.0578
(0:0045)
Separation
Probability
Manufact
-0.031
(0:0051)
-0.0762
(0:0063)
Separation
Separation Separation Separation
Probability Probability Probability Probability
High School High School
College
College
100 empl
-0.0287
-0.0205
-0.0405
-0.0333
(0:0028)
(0:0028)
(0:0074)
(0:0074)
ln(wage)
-0.0721
(0:0035)
-0.0558
(0:0068)
Table 4: Wage and Size Impact on Firm Separation Probability (Probit). 1996 Panel of
Survey of Income and Program Participation. 4-month probabilities. Controls include
gender, race, education, medium size …rm dummy, quartic in age, marital status, population density, 11 industry dummies, 13 occupation dummies. Services includes industry
codes 700 through 893. Manufacturing includes industry codes 100 through 392. Standard
errors clustered by individual. Coe¢ cients represent marginal e¤ects evaluated at the average value of the 4-month probability, which equals 0.1097 (all), 0.113 (services), 0.071
(manufacturing), 0.104 (high school) and 0.103 (college) respectively. 150,936, 55,027,
29,580, 107,435 and 15,618 observations respectively.
and when considering workers with a high school or college degree.10 In addition, the
overall occupational switching behavior (i.e. both within and across …rms) of workers in
large …rms is di¤erent than those in smaller …rms. More speci…cally, workers in larger …rms
are signi…cantly less likely to switch occupations overall (results available upon request).
2.3
Firm Size Wage Premium
Table 5 presents the well-known empirical regularity that workers in …rms with more
employees earn higher wages. In particular workers in …rms with more than 100 employees
10
The results are qualitatively and quantitatively similar in magnitude if I focus on separating and
switching occupations. In addition, wages increase upon separating and switching occupations (results
available upon request).
8
100 empl
25-99 empl
ln(wage)
All
0.1311
(0:0047)
ln(wage)
Services
0.0932
(0:0073)
0.0298
(0:0054)
0.016
(0:0086)
ln(wage)
ln(wage)
ln(wage)
Manufacturing High School
College
0.1878
0.13
0.1588
(0:0126)
(0:0056)
(0:0179)
0.0215
(0:0139)
0.0359
(0:0066)
0.0316
(0:0221)
Table 5: Wage Premium. 1996 Panel of Survey of Income and Program Participation.
Controls include gender, race, education, quartic in age, marital status, population density, 11 industry dummies, 13 occupation dummies. Services includes industry codes 700
through 893. Manufacturing includes industry codes 100 through 392. Standard errors
clustered by individual. 169,536 (all), 61,968 (services), 32,408 (manufacturing), 119,383
(highschool) and 17,203 (college) observations respectively.
earn on average 13% higher wages than workers in …rms with fewer than 25 employees.
Moreover, the wage premium is present for workers employed in the service industry, in
manufacturing, as well as when focusing on workers with a high school degree and also
those with a college degree.11
I next ask whether the observed wage premium is due to the number of …rm employees
or the larger selection of occupations in large …rms and thus I investigate how the wage
premium depends on the number of occupations in the …rm, separately from …rm size.12
The number of occupations in each …rm can only be obtained through the use of
matched employer-employee datasets and, to my knowledge, the existing U.S. matched
employer-employee datasets do not contain detailed information on workers’occupations.
This is not true however for the matched employer-employee datasets of other countries,
such as Denmark. The Integrated Database for Labour Market Research (IDA) created
by Statistics Denmark contains detailed information on all employers and all employees in
the Danish economy. The interested reader should refer to Frederiksen and Kato (2014)
for more details on the IDA dataset.13
The …rst column of Table 6 presents the wage premium for Denmark in 2007. As in
11
The wage premium is also present if I consider establishment size instead of employer size.
The model presented in the next section predicts that …rms with more occupations have more employees. In the data the correlation between number of workers and number of occupations, while positive,
is less than one (see Online Appendix), as there are other factors leading to variation in …rm size. The
fundamental mechanism in the model that generates the wage premium is the number of occupations,
rather than the number of …rm employees.
13
I am grateful to Anders Frederiksen for running the regressions below on the IDA dataset.
12
9
ln(labor income) ln(labor income)
ln(# of employees)
0.0124
-0.0114
(0:0002)
(0:0006)
4-6 occupations
0.0562
(0:0020)
7-10 occupations
0.1123
(0:0022)
11-20 occupations
0.1616
(0:0024)
21-40 occupations
0.1784
(0:0031)
>40 occupations
0.1958
(0:0042)
Table 6: Log Annual Labor Income on Firm Size and Number of Occupations in 2007.
Integrated Database for Labour Market Research - Denmark. Controls include gender,
years of schooling, quadratic in age and labor market experience and 9 industry code
dummies. Actual labor market experience. Labor market experience is constructed using
pension data from 1964 onwards. Log of annual labor income include both base pay and
variable pay components such as bonuses. 1,196,522 observations.
the U.S. data, an increase in …rm size, as measured by the number of employees, leads
to economically and statistically large increases in earnings. In the second column, I run
the same regression, but now ‡exibly control for the number of 3-digit occupations of
the …rm, by creating six groups. Increasing the number of occupations inside the …rm
leads to large and statistically signi…cant earnings increases for workers. In particular, all
else equal, workers in …rms with more than 40 occupations have almost 20% higher labor
income compared to workers in …rms with 3 occupations or less, conditional on …rm size.
The coe¢ cient on …rm size is now negative.14
Thus the results in Table 6 are consistent with the notion advanced in this paper that
a key mechanism behind the wage premium is the presence of more occupations in large
…rms rather than the number of …rm employees per se.
Summarizing, the patterns of within …rm mobility suggest that there is no typical
14
Brown and Medo¤ (1989) showed that the size premium persists even when controlling for a number
of …rm and worker characteristics. They did not control however for the number of occupations.
One explanation for the negative coe¢ cient is that in …rms with many employees, search frictions
are still present within the …rm, and that information dissemination is harder than in small …rms. For
example, in very large …rms it may be di¢ cult for a department that has an opening to have good
information about workers in other departments. In that case, workers who are in small …rms but with
many occupations are at an advantage compared to workers in big …rms with same number of occupations,
consistent with the regression results.
10
career-ladder that workers follow within the …rm. Instead, ‡ows within a …rm are o¤setting. Workers with low wages, as well as workers in large …rms are more likely to switch
occupations within their …rm. Moreover, workers in larger …rms are less likely to separate from their …rm. The size e¤ect is quantitatively large and holds even conditioning
on wages. In addition, I replicate the well-known size wage premium and also …nd that
workers in …rms with more occupations have substantially higher earnings, even when
holding constant the number of …rm employees.
3
Model
Motivated by the facts documented in the previous section, I develop a tractable, equilibrium model of within and across …rm reallocation. Given my …ndings above, I build on
the matching theory of Jovanovic (1979) and assume that workers learn about the quality
of their occupational match within a …rm. This is consistent with o¤setting ‡ows within
a …rm, documented above. There are no search frictions inside the …rm.15 Search frictions across …rms are modeled in a similar fashion to the search and matching literature:
matches between workers and …rms are determined by a matching function and the rents
to realized matches are split between workers and …rms using Nash bargaining.
In equilibrium, …rms with more occupations employ more workers. Workers in larger
…rms are more selective and abandon unpromising matches faster than workers in smaller
…rms. As a result they are more likely to switch occupations within their …rm, conditional
on wages. In addition, workers in larger …rms have more options, so the separation rate
is lower, even conditional on wages. In large …rms, across …rm reallocation is replaced by
within …rm reallocation. Workers with higher wages are less likely to switch occupations
both within, as well as across …rms. Workers in larger …rms are better matched and earn
higher wages. The within-…rm wage distribution has a similar shape as the economy-wide
wage distribution and both are empirically accurate.
I …rst describe the environment, solve for the equilibrium, then describe the implications and …nally derive the within-…rm and overall wage distributions.
3.1
The Economy
Time is continuous. There is a large mass of ex ante identical potential …rm entrants.
Firms enter the market by paying c. Each …rm, upon entering, draws the number of its
15
In the Online Appendix, I extend the model to allow for search frictions inside the …rm as well.
11
occupations, m, from a known distribution with support f1; 2:::; M g and corresponding
probabilities qm .16 Firms are destroyed exogenously according to a Poisson process with
parameter > 0, in which case all of its employees become unemployed.
There is a population of risk neutral workers of mass one. A worker can be either
employed or unemployed. Both workers and …rms have a discount rate equal to r.
There are search frictions: workers and …rms need to search for each other. Search is
undirected and an unemployed worker meets a …rm according to a Poisson process with
parameter > 0.17 All unemployed workers are identical and an unemployed worker
earns b.
When employed, a worker works in one task at a time. When he begins employment at
a …rm with m tasks, he randomly picks one of those m tasks which are ex ante identical.
The ‡ow output for worker i, in task j, in …rm l at time t is given by
dYtijl =
ijl
dt + dZtijl ;
G
where dZtijl is the increment of a Wiener process, ijl 2
; B is mean output per
unit of time and > 0. In addition G > B and productivities, ijl , are independently
distributed across tasks, …rms and workers.18
Furthermore ijl is unknown. Before starting work at a task, a worker draws his
productivity parameter, ijl , from a Bernoulli distribution, with parameter p0 , the probability that ijl = G .19 To avoid a situation where a worker never chooses to be employed,
I assume that p0 G + (1 p0 ) B > b.
At any point, a worker can leave his current task and go to another one or to unemployment. In the Online Appendix, I extend the model to allow for search frictions
inside the …rm, whereby a new task becomes available to the worker at some rate (see also
discussion in Section 3.5). There are no restrictions in the number of workers employed
in a task, i.e. there are no congestion externalities. A worker cannot return to a task.20
A worker separates from his …rm either endogenously or exogenously if his …rm exits the
16
I use the terminology “occupations” or “tasks”, but the reader may choose to think of these as
departments, teams, locations etc.
17
is the implied job …nding rate from an increasing, constant returns-to-scale matching function
m (u; f ), where u is the mass of unemployed workers and f is the mass of operating …rms.
18
In the Online Appendix, I also consider a setup where workers are hierarchically ranked, so that
abilities are positively correlated across tasks and workers learn about their unobserved productivity.
In addition, it is straightforward to extend the model to allow for hierarchies within occupations.
ijl
19
The results that follow also hold for the case where pijl
0 is not the same for all matches, but p0 is
distributed according to a known distribution with support [0; 1].
20
As noted in the Section 2, the vast majority of within …rm ‡ows, are switches to new occupations.
12
market.
Search frictions generate rents to realized matches which are split between the worker
and the …rm through the wage setting mechanism. Following the literature, I assume
that the wage is determined by generalized Nash bargaining between the …rm and each
worker separately, with 2 (0; 1) denoting the worker’s bargaining power. In Section
3.5 I discuss how the results below go through when …rms post wages. Nash bargaining
implies that the incentives of workers and …rms are perfectly aligned. In what follows,
I describe behavior from the worker’s point of view, e.g. “the worker leaves his current
task and moves to another one.”Alternatively one can also view the outcome as the …rm
reassigning the worker to another task.
While employed, workers observe output and obtain information regarding the quality
of the match in that speci…c task. Let pijl
t denote the posterior probability that the match
of worker i with task j in …rm l is good, i.e. ijl = G . In particular, a worker observes
his ‡ow output, dYtijl , and updates pijl
t , according to (Liptser and Shyryaev, 1977)
dpijl
t
G
=
pijl
t
1
pijl
t
dYtijl
pijl
t
G
+ 1
pijl
t
B
dt
;
(1)
B
where =
. The last term on the right hand side is a standard Wiener process
with respect to the unconditional probability measure used by the agents. To minimize
notation, from now on, I drop the t subscript, as well as the i, j and l superscripts.
The beliefs regarding the quality of the worker-task match follow a Bernoulli distribution. The posterior probability is thus, a su¢ cient statistic of the worker’s beliefs and a
state variable for his value function.
3.2
Equilibrium
I focus on the stationary equilibrium of the above economy. I analyze equilibrium wages
and optimal behavior of workers and …rms. In the next section I examine the model’s
implications and also derive the equilibrium …rm size.
Besides their employment status and beliefs regarding their current task, workers
also di¤er in their opportunities to work in other tasks within the …rm. The number of
remaining tasks, k, available to the worker in his current …rm, therefore, also constitutes
a state variable for the value of an employed worker.21
21
Note that the total number of tasks in the worker’s current …rm is not relevant, as employment
history in previous tasks does not a¤ect current or future payo¤s.
13
Given the process for the evolution of beliefs (equation (1)), the Hamilton-JacobiBellman equation of a worker employed in a task with posterior p and k 0 other tasks
are available to work in, is given by
rW (p; k) = w (p) +
1 2 2
p (1
2
p)2 Wpp (p; k)
(W (p; k)
U) ;
(2)
where W ( ) is his value, Wpp ( ) is the second derivative of W ( ) with respect to p and
w ( ) is his wage, which as shown below, depends only on p. The ‡ow bene…t of the worker
consists of his ‡ow wage, plus a term capturing the option value of learning, which allows
him to make informed decisions in the future (e.g. whether to leave his current task or
whether to quit to unemployment). Finally, at rate the worker exogenously separates
from the …rm.
The ‡ow value of an unemployed worker is given by:
rU = b +
M
P
qm W (p0 ; m
1)
U;
(3)
m=1
where qm is the fraction of …rms with m tasks.
The solution to the generalized Nash bargaining problem results in the linear sharing
rule:
J (p; k) = (1
) (W (p; k) U ) ;
(4)
where J (p; k) is the value to the …rm of employing a worker with posterior p and k tasks
remaining. As shown in Appendix A, using the above equation, I obtain:
Lemma 1 The worker’s wage is given by
w (p) =
where
(p) = p
G
+ (1
p)
B
(p) + (1
) rU;
(5)
is the worker’s expected output.
Note that the wage is an a¢ ne transformation of the posterior, p and does not depend
on k: As in the model of Mortensen and Pissarides (1994), wages only depend on the
current level of output and not on its future path.22
If there is an increase in the number of tasks available to the worker, then the worker
cannot be worse o¤, as he has the option of not trying them out, so W ( ) is non-decreasing
in k. Moreover, it cannot be the case that W (p; k + 1) = W (p; k), as that would imply
22
The wage equation (5) is almost identical to the wage equation in the Mortensen-Pissarides (1994)
model (see for instance equations (2.9) and (2.10) on page 42 of Pissarides, 2000).
14
that a worker can forever ignore one task and obtain the same value. This can not be
true, as the option value of trying out one more task is positive, hence:23
Lemma 2 The value of an employed worker, W ( ), is increasing in k.
A worker needs to decide when to leave his current task and move on to either another
task, or to unemployment.
Take the case of a worker with k 1. The solution to the optimal stopping problem
is given by a trigger p (k), such that the following value matching and smooth pasting
conditions are satis…ed:
W p (k) ; k = W (p0 ; k 1) ;
(6)
and
(7)
Wp p (k) ; k = 0:
Note that I allow the threshold, p (k), to depend on the number of remaining tasks in the
current …rm.
When there are no more tasks available, i.e. k = 0, the worker optimally quits to
unemployment when:
W p (0) ; 0 = U;
(8)
and
(9)
Wp p (0) ; 0 = 0:
In Appendix B, I use the boundary conditions (6) through (9) to solve for the value
function of a worker. The following proposition states the result:
Proposition 3 The value of an employed worker, W ( ), is increasing in p. A worker
optimally leaves his current task when his posterior hits p (k), de…ned as:
p (0) =
(
(
1) (rU
B
1) rU
B ) + ( + 1) (
23
G
rU )
;
(10)
Consider a worker with current posterior p < 1 and k tasks remaining. Assume W ( ) is increasing
in p (this is shown to be true below) and that one more task becomes available. Consider the following
strategy: if his current task proves unsuccessful, which, as shown below, occurs when his belief reaches
a threshold, p (k), instead of moving on to another task, he begins employment in the newly-available
task instead. In this case, he has k tasks remaining again, but a higher posterior p0 > p (k), so his value
must be higher. Since the worker leaves his current task with positive probability, then the option value
of trying out one more task is positive.
15
for k = 0 and:
p (k) =
(
( + 1) (
for k > 0, where
(3).24
=
1) (r + ) W (p0 ; k 1) (r +
G
B)
2 ((r + ) W (p0 ; k 1)
q
8(r+ )
2
r) U
(r +
B
r) U
B)
;
(11)
+ 1 and the value of unemployment is de…ned in equation
The above proposition, along with Lemma 2, immediately results in the following
corollary:
Corollary 4 The triggers p (k), are increasing in k.
In other words, the better their outside option, the more likely are workers to leave
an unpromising match.
Finally the mass of operating …rms is pinned down by the free …rm entry condition
M
P
qm Vm = c;
m=1
where Vm is the value of a …rm with m tasks and no workers.
3.3
Model Implications
In this section I discuss the model’s implications regarding wages, within …rm reallocation
and …rm separation rates. I also derive the equilibrium …rm size.
The equilibrium evolution of beliefs, employment states and remaining tasks is a positive recurrent process: starting from any posterior and any number of remaining tasks k,
(p; k) 2 (p (k) ; 1] [0; M 1], the joint process returns to (p; k) in…nitely many times, as
long as > 0.25 Therefore there exists a stationary distribution of beliefs and remaining
tasks.
I …rst show that workers are more productive, on average, in …rms with more tasks,
m. The proof consists of two steps: I …rst consider all workers with k tasks remaining and
24
A worker quits to unemployment only after exhausting all tasks available to him, i.e. W (p0 ; 0) > U .
To see this note that if a worker were to never work in the last task then this would imply that p (0) p0 .
Since p (0) < p (k) for all k > 0 (see Corollary 4 below), a worker chooses not work in the last task if and
only if he refuses to work in any task (i.e. the worker always prefers to remain unemployed). This does
not happen since by assumption p0 G + (1 p0 ) B > b.
25
An unemployed worker also returns to being unemployed in…nitely many times. The assumption that
> 0, while not necessary for the existence of a stationary equilibrium, ensures that in equilibrium not
all workers have found a good match and thus never switch occupations.
16
show that average productivity is increasing in k. I next establish that there is a higher
percentage of workers that have few tasks remaining (low k), in …rms with fewer tasks
(low m).
The following lemma, proved in Appendix C, states that the cross-sectional posterior
mean of all workers with k tasks remaining, is increasing in k.
Lemma 5 Let Fk (x) be the probability that a worker with k tasks remaining has posterior,
p, less or equal to x, i.e. Fk (x) Pr (p xjtasks remaining = k). Then
Z
1
p(k0 )
when k 0 > k.
Moreover, let
k
Z
pfk0 (p) dp >
Z
1
pfk (p) dp;
p(k)
1
p
G
+ (1
p)
B
fk (p) dp;
p(k)
denote the mean output of tasks in which the employed worker has k tasks remaining.
Then Lemma 5 immediately implies that:
Lemma 6 Workers who have more tasks available to work in, have, on average, higher
output, i.e.
k0
>
k;
whenever k 0 > k.
Intuitively, from Corollary 4, workers with many tasks remaining (high k) are more
selective, i.e. have higher p (k). This implies that when their (expected) output falls, they
are more likely than workers with few tasks remaining to switch to a new task with higher
output. As a result, on average, their output is higher than workers with few tasks.
I next show that in …rms with many tasks (high m), there is a lower percentage of
workers that have few tasks remaining (low k).
Intuitively, consider two …rms, one with m = 10 and one with m = 2. In the former
…rm, all workers start o¤ with k = 9. If their task does not work out, they move on to
k = 8 and so on. In the latter …rm, all workers start o¤ with k = 1 and then might move
on to k = 0. Comparing the steady state distribution across tasks remaining, k, in the
two …rms, it’s natural to expect a much lower percentage of workers to have k = 1 or
17
k = 0 in the …rm with 10 tasks compared to the …rm with 2 tasks. In addition, no workers
have k > 2 in the m = 2 …rm.
Formally, let sm
k be the share of workers with k tasks remaining, in …rms with m tasks
in total. The average productivity of …rms with m total tasks is given by
m
P1
sm
k
k:
k=0
I next examine how the employment share of task k, sm
k , changes as m increases.
Consider a worker who has just been hired by a …rm. Let Pr (m 1jm) denote the
probability that a worker starting o¤ at task m reaches task m 1.26 Then the probability
that a worker entering a …rm with m tasks, reaches task k is given by
Pr (m
1jm)
Pr (m
2jm
1)
:::
Pr (kjk + 1) :
Since Pr (jjj + 1) 2 (0; 1) for all j, the probability of a worker reaching task k m 1,
declines with the number of …rm tasks, m (even though the probability of being reassigned
to another task increases in the number of remaining matches). In other words, the
expected amount of time a worker spends in tasks greater than k is increasing in m.
However, conditional on reaching task k, the expected amount of time he spends in every
one of the remaining k tasks has not changed. The expected share of time a worker spends
in every task, k
m 1, while employed in the …rm, is therefore decreasing in m. By
Birkho¤’s Ergodic Theorem the following lemma holds:
Lemma 7 The steady state …rm employment share, sm
k , of every task, k
decreasing in the total number of …rm tasks, m.
m
1, is
Put di¤erently, the distribution of workers across tasks is better, in a …rst-order stochastic dominance sense, in …rms with more tasks. Lemma 7, along with Lemma 6, imply
that in …rms with a large number of tasks, fewer workers are employed in tasks where
average productivity is low, leading to the following proposition:
Proposition 8 Average …rm productivity,
m
P1
sm
k
k,
is increasing in the number of total
k=0
…rm tasks, m.
26
Ignoring shocks, the probability that a worker’s posterior belief, p0 , reaches p (m 1), before it
reaches 1, unconditionally on match quality, is given by 1 1p(mp0 1) (Karlin and Taylor, 1981). The result
that follows holds for
> 0, since Pr (jjj + 1) 2 (0; 1) for all j, when
18
is …nite.
Since wages are an a¢ ne function of expected output, then:
Corollary 9 Average wages are higher in …rms with more tasks, m.
This is consistent with the …ndings in Table 6 where workers in …rms with more
occupations earn higher wages.
It’s worth noting that since all workers start o¤ with the same wage, w (p0 ), regardless
of …rm size, the wage premium is generated through di¤erential wage growth rates in
large …rms relative to smaller ones.27 Empirically, the …rm-size wage premium is indeed
higher for workers with longer tenure. In particular, it more than doubles when comparing
newly hired workers to workers that have more than 9 years of tenure in the …rm (results
available upon request).
I next explore the framework’s implication regarding within …rm reallocation. The
probability a worker with posterior belief p, switches occupations when he has k tasks
remaining within the …rm, ignoring shocks, is given by (Karlin and Taylor, 1981)
Occ Switch Prob =
1
1
p
:
p (k)
The probability of switching occupations within the …rm is declining in p and therefore
the wage, w (p), but increasing in p (k). Corollary 4 states that p (k) is increasing in the
number of remaining tasks, k, while from Lemma 7 workers in high m …rms have higher
k on average. Thus the following proposition holds:
Proposition 10 (i) Conditional on wages, workers in …rms with more tasks, m, are
more likely to switch occupations within their …rm. (ii) Workers with higher wages are
less likely to switch occupations within their …rm.
Both predictions are consistent with the results of Table 2. Moreover, allowing for
exogenous match destruction, > 0, does not change the above result: It decreases the
probability of an occupational switch for all workers uniformly, since by assumption, it
does not depend on …rm size or wages.
I next explore the impact of occupational switching on wages. The model implies that
workers who switch occupations within a …rm see their wages increase, as they leave an
unproductive match and move on to a (potentially) more productive one. Formally, since
p0 > p (k) for all k, from Lemma 1, w (p0 ) > w p (k) :
27
It is easy to show that relaxing the assumption that p0 is the same for all matches (by allowing it
to be distributed according to a known distribution with support [0; 1]), leads to a positive initial wage
premium for workers in …rms with more tasks, m. All other results continue to hold.
19
Proposition 11 Workers who switch occupations within the …rm experience wage gains.
Indeed, this prediction is consistent with the data: Table 3 reveals that workers switching occupations within a …rm experience wage gains.
I now examine how the …rm separation rate varies with the number of occupations or
tasks, m. The probability that a worker, who just found employment in a …rm with m
tasks, separates endogenously, is given by (ignoring shocks)
Pr (m
1jm)
Pr (m
2jm
1)
:::
Pr (U nj1) ;
where Pr (U nj1) is the probability a worker starting o¤ in the last task quits to unemployment. Since Pr (jjj + 1) 2 (0; 1) 8j, the probability of endogenous …rm separations
declines with the number of …rm tasks, m. It is clear that the above result does not
change if > 0, since that shock is independent of m. The following proposition holds:
Proposition 12 Firms with more tasks, m, have lower separation rates.
Note that Proposition 12 implies that average tenure in …rms with more tasks is higher.
Thus the following corollary immediately follows:
Corollary 13 Firms with more tasks, m, have more employees.
Therefore, all the results where large …rms are de…ned to be those with more tasks,
also hold when considering the number of workers as the measure of size.
The SIPP does not contain information on the number of occupations per …rm, but
only its size (number of workers in reported in three bins). In the Online Appendix,
I use the Danish matched employer-employee data and document that …rms with more
occupations have more workers, consistent with Corollary 13 above.
The …rm separation probability of two workers with the same posterior p and therefore
the same wage, is lower for the one with higher k, based on the derivation above. From
Lemma 7, the following proposition is true:
Proposition 14 Conditional on the wage, the …rm separation probability declines in the
total number of …rm tasks m.
Again, this is consistent with the results presented in Table 4, where, conditional on
wages, the annual separation probability is approximately 6% lower for workers in large
20
…rms. Moreover, consistent with the data, wages negatively a¤ect the …rm separation
probability.
Comparing two workers who are similarly matched (and therefore earn the same wage),
the worker in the larger …rm is less likely to separate; indeed if their match doesn’t work
out, the worker employed in the larger …rm has more alternatives within his …rm and
therefore is less likely to leave his employer (Proposition 14). Indeed, as shown above
(Proposition 10 and Table 2), conditional on wages workers in large …rms are more likely
to move within the …rm, which is consistent with them having more options available.28
3.4
Within-Firm and Economy-Wide Wage Distributions
I next consider the distribution of wages in the above economy, as well as the distribution
of wages with each …rm. Both distributions are derived analytically and the resulting
expressions are presented in Appendix D, while the detailed derivations can be found in
the Online Appendix. The following proposition summarizes the …ndings:
Proposition 15 The unique distribution of wages for the above economy is given by
equations (19) and (20) of Appendix D. In addition, the distribution of wages within a
…rm is given by equations (21) and (22) also in the Appendix. Both wage distributions are
decreasing in the region [w (p0 ) ; w (1)] and feature a fat right Pareto-type tail if > 2 .
The functional form of the within …rm wage distribution is the same as the wage
distribution of the economy, though the coe¢ cients are di¤erent. This is consistent with
empirical evidence from several countries presented in Lazear and Shaw (2009). Moreover
both wage distributions have an empirically accurate shape and can replicate the fat
Pareto-type tail that we observe in the data (see also discussion in Moscarini, 2005).
3.5
Extensions
I next discuss some extensions/modi…cations of the current setup:
Search frictions inside the …rm: I relax the assumption that there are no frictions
inside the …rm. Consider instead the case where a task becomes available inside the
…rm at some rate and a worker needs to decide whether to switch to a new task. If his
beliefs regarding his current task are low enough, he is willing to switch to a new task
if one becomes available. Whether he is willing to switch is captured by some threshold
28
The model also implies that workers see their wages decline prior to an occupational switch. This is
indeed true in the data (results available upon request).
21
level of beliefs. If however a task does not become available and his belief continues to
fall, he may decide that it’s not worth waiting for a new task and instead separate to
unemployment. This will be determined by a di¤erent belief threshold. Both thresholds
depend on the number of remaining tasks, k. Alternatively, it could be the case that
the worker always prefers to remain in his current task waiting for another one, rather
than quit to unemployment. In the Online Appendix I solve for these thresholds and
characterize optimal worker behavior when there are search frictions inside the …rm.
Human capital accumulation: I also extend the model to allow for the possibility
of human capital accumulation. In particular, there are now two types of workers: workers
who have accumulated human capital and workers without human capital. Workers who
do not have human capital, accumulate it at some rate, . Expected output produced by
(p), where > 1, while workers without human
workers with human capital is now
capital produce (p) as before. In the case of a separation, a worker loses the human
capital he has accumulated.
Workers behave di¤erently depending on their human capital level. More speci…cally,
there are now two threshold, pnohc (k) and phc (k). The solution of the model in this case,
which is substantially more complicated, is presented in Appendix E, whereas detailed
derivations are contained in the Online Appendix.
On-the-job search: The model can be extended to allow for on-the-job search. In
particular, assume that, following an appropriate modi…cation of the matching function,
employed workers are allowed to contact other …rms at rate , where > 0 captures
the e¤ectiveness of employed workers’search technology relative to unemployed workers.
Search is costless and unobserved by the …rm.
If an employed worker meets another …rm, he chooses the …rm where his value is the
highest when receiving the wage resulting from Nash bargaining. In the Online Appendix, I show that this is the equilibrium outcome of an ascending auction in which the
current and poaching …rms place bids to attract the worker. I also discuss there how the
assumptions behind the auction preserve the convexity of the payo¤ set, thus overcoming
Shimer’s (2006) critique.
An employed worker with current belief p and k tasks remaining, who contacts a
…rm with m tasks, switches to the new …rm if and only if W (p0 ; m 1) > W (p; k) or
p < potj (k; m), where the implicit threshold potj (k; m), depends on both k and m. An
employed worker’s value function now becomes
22
1 2 2
p (1 p)2 Wpp (p; k)
(W (p; k) U )
2
M
P
qm max fW (p0 ; m 1) ; W (p; k)g W (p; k) ;
rW (p; k) = w (p) +
+
m=1
where the last term captures the discrete gain to the worker if he meets a …rm where his
value is higher. The value of the …rm, J (p; k), is similarly modi…ed to capture its loss
when a worker moves to another …rm.
Following similar calculations as in the model without on-the-job search (see steps in
Appendix A), the wage function now becomes
w (p) =
(p) + r (1
)U
M
P
qm I fW (p0 ; m
(12)
1) > W (p; k)g ((1
) (W (p0 ; m
1)
W (p; k)) + J (p; k)) :
m=1
Note that the worker’s wage now has an additional term compared to the case with no
on-the-job search (equation (5)). E¤ectively, his wage is reduced by an amount proportional to his search intensity. When the worker leaves his current …rm for another …rm,
the separation is no longer bilaterally e¢ cient, as there are lost rents for the incumbent
…rm. The worker compensates his …rm by an amount equal to the weighted average of
M
P
the expected worker’s gains,
qm I fW (p0 ; m 1) > W (p; k)g W (p0 ; m 1) W (p; k),
m=1
and the …rm’s losses, J (p; k), multiplied by his job …nding probability.
Unlike the model without on-the-job search, it is not possible to solve for the decision thresholds p (k) and potj (k; m) analytically, but instead they need to be computed
numerically.
Directed search and wage posting: In the main model above wages are determined
by generalized Nash bargaining between the worker and the …rm. Li and Tian (2013)
modify the setup developed in present paper, to allow for …rms to post wages (rather
than bargain with the workers). In addition, workers now choose to which …rm they
should apply (directed search). In equilibrium, workers are indi¤erent across …rms with
di¤erent number of occupations, m. That setup as well can generate the observed sizewage premium and also a negative relationship between …rm size and worker separation
rates.
23
mL = 2 mL = 3 mL = 3 mL = 4 mL = 4 mL = 5 mL = 5
Parameters: mS = 1 mS = 1 mS = 2 mS = 1 mS = 2 mS = 1 mS = 2
p0
0.21
0.296
0.118
0.261
0.192
0.309
0.255
0.304
0.139
0.28
0.108
0.245
0.12
0.172
L
q
53.81% 51.97% 58.55% 50.18% 57.12% 50.38% 56.73%
0.058
0.072
0.072
0.073
0.08
0.079
0.085
G
27.61
41.56
46.34
60.16
37.57
44.20
41.52
B
4.37
-4.07
4.15
-8.81
2.37
-5.94
-1.98
Table 7: Calibrated Parameters
3.6
Calibration
I next calibrate the model to show that it can replicate some of the key facts presented in
this paper. I use information on worker reallocation and the level of wages to calibrate the
model’s parameters and then check some additional moments, such as the wage premium.
A number of observable characteristics such as education, gender and race have been
found to impact wages (Abowd et al., 1999) and occupational mobility (Kambourov and
Manovskii, 2008). Given that, I restrict my analysis to the largest subsample in my data
along these dimensions: white males with a high school education.
Following the size partition in the data, I divide …rms to those with more than 100
employees and those with fewer than 100 employees. To match the SIPP’s sampling
procedure, I sample the simulated data every four months. I next describe the calibration
procedure. Appendix F contains more details.29
I …rst discuss the parameters that are calibrated independently. The discount rate, r,
is set to 1.64% which equals approximately 5% annually. The job-…nding probability, , is
calibrated as in Shimer (2012), leading to a value of 0.78. Since the SIPP contains weekly
employment information, time aggregation is not a concern. Following Shimer (2005),
the Nash bargaining coe¢ cient, , is set to 0.72. Finally, the value of home production
b, is set to $3.42 which is 30% of average wage in the sample. In what follows I consider
di¤erent combinations for the number of occupations in large and small …rms, mL and
mS . As it turns out, smaller numbers are better able to match the moments.
I need to calibrate the following six parameters: the initial belief, p0 , the signal to noise
ratio, , the exogenous separation rate , the probability an unemployed worker contacts
29
Even though wages in the SIPP are top-coded at $30, this a¤ects less than 1% of observations in the
sample.
24
mL = 2
Data mS = 1
62.94% 62.90%
$11.41 $11.41
$9.50
$9.48
8.75% 8.63%
12.41% 12.58%
-0.032 -0.038
Targeted Moments:
Empl Share Large Firms
Mean Wage
Initial Wage
Separation Prob Large
Separation Prob Small
Large Coe¢ cient
Other Moments:
Wage Gains upon Switch
4.97% 4.90%
Wage Coe¢ cient
-0.0095 -0.0126
Wage Growth after Switch 8.22% 7.09%
% of Wage Pr Explained
mL = 3
mS = 1
62.89%
$11.40
$9.62
8.32%
13.08%
-0.045
mL = 3
mS = 2
62.89%
$11.41
$9.40
9.29%
11.16%
-0.019
mL = 4
mS = 1
63.03%
$11.42
$9.44
7.99%
13.65%
-0.054
4.02%
-0.0109
7.79%
3.38% 2.74%
-0.0056 -0.0091
10.36% 9.76%
mL = 4
mS = 2
62.93%
$11.41
$9.42
9.12%
11.69%
-0.025
mL = 5
mS = 1
62.97%
$11.41
$9.71
8.17%
13.76%
-0.052
mL = 5
mS = 2
62.96%
$11.41
$9.43
9.03%
11.77%
-0.026
2.42% 1.53%
-0.006 -0.0101
11.71% 8.79%
2.32%
-0.0056
11.97%
12.50% 21.79% 13.33% 30.20% 19.13% 34.63% 20.78%
Table 8: Moments
a large …rm rather than a small one, q L ,30 the mean output in a good match, G , and the
mean output in a bad match, B . To do so, I use six moments. In particular, I target
the share of employment in large …rms, the separation rates of large and small …rms, the
average wage level and the average initial wage level. The last moment targeted is the
coe¢ cient of the large …rm dummy in the linear probability regression of the probability
of …rm separation on wages and …rm size. The calibrated parameters are presented in
Table 7.
Informally, identi…cation works as follows: the share of employment in large …rms
pins down q L . G and B are pinned down by the average wage and initial wage levels.
The separation rate of small …rms and the coe¢ cient on large …rms pin down p0 and .
In particular, workers in small …rms separate more often because they are both worse
matched (lower belief) and have fewer options within the …rm; when conditioning on
wages (and thus beliefs), workers in smaller …rms still separate more often because of fewer
options. The initial belief, p0 , determines whether or not separation will (eventually) take
place: if p0 is very high (close to 1), few workers separate; conversely, if p0 is very low
almost all matches eventually dissolve. Thus the overall separation rate of small …rms,
for a given , pins down p0 . On the other hand, , determines how fast a worker learns
and thus separates. The di¤erence in the separation rates, conditional on the wage, for a
30
Note that I don’t assume that workers contact each …rm with the same probability, but allow for the
possibility of workers oversampling large …rms relative to small.
25
35
mL=2, mS=1
mL=4, mS=2
30
25
20
15
10
0.2
0.3
0.4
0.5
Lamda
0.6
0.7
0.8
Figure 1: Fraction of Wage Premium Explained.for Di¤erent Values of Lamda
given p0 , pins down . Finally, the separation rate of large …rms pins down (conditional
on p0 and ).
Table 8 presents the targeted moments, as well as its empirical counterparts. The
model, especially the speci…cations with low values of mL and mS , matches the targeted
moments well. More speci…cally, all speci…cations match the wage levels and the employment shares well. Speci…cations with low values for mL and mS are better able to match
the separation probabilities and the coe¢ cient on the large …rm dummy.
In addition, Table 8 reports three moments that are not targeted in the calibration:
the wage gain that a worker enjoys when switching occupations within the …rm; the wage
coe¢ cient in the separation regression conditional on size; and the annual wage growth
rate in the …rst year following an occupational or …rm switch. Again the speci…cations with
low values of mL and mS match well these non-targeted moments, whereas larger values
are somewhat less successful.31 This suggests that a small number of occupations is a
better approximation.32;33 Finally, in all speci…cations, the fraction of the wage premium
31
Introducing …rm-speci…c human capital in the mL = 2 and mS = 1 calibration leads to similar results.
The solution of the model with human capital is presented in Appendix E and detailed derivations are
contained in the Online Appendix.
32
The Danish data suggests that a small number of occupations is a good approximation for …rms with
fewer than 100 employees.
33
Large …rms, have a number of di¤erent occupations, ranging from janitors to accountants. For the
calibration of this setup however, I am not interested in the total number of occupations in each …rm,
but rather the number of relevant occupations for each worker, which is smaller.
Moreover note that even though the worker does not keep track of past occupations, in equilibrium he
would prefer a new occupation to one he’s already tried out since p0 > p (k) for all k. Thus mL and mS
should be seen as the number of “new”occupations a worker encounters in a …rm, rather than the …rm’s
26
explained ranges from 12.50% to 34.63%. However besides search frictions, there are
arguably other impediments to switching …rms, such as administrative costs and adverse
selection (Acemoglu and Pischke, 1998), whose impact is the same as a decrease in the
job …nding rate. In addition, the job …nding rate ‡uctuates signi…cantly (Elsby et al.,
2009, Shimer, 2012). Given these concerns, it is instructive to consider how the wage
premium varies with the job …nding rate. Figure 1 presents the fraction explained of the
observed wage premium for other values of the job …nding rate, , for the speci…cations
with mL = 2; mS = 1 and mL = 4; mS = 2 which …t the data well. As expected, as
falls, the fraction of the premium explained increases. Workers in larger …rms are now
much better matched than their counterparts in smaller …rms.
4
Conclusion
This paper links the reallocation of workers across occupations but within …rms, and the
reallocation of workers across …rms. Unlike a large literature that has studied frictional
labor markets, while ignoring within …rm mobility, this paper considers the equilibrium
interaction of the two forms of reallocation in the presence of frictions.
I …rst investigate empirically worker mobility within and across …rms and how it’s
a¤ected by the size of the …rm and wages. Workers in large …rms replace across …rm
mobility with within …rm mobility: Workers in large …rms are less likely to separate,
but more likely to switch occupations within their …rm, even conditional on wages. High
wage workers are less likely to reallocate either within a …rm, or across …rms. Moreover, I
…nd that occupational mobility within …rms is not consistent with a career-ladder model.
Flows within a …rm are largely o¤setting. I also show that workers in …rms with more
occupations have substantially higher earnings, even when holding constant the number
of …rm employees.
I then show that bringing together within …rm and across …rm reallocation along with
labor market frictions, naturally accounts for the di¤erences in worker ‡ows and wages
across …rms of di¤erent sizes. I introduce a tractable equilibrium model of within …rm
and across …rm mobility and prove that it is consistent with the observed facts.
The tractability of the model implies that it can be extended in several dimensions: In
the Appendix and Online Appendix, I consider frictions within the …rm, human capital
total number of occupations.
Finally, while in general the level of within …rm mobility is overpredicted in the model, the combinations
of mL = 2 and mS = 1, as well as mL = 3 and mS = 1 are closer to their empirical counterpart than
higher values of mL and mS (3.42% and 4.99% respectively compared to 2.69% in the data).
27
accumulation and on-the-job search. In addition, future work can consider ex-ante worker,
as well as …rm heterogeneity, which are relatively straightforward to introduce in the
model. The contribution of the present paper is that within and across …rm reallocation
should no longer be viewed in isolation.
Appendix
A
Proof of Lemma 1
The value to the …rm of employing a worker with posterior p and k is given by
rJ (p; k) =
(p)
Multiplying eq. (2) by 1
(1
) r (W (p; k)
1 2 2
p (1 p)2 Jpp (p; k)
J (p; k) :
2
and subtracting (1
) rU leads to
(13)
w (p; k) +
U ) = (1
1
(1
) 2 p2 (1 p)2 Wpp (p; k) (14)
2
) (W (p; k) U ) (1
) rU:
) w (p; k) +
(1
Multiplying eq. (13) by , subtracting eq. (14) from it and using the surplus sharing
condition, eq. (4), leads to
w (p; k) =
(p) +
1 2 2
p (1
2
p)2 ( Jpp (p; k)
(1
) Wpp (p; k)) + (1
) rU:
(15)
Finally, by taking the second derivative with respect to p in the surplus sharing condition, eq. (4), and using eq. (15), results in the wage equation, (5).
B
Proof of Proposition 3
The surplus of the match between the …rm and the worker, S (p; k), is given by
S (p; k) = W (p; k) + J (p; k)
U:
Substituting in for W (p; k) and J (p; k) leads to
(r + ) S (p; k) =
(p)
rU +
28
1 2 2
p (1
2
p)2 Spp (p; k) :
The general solution to the above di¤erential equation is given by
S (p; k) =
where
=
q
8(r+ )
2
1
(p) rU
+ K1k p 2
r+
1
2
1
1
1
1
1
p) 2 + 2 + K2k p 2 + 2 (1
(1
1
2
p) 2
;
+ 1 and K1k and K2k are undetermined coe¢ cients that depend on k.
1
1
When p ! 1 however, limK2k p 2 + 2 (1
p!1
1
p) 2
1
2
1
= K2k 1 lim (1
1
2
p) 2
p!1
= +1 which
follows from > 1. Since the present discount sum of the output produced by a good
G
match is given by r+ < 1, it must be the case that K2k = 0, 8k. Thus
1
(p) rU
+ K1k p 2
r+
S (p; k) =
1
2
1
1
p) 2 + 2 ;
(1
where K1k is an undetermined coe¢ cient. Moreover
G
Sp (p; k) =
B
+ K1k
r+
1
2
1
2
p p
1
2
1
2
(1
p)
1
+ 12
2
:
Consider the case where k = 0. Using equation (4), one can rewrite the value matching
and smooth pasting conditions (equations (8) and (9) respectively), in terms of S ( ) and
use them to pin down p (0) and K10 .
From equation (9), one immediately obtains
G
K10
=
B
r+
1
2
1
2
1
1
1
p (0) 2 + 2
p (0)
1
p (0)
1
2
1
2
:
Substituting for K10 into equation (8) and solving leads to equation (10).
Similarly in the case where k > 0, equation (7) leads to
G
K1k =
r+
B
1
2
1
2
1
p (k)
1
1
p (k) 2 + 2
1
p (k)
1
2
1
2
:
Substituting for K1k into equation (6) leads to equation (11).
To show that the value function is increasing in p, note that straightforward calculations show that Spp (p; k) > 0, and therefore Wpp (p; k) > 0, for all p and k. Given that
Wp p (k) ; k = 0, this implies that Wp (p; k) > 0 for all p > p (k).
29
C
Proof of Lemma 5
R1
In this appendix I want to show that p(k) pfk (p) dp is increasing in p (k), and therefore k
(Corollary 4). Intuitively the higher boundary does not let the process move away from
p0 towards zero and shoots it back to p0 sooner. I prove that this holds for the more
general case where each worker draws his prior, p0 from some distribution with support
[0; 1] and known density h ( ). In this case, a worker begins employment at task, as long
as his draw is greater than p (k). This implies that workers who enter the distribution,
h(p0 jp0 >p(k))
do so at various priors according to 1 H p(k) .
( )
First consider the set of all workers with k tasks remaining who draw the same prior,
p0 2 p (k) ; 1 . Let pt be a di¤usion process that starts at p0 2 (0; 1), evolves according
to equation (1), while at a Poisson rate > 0, it returns to p0 . Finally, let p (k) 2 (0; p0 ),
be a re‡ective boundary, such that when the process hits it, it immediately returns to p0 .
Moreover de…ne
Z
1
I p0 ; p (k) =
p(k)
p0
(p) dp;
pgp(k)
p0
where gp(k)
(p) is the steady state density of the above process. Then
Z
1
p(k)
pfk (p) dp =
Z
1
I p0 ; p (k)
h p0 jp0 > p (k)
h(p0 jp0 >p(k))
is higher when p (k), and therefore k
p(k)
For a given p0 > p (k), the weight
1 H (p(k))
1
H p (k)
dp0 :
(Corollary 4), is higher. Moreover, if I p0 ; p (k) is increasing in p (k) and p0 and therefore
k, then the left hand side is increasing in k.
I proceed to show that I p0 ; p (k) is increasing in p (k) and p0 .
I …rst want to prove that if p1 < p2 , then I p0 ; p1 < I p0 ; p2 .
De…ne
Tp = inf ft > 0 : pt = pg :
Call pit the di¤usion with boundary pi . From Athreya and Lahiri (2006), Theorem
30
14.2.10, part (i),34 it is the case that
I p0 ; p =
1
Ep0 Tp
Ep0
Z
Tp
(16)
ps ds :
0
Since
Ep0 Tp1 = Ep0 Tp2 + Ep2 Tp1 ;
and
Ep0
Z
Tp1
p1s ds
Z
= Ep0
0
Tp2
p2s ds
+ Ep2
0
Z
Tp1
p1s ds ;
0
straightforward algebra implies that it su¢ ces to prove that
1
Ep2 Tp1
Z
Ep2
Tp1
p1s ds
< I p 0 ; p2 :
(17)
0
De…ne a process Yt which starts from p2 and evolves like p1t with the only di¤erence
that whenever it reaches p1 , it is resets to p2 , rather than p0 . On the contrary, p2t starts
from p0 . Run Yt and p2t with the same Brownian motion, Wt , and the same Poisson times
of resetting. Now, when Yt and p2t meet, they continue together until p2 is hit. Then p2t
jumps to p0 , while Yt continues. Moreover, since p2t starts above Yt , it is always true that
Yt p2t (in the case of shock, p2t jumps to p0 and Yt jumps to p2 ). However for a positive
proportion of time the inequality is strict. Thus
1
lim
t!1 t
Z
0
t
1
Ys ds < lim
t!1 t
Z
t
p2s ds;
0
which using eq. (16) above, leads to eq. (17).
I also need to prove that if p30 < p40 , then I p30 ; p < I p40 ; p . Now call pit the di¤usion
starting o¤ from pi0 . As before, since
Ep40 Tp = Ep40 Tp30 + Ep30 Tp ;
and
Ep40
Z
0
Tp
p4s ds
= Ep40
Z
Tp3
0
0
34
p4s ds
+ Ep30
Z
Tp
p3s ds ;
0
Here set f (Xj ) = Xj . The theorem states the proof in the case where Xj is a discrete time process.
To prove it in the continuous time I follow the proof of the discrete case. The sums Yi in that proof are
replaced by integrals and the sequence of regeneration times in this case is the times where pit hits pi
(i = 1; 2). In each of these times the di¤usion pit restarts and the trajectories between these times are
i.i.d.
31
straightforward algebra implies that it su¢ ces to prove that
I p30 ; p <
R Tp30
Ep40
0
p4s ds
(18)
:
Ep40 Tp30
Using a similar argument as above, de…ne a process Zt which starts from p40 and evolves
like p4t with the only di¤erence that whenever it resets to p40 whenever it reaches p30 rather
than p. On the contrary, p3t starts from p30 and resets at p. Running Zt and p3t with the
same Brownian motion, Wt , and the same Poisson times of resetting implies as above
that35
Z
Z
1 t
1 t 3
ps ds < lim
Zs ds;
lim
t!1 t 0
t!1 t 0
which using eq. (16) above, leads to eq. (18).
Since shocks are i.i.d. across workers, by Birkho¤’s Ergodic Theorem time averages
equal to space averages. Therefore by proving that the mean of the above system is
increasing in p (k), I also prove that the mean of the cross-sectional distribution of workers
with k task remaining is also increasing in p (k).
D
Within-Firm & Economy-Wide Wage Distributions
Let z (w) denote the wage density of employed workers. Then as derived in the Online
Appendix, the economy-wide steady-state wage distribution is the following:
For w 2 w p (0) ; w (p0 )
2
z (w) =
G
B 2
B
[ w
+ (1
) rU
1
G
w
(1
2
) rU
M
X1
Ck1 (19)
k=0
w
B
+ (1
) rU
2
G
w
(1
) rU
1
M
X1
1
Ck1
k=0
p (k)
p (k)
and for w 2 [w (p0 ) ; w (1)]
z (w) =
2
G
B 2
w
B
+ (1
) rU
1
G
w
(1
) rU
2
M
X1
Ck3 ;
k=0
35
(20)
Essentially I now have p40 in the place of p0 , p30 in the place of p2 , and p in the place of p1 . In the
place of Yt I have p3t and in the place of p2t I have Zt .
32
2
1
];
where
1 2
[p0 (1
2
1 p (k) 2 1
+
p0
p (k)
Z p0
p 1 (1 p)
Ck3 = Ck1 f
1
p0 )
1
(1
2
( 1
p0 )
(
1
dp +
p(k)
1
2
2
1
f
2
(1
2
2 ) p (k)
( 1
1
+2
p (k)
p (k)
1
p (k)
+ 3p0 ) p0 (1
+ 3p0 )
g
1
p0 )
+
3p0 )]
2
Z
1
p0
2
p
(1
p)
1
dp
p(k)
Z
1
1
p
(1
p)
2
dpg 1 ;
p0
and Ck1 is determined by the solution of a M by M system is linear system of equations,
given by equation (5) of the Online Appendix.
Similarly, let zm (w) denote the wage density of workers employed in a …rm with m
tasks. As derived in the Online Appendix, the within-…rm wage distribution, for a …rm
with m tasks, is the following:
For w 2 w p (0) ; w (p0 )
2
zm (w) =
B 2
G
B
[ w
+ (1
) rU
1
G
w
(1
2
) rU
m
X1
Ckm1 (21)
k=0
B
w
+ (1
2
) rU
G
w
(1
1
) rU
m
X1
1
Ckm1
k=0
p (k)
p (k)
and for w 2 [w (p0 ) ; w (1)]
zm (w) =
2
B 2
G
B
w
+ (1
) rU
1
G
w
(1
) rU
2
m
X1
Ckm3 ;
k=0
(22)
where
Ckm3 = Ckm1 f
+
1
Z
p (k)
p (k)
1 2
[p0 (1
2
2
p(k)
p
p0
(1
1
( 1
+ 3p0 )
1
p0
1
p0 )
p)
1
(1
2
p0 )
dp +
(
1
+2
p (k)
p (k)
33
2
3p0 )]
1
Z
p0
p(k)
p
2
(1
p)
1
dp
2
1
];
1
2
2
1
f
2
(1
2
2 ) p (k)
( 1
1
1
p (k)
+ 3p0 ) p0 (1
p0 )
g
1
+
Z
1
p
1
(1
p)
2
dpg 1 ;
p0
and Ckm1 is given by the m by m linear equation system given by equations (11) and
(12) of the Online Appendix. Note that the above expressions give the solution for the
within-…rm wage distribution, for a given m. In other words, one needs to solve a di¤erent
system of linear equations for every m.
It’s worth noting that the functional form of the within-…rm wage distribution is
the same as the wage distribution of the economy, though of course the coe¢ cients are
di¤erent. This is consistent with empirical evidence from several countries in Lazear and
Shaw (2009).
Finally, as demonstrated in the Online Appendix, both the overall wage distribution,
as well as the within-…rm wage distribution are declining from w (p0 ) until the highest
possible wage, w (1), and they also feature a fat Pareto-type tail, as long as > 2 . The
latter is the same condition as in Moscarini (2005). The intuition is that if the speed of
learning ( ) if fast enough relative to the rate at which workers get replaced ( ), then
“too many” workers will …nd out that they’re in a good match and receive very high
wages. Conversely, if learning is relatively slow compared to the replacement rates, then
relatively few workers will reach those high levels.
E
Human Capital
The derivations of the two threshold functions, pnohc (k) and phc (k) can be found in the
Online Appendix. Below are the resulting expressions.
For the case of a worker who has not accumulated human capital, the threshold
nohc
p
(k) is implicitly de…ned by
lim[
p!1
+
1
1
1
1
2q
pq (1
1 q
p (1
2q k3
g (W )
1
q
q
pnohc (k)
1 q
1
p)1
p)
1
[ k1
q
pnohc (k)
pnohc (k)
q
1
!q
1
0
@
34
pnohc (k)
1
q 1
1
!q
pnohc (k)
pnohc (k)
!
q
1 p
p
q
1
pnohc (k) W
1
q 1
1 p
A
p
k2
+
+
1
q (q
1
2q
1
1
@
1)
1
p1
q
nohc
q+p
(k)
pnohc (k)
q
p)q [k1
(1
1
+q
1
1
0
@ 1
1
q (q
1)
g (W ) =
k2 =
q
p
p
G
rU
r+
1
1
2r+ +
r+ +
B
hc
rU +
r+ +
G
hc
=
pnohc (k)
1
G
k1 =
=
s
B
1
q+p
p
1
1
q 1
p
1
A]
p
q W
! q!
;
2
nohc ;
K1k ;
q
1
pnohc (k)
pnohc (k)
!1 q 1
nohc
(k)
1 p
A
nohc
p
(k)
! q
!!
q pnohc (k)
1 pnohc (k)
]]
pnohc (k)
pnohc (k)
p
1 q
q
!q
pnohc (k)
1
p
q
p
nohc
(k)
pnohc (k)
p
rU
+
r+ +
r+ +
k3 =
p
1
p
where
1
1
1
q
p
G
=
1
p)q pnohc (k)q
(1
1 1
p
2q k3
+g (W )
+k2
0
B
r+
8 (r + )
2
hc
rU
r+
B
!
+ 1;
nohc
=
1
; q=
2
G
;
r+ +
r+ +
1
;
1
2
1
2
r
hc ;
4 + k3
k3
!
;
B
=
:
Finally for the case where k = 0, then W = 0, whereas for the case where k > 0,
W = Ep0 S nohc (p0 ; k
1) ;
where
S nohc (p; k) =
1
1
+
2q
1
1
pq (1
p)1
1 q
p (1
2q k3
q
pnohc (k)
q
q 1
pnohc (k)
1 q
0
!q
nohc
1
p
(k)
1
@
p)1 q [ k1
q 1
pnohc (k)
35
1
pnohc (k) W
1
1
p
p
q 1
1
A
g (W )
+
+
1
1
q
0
pnohc (k)
pnohc (k)
q + pnohc (k)
1
@
k2
q (q 1)
pnohc (k)
1
1
2q
1
1
p1
q
1
+q
1
q (q
0
@ 1
1
1
1
p) [k1
q
1
p
1)
pnohc (k)
pnohc (k)
q
1 q
p
q
p
1
q
!
1
q+p
p
pnohc (k)
1
1
p
q 1
1
A]
p
q W
! q!
pnohc (k)
pnohc (k)
!1 q 1
1 pnohc (k)
A
pnohc (k)
! q
!!
1 pnohc (k)
1 pnohc (k)
q nohc
]:
pnohc (k)
p
(k)
p
p
q
!q
pnohc (k)
1
1
p
q
p
p
1
q
(1
1
1
p)q pnohc (k)q
(1
1 1
p
2q k3
+g (W )
+k2
q
!q
p
p
1
For the case of a worker who has accumulated human capital, we have
phc (0) =
(
hc
(
1) (rU
1) rU
B) + (
hc
B
hc
+ 1) (
G
rU )
;
and
( hc 1) (r + ) Ep0 V hc (p0 ; k 1) (r +
r )U
hc
2 (r + ) Ep0 V (p0 ; k 1) + 2 (r +
r ) U + hc ( G
phc (k) =
where:
S hc (p0 ; 0) =
S hc (p0 ; 0) =
(p0 )
r+
1 1
2 2
G
K10 =
=
r+
rU
G
B
rU
1
+ K10 p02
1
2 hc
1
1
1
2
1
2 hc
p0 ) 2 + 2
(1
hc
B)
G
+
B
+
;
;
r+
1
hc
B
r+
G
K1k
(p0 )
r+
B
B
1
2
1
2
1
2
1
2
1
1
phc (0) 2 + 2
phc (0)
hc
1
hc
1
hc
p (k)
36
phc (0)
1
1
1
1
phc (0) 2 + 2
phc (0)
hc
1
phc (k) 2 + 2
1
p02
1
2 hc
hc
1
phc (0)
hc
1
phc (k)
1
1
2
1
2 hc
1
2
1
2 hc
1
p0 ) 2 + 2
(1
;
:
hc
;
F
Calibration Details
The model is calibrated in two stages. First, some parameters are calibrated independently
(r; ; and b). Second, I pick a combination of mL and mS and calibrate the remaining
six parameters (p0 ; ; ; q L ; G and B ) by targeting the six moments described in the
main text.
In order to simulate the model, I …rst need to compute the worker’s optimal decision
rules. In particular, the triggers, equations (10) and (11), are a function of the value of
an unemployed worker, equation (3), which in turn depends on the value of an employed
worker, equation (2). I therefore use a …xed point in order to compute the value of
unemployment and the equilibrium triggers.
I simulate a discrete-time version of the model presented above, by discretizing the
model presented in Section 3. I use the Central Limit Theorem to approximate the Wiener
process of equation (3) by the sum of repeated Bernoulli trials. Similarly the exogenous
destruction shock, , is approximated by a Poisson distribution. I exploit the ergodicity
of the setup and simulate a single worker for 4 million periods each time.
The weighting matrix used to match the six moments is the inverse of the empirical
variance-covariance matrix of these moments. This matrix is obtained by bootstrapping
the data 10,000 times.
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