OSIPP Discussion Paper : DP-2007-E-002

The Persistence of Differences in Productivity, Wages, Skill Mixes and Profits
Between Firms
March 27, 2007
Katsuya Takii
Associate Professor, Osaka School of International Public Policy (OSIPP)【キーワード】J24, L25, Organization Capital, Assignment, Persistence【要約】In this paper, we construct a dynamic assignment model that can provide a unified
explanation of several observed features of persistent differences in productivity, wages, skill mixes
and profits between firms. Large organization capital (high firm-specific knowledge) attracts
skilled workers, who can create further organization capital in the future. This positive feedback
brings about persistent differences in these variables. We also analyze how the real and perceived
values of a firm's organization capital interactively influence persistence.

We estimateparameters and simulate the model. Our results show that a positive assignment mechanism
accounts for a large part of the observed persistence.
I would like to thank R. Anton Braun, Julen Esteban-Pretel, Yuichi Fukuta, Shigeru Iwata, Charles
Yuji Horioka, Hidehiko Ichimura, Boyan Jovanovic, Akihisa Shibata, Futoshi Yamauchi andseminar participants at the Contract Theory Workshop, Georgetown University, the Kansai
Macroeconomics Workshop, Kobe University, Osaka University, Osaka Prefecture University, theResearch Institute of Economy, Trade and Industry, and Tokyo University for helpful comments.
Financial support from the Murata Science Foundation and the Japanese Ministry of Education,
Culture, Sports, Science and Technology (a Grant-in-Aid for Scientific Research) are gratefully
acknowledged. Osaka School of International Public Policy, Osaka University, 1-31,

Machikaneyama, Toyonaka, Osaka, 560-0043, Japan. E-mail: takii@osipp.osaka-u.ac.jp1

1

Introduction

Why are some firms persistently more productive than others? Evidence repeatedly
reveals that there are substantial and persistent diﬀerences in productivity between
plants and between firms [e.g., Baily et al. (1992)]. Apparently, productivity is not
the only variable that exhibits persistent diﬀerences. Evidence also shows that skill
compositions and wage payments exhibit persistent diﬀerences between firms [e.g.,
Haltiwanger et al. (2000)]. Moreover, persistent diﬀerences in profits are pervasive[e.g., McGahan (1999)].
The coexistence of persistent diﬀerences in these variables is not coincidental.
Productive firms employ skilled workers and pay high wages [e.g., Haltiwanger etal. (1999)]. In addition, skills and the market value of a firm are positively correlated [Abowd et al. (2004)]. Evidence implies that the persistence of diﬀerences in
productivity, skills, wages and profits may have the same source.
Although economists pay it little attention, there is another well-known aspectof productivity diﬀerences.

In Table 1, future relative productivity is regressedon present and past relative productivity; relative productivity is estimated by thelogarithm of labor productivity relative to an industry and year average. The tableshows that after controlling for current relative productivity, the sequence of past
relative productivity levels still influences future relative productivity. This meansthat even a previously lower ranked firm that suddenly climbs up the ladder wouldfind it diﬃcult to maintain its success in the future. That is, one-oﬀ large temporalshocks are less likely to account for continuous success. To predict the movementof future relative productivity, we need to know the history of relative productivity.
However, why does current relative productivity fail to summarize an influence fromits past?

A useful theory of productivity diﬀerences must explain not only why

productivity diﬀerences persist between firms, but also why a firm’s history mattersfor future persistence.
In this paper, we aim to provide a unified explanation for these findings.
2

We

The Dependent variable is D ln yt+1 .
constant

D ln yt

D ln yt−1

D ln yt−2

D ln yt−3-0.0010.671∗∗

0.134∗∗

0.060∗∗

0.048∗∗(0.002)(0.007) (0.009)(0.009)(0.007)

Adjusted-R2

0.763# of observations 21154
Table 1: AR 4

The variable, D ln yt , is the logarithm of labor productivity relative to an industry
average in year t. The variables are constructed from an industry annual dataset in
COMPUSTAT for the period from 1975 to 2004. Data construction is explained inAppendix 1. Standard errors are in parentheses. ∗∗ means significant at the 0.5
percent level.
develop a dynamic assignment model for the relationship between the skills of workers
and firm-specific knowledge, which we term a firm’s organization capital.1 We also
examine how the assignment model explains the observed evidence.
Two assumptions are emphasized: organization capital and skills are complementary to each other; and skill is an input for the accumulation of organization capital.
Because skill is complementary to organization capital, skilled workers are assigned
to firms with large organization capital. On the other hand, because skill is an input
for the accumulation of organization capital, the employment of skillful workers promotes the accumulation of organization capital. Hence, the dynamics exhibit positive
feedback. We investigate the extent to which this positive feedback mechanism raises
the persistence of a firm’s organization capital. Given that the productivity, wages,
skill composition and profits of a firm are shown to be strictly increasing functionsof organization capital, the model can explain not only the persistence of these variables, but can also explain positive correlations between productivity, skills, wages
1

More specifically, we define organization capital as all types of intangible assets embodied in

an organization. It might consist of organizational structure, daily practices, routines, information
held by an organization, corporate culture, reputation and so on.

3and profits.
The theory predicts that a rise in the heterogeneity of skills increases the persistence of organization capital. When the variance of skills is high, the top organization
has the most advantages because it can attract the best workers who can provide thefirm with the best knowledge and promote the accumulation of organization capital.
Hence, the larger is the variance of skill, the longer the top organization can enjoy its
relative advantage. It is shown that when there are no idiosyncratic shocks, every
firm’s rank stays the same forever and firms’ relative advantages (and disadvantages)
persist indefinitely.
Although this simple logic can explain why productivity diﬀerences between firms
persist and why there are positive correlations between productivity, wages, skills andprofits, it cannot explain why current relative productivity fails to summarize theinfluence of the past. In order to explain this evidence, we need a third assumption:organization capital is unobservable.

When organization capital is unobservable,

there is assignment between the quality of workers and perceived organization capital.
If a firm’s organization capital is believed to be high, this belief attracts skillful
workers, who provide the firm with better knowledge.

Because the current belief

is constructed from past observations, historical observations can influence futureorganization capital by changing perceptions about its current level. This mechanism
may explain why a firm’s history is important.
The model of unobserved organization capital has further theoretical and empirical advantages. Theoretically, this extension allows us to analyze the dynamic
interaction between a firm’s real capacity and its perceived capacity.

On the one

hand, when organization capital is believed to be high, this belief attracts skilledworkers who can develop superior knowledge in the firm. On the other hand, if actual organization capital is high, the firm’s current performance is likely to be good,which generates the perception that the firm has a high level of organization capital.
Hence, we can analyze how this dynamic positive feedback between actual capacity4and perceived capacity influences the persistence of productivity, skill mixes, wage
payments and profits.
In particular, we examine how the noisiness of information influences persistence.
Suppose that a firm’s current output reveals information about its organization capital. If the revealed information is noisy, managers learn little from the new observations, and thus do not change their beliefs drastically. If there is assignment between
these beliefs and the quality of workers, the quality of assigned workers changes little
and, therefore, so does accumulated actual organization capital. In particular, when
output has no predictive power for organization capital, the belief never changes. In
that case, we show that the firm’s rank remains the same on average forever andthat the dynamics of actual organization capital exhibit temporal deviations fromthe constant belief.
Extending the model to incorporate unobserved organization capital has anotheradvantage. It allows us to structurally estimate the parameters of our model fromproductivity dynamics.

Exploiting this additional advantage, we diﬀerentiate twosources of persistence — positive assortative assignment and noisy information — fromothers by using an industry annual dataset from COMPUSTAT covering 1970 to
2004.
The estimated parameters are all significant and their signs are consistent withtheoretical predictions. In particular, our empirical results show that two-year lagged
relative wages have a positive impact on current relative productivity after controllingfor one- and two-year lags of relative productivity; note that, in this paper, “relative”
refers to the logarithm of each value relative to industry and year averages. This evidence is consistent with the hypothesis that skilled workers help firms to accumulate
assets and raise future productivity. Our empirical results also show that perceived
relative productivity, which is constructed by using sequences of past relative productivity, has a positive impact on future relative productivity. This is consistent
with the hypothesis that people learn about a firm’s capacity from its past perfor-5

mance and form beliefs that influence its employment of skilled workers and future
performance.
By using the estimated parameters, we simulate not only the autocorrelationsof relative productivity, relative wages and expected relative profits per worker, butalso the correlation between relative productivity and relative wages. All simulated
autocorrelations replicate the observed autocorrelations quite well.

The model isalso able to explain the observed high correlations between relative productivity andrelative wages.

That is, our model can quantitatively account for not only thepersistence of these variables, but also the high positive correlation between relative
productivity and relative wages.
We use our model to conduct two counterfactual experiments. They show that if
there were no skill diﬀerence between workers and, therefore, if there were no assignment problem, firms’ relative advantages (disadvantages) would almost disappear in
about five years. In addition, the correlation between relative productivity and relative wages would diminish substantially, while even if output perfectly predicted thelevel of organization capital, there is only a minor influence on variables’ persistence
and the correlation between relative productivity and relative wages . These exercises consistently suggest that a positive assignment mechanism accounts for a largepart of the observed persistence of variables. The diﬃculty of estimating organization
capital plays only an auxiliary role.
It has long been recognized that an individual firm possesses particular resources[e.g., Kaldor (1934), Robinson (1934) and Lucas (1978)]. As a source of its specificresources, many economists emphasize the importance of firm-specific knowledge accumulated through experience [e.g., Penrose (1959) and Rosen (1972)].

Prescottand Visscher (1980) refer to this accumulated specific knowledge as a firm’s organization capital. Recently, interest in organization capital has reemerged. Jovanovic
and Rousseau (2001), Atkeson and Kehoe (2005) and Samaniego (2006) quantify themacroeconomic eﬀects of organization capital. Faria (2003) explains merger waves

6

by using a model of assignment between organization capital and skills. However,no paper has addressed the question of why some firms succeed in accumulatingorganization capital, whereas others do not. This is the main aim of this paper.
Unlike previous researchers, we model organization capital as a form of the vintage human capital analyzed by Chari and Hopenhayn (1991). For any organization,
ancestors determine a particular routine, culture, organizational structure, set ofrules and how to arrange machines and structures that successors inherit and modify.

Hence, the workers employed in the past influence the organization’s future.

This modeling strategy allows us to investigate how the assignment of workers toorganizations has long-run eﬀects on organization.
Positive assortative assignment models also have a long history. Becker (1973)
originally derived a condition for positive assortative matching in a marital market.
Sattinger (1979) analyzed a positive assortative assignment equilibrium between physical capital and skills. More recently, economists have rediscovered the importanceof assignment models [e.g., Kremer (1993), Costrell and Loury (2004) and Shimer(2005)]. However, in most papers, the distribution of assigned variables is treated asgiven.
Notable exceptions are Acemoglu (1997) and Jovanovic (1998). Acemoglu (1997)
endogenizes the distribution of skills and physical capital and Jovanovic (1998) endogenizes the distribution of skills and technology. Both authors examine persistent
income inequality. Unlike them, we endogenize the distribution of organization capital and examine persistent diﬀerences in productivity, skills, wages and profits.
A key assumption behind positive assortative matching is that organization capitaland skills are complementary.

There is supporting evidence for this assumption.

Chandler (1977) reports historical evidence that the development of administrative
hierarchies is essential for monitoring and coordinating resources in modern business
firms.

Chandler (1977) demonstrates that this organizational structure demands

skilled workers for processing information. Recent evidence obtained by Bresnahan,

7Brynjolfsson and Hitt (2002) shows that reorganization associated with IT investment
demands more skilled workers.

For example, the use of flexible machinery often

requires workers to have greater discretion, which in turn requires them to have data
analysis skills and problem solving abilities.
Learning is another important feature of the model.

As Jovanovic (1982) ex-

plains, a firm gradually learns its own productive capacity.

However, unlike Jo-

vanovic (1982), we assume that a firm’s productive capacity is a product of its active
investment and is aﬀected by uncertainty arising from that investment. Ericson andPakes (1995) incorporate similar dynamics for productive capacity. Hence, our model
can be seen as a hybrid of the passive learning model of Jovanovic (1982) and theactive investment model of Ericson and Pakes (1995).
Several models generate an equilibrium distribution of wages [e.g., Burdett andMortensen (1998)] and an equilibrium distribution of productivity [e.g., Eeckhout
and Jovanovic (2002)]. These researchers show how ex ante homogeneous agents cangenerate ex post heterogeneity. Unlike them, we assume an ex ante heterogeneity ofskills and then generate distributions of organization capital, productivity, wages andprofits. An advantage of our approach is that our unique stationary distribution is
globally stable. Hence, our distribution is robust to disturbances.
The paper is organized as follows.

In the next section, we set up a dynamic

positive assortative assignment model under the assumption that organization capitalis perfectly observable. In this section, we clarify the mechanism through which skill
diﬀerences enhance persistence in the model.

In Section 3, we extend the model

to incorporate the imperfect observation of organization capital.

We analyze how

perceived capacity and actual capacity have interactive eﬀects on persistence.

In

Section 4, we discuss the identification of the parameters from the data and report
our empirical results. In Section 5, we simulate our model by using the estimatedstructural parameters. In Section 6, we discuss extensions and conclude the paper.

8

2

A Dynamic Assignment Model in which Organization Capital is Observable

In this section, it is assumed that organization capital is observable. We establish apositive assortative assignment equilibrium between organization capital and skills.
The assumption that organization capital is observable is used to clarify the intuition
about how an assignment model can explain persistent diﬀerences in productivity,
wages, skills and profits. Our standard assignment model is based on that of Sattinger (1979). We extend the model to incorporate dynamics by endogenizing the
distribution of organization capital.
The economy is represented by a continuum of workers and firms. The populationof both firms and workers is normalized to unity. Each firm has organization capitalof kto , and a set of jobs, the total mass of which is also normalized to unity.

Weassume that the ith job in a firm that has organization capital of kto employs one
worker who has quality of qit and produces output of yit according the following
production function:
yit = A (kto )α qitψ , α > 0, ψ > 0,
where A, α and ψ are constant parameters. Because the total mass of jobs is 1, we
R1interpret 0 yit di as both a firm’s total output and its labor productivity.
Assume that the ith job pays competitive wages of w (ln qt ). As discussed later,

the employment decision is made for each job and job supervisors are assumed tomaximize the profits made from the job. The profit maximization problem by the
managers in the ith job is written as
χi (ln kto ) = arg max {exp [ln A + α ln kto + ψ ln qit ] − w (ln qit )} , ∀i, ln kto .
ln qit(1)

We express wages and policy as functions of ln kto and ln qt to simplify the algebra.
Suppose that ln kto is normally distributed with a mean of µkt and a standard
deviation of σ kt at the date t. Assume also that ln qt is normally distributed with a9mean of µq and a standard deviation of σ q at any date. For simplicity, we assume
that jobs and workers have reservation values of 0. Because the number of jobs isthe same as the number of workers, nobody chooses the outside option and every
agent can find a partner. Hence, these assumptions make it possible to focus on theassignment problem.
We focus on the positive assortative equilibrium.

This means that the top x

percent of ln kto is assigned to the top x percent of ln qt for any x. Let Φ (·) denoteo−µkt
the standard normal distribution. Given that ln kσt kt
andln qt −µq
σq

are distributed as

standard normal variables, a positive assortative equilibrium implies that¶µ¶µχi (ln kto ) − µqln kto − µkt=1−Φ, ∀i, ln kto .
1−Φ
σ kt
σq

(2)

Equations (1) and (2) characterize a static market equilibrium.
Definition 1 A market equilibrium with observed organization capital consists of
χi (·) and w (·) that satisfy equations (1) and (2).
We aim to find a policy function and a wage function that are consistent with thisdefinition of equilibrium. Equation (2) states that the policy function must satisfy
χ (ln kto ) ≡ χi (ln kto ) =σq[ln kto − µkt ] + µq , ∀ ln kto .
σ ktHence, all jobs in a firm are filled by workers of the same quality. This policy functionmeans that, in equilibrium, highly qualified workers must be assigned to a firm thathas a high level of organization capital. For this policy function to be consistent withthe definition of equilibrium, the policy function must solve equation (1). Consider£¤
a firm that has organization capital of ln kto = σσktq ln qt − µq + µkt ≡ χ−1 (ln qt ). For
all jobs in this firm, ln qt must be the optimal choice. Hence, marginal cost at ln qt
must be equal to the marginal product of ln qt , as follows:¡¢
w0 (ln qt ) = ψy χ−1 (ln qt ) , ln qt , ∀ ln qt ,10

where y (ln kto , ln qt ) = exp (ln A + α ln kto + ψ ln qt ). Moreover, because the reservation value of workers is 0, w (−∞) = 0. The following wage function is derived fromthe marginal condition and the boundary condition:w (ln qt ) =ψσqy (χ−1 (ln qt ) , ln qt )
ασ µ∞
.
ψσq
1 + ασµ∞It is easy to check that the second-order condition is satisfied by this wage function.
Hence, the policy function and the wage function are consistent with the definition ofequilibrium. By construction, the equilibrium is unique. Note that wage payments
increase in ln qt , which is also an increasing function of ln kto . Hence, a firm that has
high organization capital pays high wages.
The firm’s profits are strictly increasing in ln kto .
Z 1
y (ln kto , χ (ln kto ))o[y (ln kto , χi (ln kto )) − w (χi (ln kto ))] di =.
π (ln kt ) =ψσq
1 + ασ0µ∞Hence, labor productivity, the quality of workers, wage payments and profits per
worker are positively correlated, and if ln kto is persistent these variables are persistent. To determine what influences the persistence of these variables, we analyze thedynamics of ln kto below.

Dynamics: Following Atkeson and Kehoe (2005) and Samaniego (2006), we assume
that organization capital is acquired by learning by doing.

In the spirit of Arrow(1962), learning by doing is modeled as an unintended result of production.
More specifically, we assume that an individual worker cannot change a particularroutine or culture in a firm, but a group of workers can.

Because top managers

cannot evaluate the qualities of individual workers, they must rely on evaluation by
supervisors in each job. Although supervisors can evaluate the quality of each worker
with respect to production in a particular job, they are unaware of how interaction
between individual workers can change the firm’s routines or culture. We implicitly
11assume that communication cannot perfectly resolve this issue.

Because skilledworkers are likely to learn the mechanism of production well and have better ideas,
the employment of skilled workers has indirect external eﬀects that are not initiallyacknowledged.
We model this process by assuming that the average quality of employed workers
improves organization capital in the next period:2okt+1= B (kto )φ (qte )γ eεt , 0 ≤ φ < 1, γ > 0,

where qte =

R1
0

(3)

qit di, B, φ and γ are constant parameters and εt is a random variable,
2

which is normally distributed with a mean of − σ2ε and a standard deviation of σ ε . The
parameter φ measures the technological persistence of organization capital. Becausesome organization capital depreciates, we assume that a fraction, φ, of organization
capital can be carried over to the next period.
The assumption about learning by doing might be unreasonable if top managers’
talents are the most influential inputs for creating organization capital. When a firm
employs top managers, it expects them to change the firm’s structure and norms.
Hence, the firm’s maximization problem must also take into account equation (3).
In Appendix 2, we assume that a firm solves a dynamic optimization problem bytaking into account the dynamics of organization capital. It constructs a recursive
positive assortative equilibrium and examines its properties.

This shows that the

dynamics of organization capital are the same as those obtained when one assumes
2

Given that every job is filled by workers of the same quality, equation (3) generates dynamics

that are the same as those from the transition equation,
φ

γ

o
kt+1
= B1 (kto ) 1 (yt ) 1 eεt ,owhere yt = A (kto )α qtψ , and B1 , ψ 1 and γ 1 are parameters. This equation implies that ln kt+1
isexpressed as a weighted sum of {ln yt−s }ts=0 . As discussed by Bahk and Gort (1993), in empiricalstudies, cumulative gross output is used as a proxy of experience accumulated through learning by
doing. Hence, our assumption is consistent with the standard learning-by-doing assumption.

12

that there is learning by doing. Diﬀerences arise in the wage and profit functions.
We maintain the learning-by-doing assumption for three reasons. First, as convincingly argued by Simon (1997) and Nelson and Winter(1982), it is reasonable toassume that an individual in a firm would find it hard to change a firm’s routines
or culture. Second, the learning-by-doing assumption simplifies the model, but conveys the main logic of the paper. Hence, most of our analysis avoids the technicaldiﬃculties associated with dynamic optimization problems.

Third, the wage andprofit functions derived on the basis of the learning-by-doing assumption are useful for our empirical work.

We discuss the unique outcomes generated by the thelearning-by-doing assumption later.
Because all jobs in a firm are filled by workers of the same quality, in which case,ln qt = σσktq (ln kto − µkt ) + µq , in equilibrium, the dynamics of organization capital canbe written aso= ln B + φ ln kto + γ
ln kt+1∙¸
σqo(ln kt − µkt ) + µq + εt .
σ kt

(4)

o
Because ln kto and εt are normally distributed, ln kt+1
is also normally distributed.

The dynamics of µkt and σ 2kt can be derived from equation (4) as follows:
sµ
¶2
2
γσ q
σε
µkt+1 = ln B + φµkt + γµq − , σ kt+1 =φ+
σ 2kt + σ 2ε .
2
σ ktThese two equations characterize the dynamics of the aggregate state variables. By
using the dynamics of µkt , equation (4) can be rewritten as
¶
µ
γσ q
o
(ln kto − µkt ) + ε∗t ,ln kt+1 − µkt+1 = φ +σ kt
2

where ε∗t = εt + σ2ε is normally distributed with a mean of 0 and a standard deviationof σ ε . This equation implies that when ln kto is larger than its industry mean µkt , theqfraction, φ + γσof this relative advantage is carried over to the next period. The paσ kt
qrameter φ captures the assumed persistence, and the second term, γσ, represents theσ kt

result of positive assortative assignment. When the ratio of the standard deviation of13

skills to that of organization capital is large, organization capital is more persistent.
When the ratio is large, the firm with the most organization capital derives the mostbenefits because this leading firm attracts the most talented workers, who provide
the firm with the best knowledge. Therefore, relative advantages persist longer.
This intuitive result is analyzed more rigorously below. First, we show that the
distribution of ln kto converges to a stationary distribution.
Proposition 2 The mean, µkt , and standard deviation, σ kt , of ln kto converge to thestationary points, µk∞ and σ k∞ , respectively, where µk∞ and σ k∞ are
2

µk∞
σ k∞

ln B + γµq − σ2ε
,
=
1−φ
q
¡
¢
γφσ q + (γσ q )2 + 1 − φ2 σ 2ε=.
1 − φ2(5)(6)

Moreover, when the distribution converges to the stationary distribution, the dynamics
of organization capital follow a first-order autoregressive (AR(1)) process, as follows:
¶
µ
γσ q
o
D ln kt+1 = φ +D ln kto + ε∗t ,(7)
σ k∞where D ln kto = ln kto − µk∞ .
The proposition states that there is a unique, globally stable stationary distribution. Hence, an economy converges to the stationary distribution in the long run.
In the remainder of this section, we discuss the dynamics of equation (7). Because
persistence is only influenced by φ+ σγσk∞q , we refer to this as the persistence parameterin what follows.

Persistence when σ ε = 0: In order to have a clear intuition, we first analyze adeterministic model.

14

Proposition 3 Suppose that σ ε = 0. Then, there is still a stationary distribution,µk∞ =ln B+γµq
γσq, σ k∞ = 1−φ, and the level of organization capital remains constant:1−φoln kt+1= ln kto .

The proposition states that, if there is no shock, the ranking of organization capitalis constant.

When there are no idiosyncratic shocks, the top organization always

attracts the best workers, who, in turn, equip the firm with the best knowledge.
Hence, it remains at the top and maintains exactly the same level of organization
capital in the long run.
Note that σ k∞ is not 0. Hence, organization capital varies between firms and so
too, therefore, do productivity, wages, skill mixes and profits. Because σ q is alwaysqpositive, when σ kt is small, the assignment eﬀect, γσ, is large. Hence, a firm withσkt

a high level of organization capital benefits substantially. This mechanism increases
σ kt . Ultimately, σ k∞ does not converge to 0.

Persistence when σ ε > 0. When we introduce idiosyncratic shocks into the accumulation of organization capital, reversion to the mean occurs. Idiosyncratic shocks
make changes in rankings possible.

A firm that receives a positive shock climbs

the rankings, which enables it to attract higher quality workers. This means that
top organizations cannot remain the best. Top organizations might slip down the
rankings.
Substituting equation (6) into the persistence parameter reveals that this paraq
meter is a strictly increasing function of γσ
.
σε

φ+

γσ q
= φ+
σ k∞

´
³
γσ q
d φ + σk∞
q
d γσ
σε

φ+

1 − φ2∈ (φ, 1) ,¡¢ ³ γσq ´−2
2
1+ 1−φ
σε

r

µ
¶
µ
¶
γσ q
γσ q
φ+
= 1, γσlim
φ+= φ.
> 0, γσlim
q
q
σ k∞
σ k∞
→∞
→0
σε
σε
15

(8)

q
When σ ε > 0, γσ
is finite. Therefore, the persistence parameter is always less thanσε1.

This means that equation (7) is covariance stationary.

Hence, the dynamics

exhibit reversion to the mean and eventually initial advantages disappear. In thiscase, it is easy to show that
£
¤ µ
¶j
o
E D ln kto D ln kt−jγσ q
ρln kj ≡
= φ+
.
σ 2k∞
σ k∞
That is, the larger the persistence parameter, the larger is the autocorrelation. Hence,q
raises the autocorrelation.
an increase in γσσεq
Two remarks are worth making. First, the parameter γσrepresents the imporσεtance of skill variation relative to the reshuﬄing eﬀect. Hence, the importance of skill
variation to persistence is not absolute. Small skill variations can induce substantialqpersistence in variables when an industry is stable. Second, when γσ
is infinite, theσεpersistence parameter converges to unity. Hence, the results in Proposition 3 can beseen as the limit of this general case.

3

A Dynamic Assignment Model in which Organization Capital is not Observable

In the previous section, we provided a clear intuition about how assignment influences the persistence of variables and explained why productivity, skills, wages andprofits are positively correlated. However, our analysis did not explain how historical
relative productivity influences future relative productivity. Because the dynamics
of organization capital in the previous section were represented by an AR(1) process,once we control for a current value, past values do not influence future ones.
In this section, we examine the same dynamic assignment model but incorporate
organization capital that is not observable.

We show that past organization capi-

tal can influence future organization capital even when current organization capitalis controlled for.

We also analyze how perceived organization capital and actual
16

organization capital have interactive eﬀects on the persistence of capital.
Assume that kto cannot be directly observed, but can be inferred from the realizations of output. To capture this idea, we add a firm-specific idiosyncratic shock to
the production function for each job as follows:
yit = eut A (kto )α qitψ , α > 0, ψ > 0,(9)2where ut is normally distributed with a mean of − σ2u and a variance of σ 2u .
When employment decisions are made about the ith job, output is not realized.
Hence, a decision must be based on a conditional expectation given the prior belief
about the level of organization capital. We assume that the prior distribution of ln kt
is normally distributed with a mean of µkt and a variance of σ 2kt . Then the expected
output from the job is¶µα2 σ 2kt+ ψ ln qit .
E [yit |µkt , ln qit ] = exp ln A + αµkt +2(10)

Similarly to the previous section, all firms are assumed to have the same σ kt at date t.
However, we deviate from the previous section by assuming that the belief, µkt , diﬀers
between firms. Given that all agents in an economy receive the same information,these agents hold the same beliefs about a firm’s organization capital. That is, thebelief, µkt , characterizes a firm’s position in the economy.

It is assumed that the

belief, µkt , is normally distributed with a mean of µekt and a standard deviation ofσ µt . We examine a positive assortative assignment equilibrium between a belief, µkt ,and a skill, ln qt .
Similarly to the previous problem, by assuming that the wage is a function ofln qt , ŵ (ln qt ), we can define a market equilibrium as follows.
Definition 4 A market equilibrium with unobserved organization capital consists ofa χ̂i (·) and a ŵ (·) that satisfy the following two equations:χ̂i (µkt ) = arg max {E [yit |µkt , ln qit ] − ŵ (ln qit )} , ∀i, µkt .
ln qit

17

(11)

µ

µkt − µekt
1−Φ
σ µt

¶

µ

χ̂i (µkt ) − µq
=1−Φ
σq

¶

, ∀i, µkt .

(12)

Applying the same logic used in the previous section, we can derive a policy
function and a wage function that are consistent with this definition of equilibrium.
Because the derivation is as before, the proof of the following theorem is omitted.
Theorem 5 There exists a unique equilibrium, which is characterized by the following
functions:σqχ̂ (µkt ) ≡ χ̂i (µkt ) =(µkt − µekt ) + µq ,
σ µt
£
¤
ψσ q
E yt |χ̂−1 (ln qt ) , ln qt
ασµ∞ŵ (ln qt ) =,ψσq
1 + ασ
µ∞

(13)
(14)

´
³
¡
¢
α2 σ2
where E [yt |µkt , ln qt ] = exp log A + αµkt + 2 kt + ψ ln qt and χ̂−1 (ln qt ) ≡ σσµtq ln qt − µq +µekt . Moreover, the expected profits of the firm areπ̂ e (µkt ) =E [yt |µkt , χ̂ (µkt )]ψσ q
1 + ασµ∞.

(15)This theorem shows that the levels of skill and expected profits are strictly increasing functions of µkt and that the wage function is a strictly increasing function ofln qt . Hence, the dynamics for skills, wages and expected profits follow the dynamics
of µkt . On the other hand, labor productivity, ln yt , is strictly increasing in ln kto andln qt . Hence, the dynamics of labor productivity are influenced by the dynamics ofln kto and µkt . To understand the dynamics of productivity, wages, skills and profits,
we analyze the dynamics of ln kto and µkt below.

Dynamics: Similarly to the analysis of the previous section, we can derive thedynamics of ln kto by substituting equation (13) into equation (3) as follows:
¸
∙
σq
o
o
e
ln kt+1 = ln B + φ ln kt + γ(µ − µkt ) + µq + εt .
σ µt kt18(16)

To derive the dynamics of µkt , we must describe the information structure of themodel.

After the firm employs a worker, output is produced.

From the realized

output, the firm knows eut (kto )α . Hence, a firm uses a signal, st ≡ ln kto + u∗t , to³´
2
infer ln kto , where u∗t = α1 ut + σ2u is normally distributed with a mean of 0 and£¤oa standard deviation of σαu . Because µkt+1 = E ln kt+1
|st , µkt , σ kt and σ kt+1 =q£¤oV ar ln kt+1
|st , µkt , σ kt , the dynamics of µkt and σ kt can be written as follows:µkt+1 =
σ kt+1ln B + φE [ln kto |st , µkt , σ kt ] + γq=φ2 (1 − ht ) σ 2kt + σ 2ε ,

∙

¸
σ2
σq
e
(µkt − µkt ) + µq − ε ,
σ µt
2

(17)
(18)

where
E [ln kto |st , µkt , σ kt ] = (1 − ht ) µkt + ht st = (1 − ht ) µkt + ht (ln kto + u∗t ) , (19)
³
´2
ασkt
σu

ht =

1+

³

ασkt
σu

(20)

´2 .

Equation (19) shows that E [ln kto |st , µkt , σ kt ] is a weighted average of the priorbelief, µkt , and new information st , where the variable ht is the weight on new information. As shown in equation (20), ht is negatively related to σ u . If the varianceof temporal shocks is large, it is diﬃcult to infer ln kto from st and thus place a small
weight on st . In this way, the variable ht measures the reliability of new information.3
Because σ kt is the same in all firms, equation (18) shows that σ kt+1 is also the same inall firms. Similarly, because µkt and st are normally distributed, equation (17) showsthat µkt+1 is also normally distributed.
3

Hence, the normality of the distribution

In fact, ht can be also rewritten as follows:
ht = 1 −E [V ar [ln kto |st , µkt , σ kt ]].
σ 2ktThis equation shows that ht would be larger if the average conditional variance were smaller relative
to the prior variance. It measures the accuracy of information, as previously used by Takii (2003,
2007), as a tractable measure of prediction ability.

19

is preserved. The following mean and standard deviation of the belief in the next
period can be derived:µekt+1σ µt+1=σ 2εeln B + φµkt + γµq − ,

2
sµ
¶2
γσ q
φ+
=
σ 2µt + φ2 ht σ 2kt .
σ µt

(21)
(22)

The derivation of equation (22) is given in Appendix 1. Furthermore, by substituting
equations (19) and (21) into equations (16) and (17), we can also rewrite the dynamics
of ln kto and µkt as follows:
γσ q(µ − µekt ) + ε∗t ,(23)
σ µt kt
¸
∙
γσ q
o
e
(µkt − µekt ) + φht u∗t ,(24)= φht (ln kt − µkt ) + φ (1 − ht ) +σ µto− µekt+1 = φ (ln kto − µekt ) +ln kt+1µkt+1 − µekt+1
2

where ε∗t = εt + σ2ε is normally distributed with a mean of 0 and a standard deviationof σ ε .
Equation (23) shows the dynamics of ln kto .

The first term of equation (23) isinfluenced by technological persistence, φ. That is, if organization capital is above
average, the fraction φ of this relative advantage is carried over to the next period. On
the other hand, the second term is influenced by positive assignment. If organization
capital is believed to be above average, the firm attracts skilled workers that help thefirm accumulate further organization capital.
Equation (24) shows the dynamics of µkt . The first term captures how new information influences the dynamics of the belief. Managers know that the fraction φ ofcurrent organization capital aﬀects the next period’s organization capital. However,current organization capital is not observable and must be inferred from current output. High output can be the result of either a large temporal shock or a high levelof organization capital. Because managers put a weight ht on new information, thefraction φht of current organization capital is believed to be translated into the next
period’s level. New information incorporates noise. Hence, the φht portion of u∗t20

also influences the posterior belief. This eﬀect is captured by the third term, φht u∗t ,in equation (24).
The second term of equation (24) captures the eﬀect of the prior belief on theposterior belief.

There are two separate eﬀects.

Because there is assignment be-

tween the prior belief and worker quality, the higher the level of organization capitalis believed to be, a priori, the higher is the quality of workers that the firm canemploy. Given that skilled workers help the firm to accumulate organization capital,organization capital in the next period is believed to be high. This assignment eﬀectq
is captured by γσin the second term. On the other hand, because output provides
σ µt

only noisy information about organization capital, a weight of 1 − ht is placed on
the prior belief. Because the fraction φ of current organization capital is translated
into organization capital for the next period, the fraction φ (1 − ht ) of the prior belief
influences the posterior.

qof the prior belief
Overall, the fraction φ (1 − ht ) + γσ
σµt

influences the posterior.
The equations (23) and (24) provide some intuition about the dynamics of ln kto
and µkt . First, the interpretation of equation (23) is that ln kto exhibits reversion to
the belief µkt and the speed of the reversion is influenced by the constant parameter
φ. Hence, assignment does not influence the persistence of ln kto unless it aﬀects µkt .
Second, given equation (24), the smaller is ht , the less is µkt subjected to two typesof shock, ε∗t and u∗t . Hence, the ranking of µkt is less likely to change. Because thereis assignment between µkt and ln qt , fewer changes in the rankings imply that µkt
becomes more persistent. That is, the noisier is the information, the more persistentis the belief.
More importantly, equation (24) implies that the current belief, µkt , can be influenced by sequences of real organization capital, {ln kso }t−1s=0 . Because current beliefs
influence the next period’s actual levels of organization capital according to equation (23), past sequences of actual organization capital can influence future values.
It is shown later that this mechanism can explain why higher order lags matter for21

productivity dynamics.
To confirm these arguments, we first show that this economy converges to thestationary distribution. Then, we analyze the dynamics of organization capital in an
aggregate economy that reaches the stationary distribution.
Proposition 6 The aggregate economy converges to a unique stationary distribution.
2

µe∞
σ µ∞
σ 2k∞

ln B + γµq − σ2ε
=
1−φ
q
¡
¢
φγσ q + (γσ q )2 + 1 − φ2 φ2 h∞ σ 2k∞¡¢=
1 − φ2
q£
¡
¡
¢
¢ ¤2
α2 σ 2ε − 1 − φ2 σ 2u +α2 σ 2ε − 1 − φ2 σ 2u + 4α2 σ 2ε σ 2u
=
2α2
´2
³

h∞ =

ασk∞
σ 2u

1+

³

ασ k∞
σ 2u

´2

Moreover, the dynamics of an individual firm in the stationary distribution are described by the following vector autoregression (VAR):
kt+1 = Mkt + ξt ,

(25)

where
⎡

M=⎣

φ,

γσq
σ µ∞

q
φh∞ , φ (1 − h∞ ) + σγσ
µ∞

⎤

⎡

⎦ , kt = ⎣and D ln kto = ln kto − µek∞ and Dµkt = µkt − µek∞ .

D ln kto
Dµkt

⎤

⎡

⎦ , ξt = ⎣ε∗tφh∞ u∗t⎤⎦

Because the stationary distribution is unique and globally stable, the economy
converges to the stationary distribution in the long run. We investigate the properties
of equation (25) and discuss what influences the persistence of organization capital.

22

Persistence when σ u = ∞: It is instructive to start with an extreme case, in which
σ u = ∞. In this case, information is too noisy and the firm can learn nothing about
the level of organization capital. Hence, h∞ = 0 and the firm only puts weight on
its prior belief. Therefore, its belief never changes. By substituting σ u = ∞ andh∞ = 0 into equations (18), (22), (23) and (24), the following proposition is easily
derived. It shows that the movement of organization capital reverts to this constantbelief.
Proposition 7 Suppose that σ 2u = ∞. Then, the dynamics of the economy are given
by
2

µe∞ =
and

ln B + γµq − σ2ε
γσ q
σ 2ε
, σ µ∞ =, σ 2k∞ =1−φ
1−φ
1 − φ2oln kt+1= φ ln kto + (1 − φ) µkt + ε∗t , µkt+1 = µkt .

Because the firm cannot learn about its own organization capital, the firm neverchanges its own belief. Hence, the belief is constant. Because assignment is based on
this belief, the firm that is believed to have a high level of organization capital attracts
good workers and maintains its position. Moreover, as actual organization capital issubjected to shocks, the movement of organization capital temporally deviates fromthe firm’s own belief. However, the level of organization capital remains the sameon average because of the constant belief.

Persistence when σ u ∈ (0, ∞), σ ε ∈ (0, ∞) and φ ∈ (0, 1). Let us examine a moregeneral case. First, we analyze the stability of equation (25). Then, we analyze what
influences persistence. The following lemma provides the conditions for stability.
Lemma 8 Let λ1 and λ2 denote the eigenvalues of the matrix M. Then, equationq(25) is covariance stationary if λ1 = φ + σγσ< 1 and λ2 = φ (1 − h∞ ) < 1.
µ∞23Note that λ2 is less than unity. This means that stability is guaranteed if λ1 < 1.
The eigenvalue λ1 corresponds to the persistence parameter of the previous section. Itqconsists of assumed persistence, φ, and the assignment eﬀect, σγσ. Hence, similarlyµ∞to the analysis of the previous section, if the assignment eﬀect is not too strong, thedynamics eventually converge to the mean.
q
Clearly, σγσand h∞ are endogenous variables. Thus, there are more fundamentalµ∞

conditions for stability. The following lemma describes the relationship between theqq
σu
endogenous variables, h∞ and σγσ, and the exogenous variables, ασ
and γσ
.
σε
µ∞
ε
q
σu
∈ (0, ∞) and γσ∈ (0, ∞). There exists a
Lemma 9 4 Suppose that φ ∈ (0, 1), ασσεεfunction η (·) and Σ (·, ·) such thath∞ = η
where η

0

³

σu
ασε

´

< 0, lim

σu
→0
ασ ε

η

³

σu
ασε

´

µ

σu
ασ ε

¶

∈ (0, 1) ,= 1 and lim

σu
→∞
ασ ε

η

³

σu
ασε

´

= 0, and

¶
µ
γσ q
γσ q
(29)
=Σ
, h∞ ∈ (0, 1 − φ) ,
σ µ∞
σε
³
´
³
´
´
´
³
³
γσ q
γσq
γσq
q
γσ q
γσ q
where Σ1 γσ
>
0,
Σ
<
0,
lim
=
0,
lim
,
h
,
h
Σ
,
h
Σ
,
h
∞
2
∞
∞
∞ =
→0
→∞
σε
σε
σε
σε
³
³σε
´ σε
´
q
q
, h∞ ∈ (0, 1 − φ) and limh∞ →0 Σ γσ
, h∞ = 1 − φ.
1 − φ, limh∞ →1 Σ γσσεσε4Explicit solutions for η (·) and Σ (·, ·) in the Lemma 9 can be derived.

Lemma 9 describes

results based on the following explicit solutions. Technical parts of the derivation are in Appendix
1.
η

µ

µ

σu
ασ ε

ασ k∞
σu

¶

¶2

³

=

1+

=

³

and
Σ

ασ k∞
σu

³

σu
ασ ε

µ

´2

ασ k∞
σu

´−2

(26)

´2 , where¡¢− 1 − φ2 +

γσ q
, h∞
σε

¶

=
φ+

s∙
³

r

σu
ασ ε

´−2

2

¢
¡
1 − φ2

¸
³
´−2
¡
¢ 2
σu
− 1 − φ2
+ 4 ασε2 )φ2 h∞1 + (1−φ1−φ2 +φ2 h∞

24

³

γσ q
σε

´−2 .

,

(27)

(28)

σu
The first part of this lemma shows that h∞ and ασhave a one-to-one relationεship. Hence, in the steady state, without loss of generality, h∞ can be treated as an
exogenous parameter.
σu
The parameter ασrepresents the standard deviation of noise relative to that ofεshocks on the accumulation of organization capital . If the standard deviation of anoise term is relatively large, firms cannot learn much and h∞ is small. If the noise
term has a relatively small variance, the firm can learn a lot and h∞ is large.
qq
The second part of the lemma shows that for a given h∞ and φ, σγσand γσσεµ∞qexhibit a one-to-one relationship. As explained in the previous section, if γσ
isσε

large, the top organization derives the greatest benefit from a positive assortative
assignment. Hence, the dynamics exhibit more persistence.
qMore interestingly, σγσ
is decreasing in h∞ . When information is noisier, rationalµ∞agents rely more on their prior beliefs to make inferences about the current level oforganization capital. Therefore, the posterior and prior beliefs are similar. Hence,µkt is more persistent. Because the dynamics of actual organization capital exhibit
reversion to the firm’s own belief, noisy information makes actual organization capital
more persistent.
Note that the properties of the function Σ (·, ·) imply that when h∞ converges toσu0, the persistence parameter, λ1 , converges to 1. This means that when ασis large,εthe dynamics can be approximated by the previous results, with σ u = ∞.

qNote also that equation (29) implies that λ1 = φ + σγσ< 1. Hence, the followingµ∞

proposition can be stated.
q
σu
Proposition 10 Suppose that φ ∈ (0, 1), and that ασ
and γσare finite. Equationσεε(25) is covariance stationary.
The two eigenvalues, λ1 and λ2 , are important determinants of the persistenceof the stochastic process.

Let ρln kj denote the autocorrelation between current

organization capital and organization capital j periods before. In addition, let ρµj
25

denote the autocorrelation between a belief about current organization capital ando
E [D ln kto D ln kt−j]a belief about organization capital j periods before: ρln kj ≡, ρµj ≡V ar(D ln kto )E [Dµkt Dµkt−j ]. The following proposition derives the autocorrelation of ln kt and µkt .
σ2µ∞

The proof is established in Appendix 1.
Proposition 11 The autocorrelations of ln kto and µkt are functions of λ1 and λ2
ρln kj = (1 − ω) λj1 + ωλj2 , ρµj = λj1 ,¢γσ q ¡
2
1
−
λ
1
σµ∞
¡ 2
¢.
where ω =(λ1 − λ2 ) φ h∞ + 1 − λ21

The above proposition states that the autocorrelation of organization capital canbe expressed as a weighted average of λj1 and λj2 ; the autocorrelation of the belief
is λj1 .

Note that λ1 > λ2 .

Hence, the following corollary results from the above

proposition.
q> 0. The autocorrelation of the belief about a firm’s
Corollary 12 Suppose that γσσεorganization capital exceeds that of its actual organization capital: ρµj > ρln kj , ∀j.
The corollary states that the belief is more persistent than is the organization capital itself. Because idiosyncratic shocks directly influence the realization of randomvariables, the variance of the realized random variable is generally larger than thevariance of the conditional expectation. The same logic applies in this case. Given
that the belief is less volatile than is actual organization capital, the autocorrelationof the belief exceeds the actual value. This corollary confirms this intuition.
q
σu
Next, we show how the parameters γσand h∞ (or ασ) aﬀect the autocorrelations.
σεεThe proof is established in Appendix 1.
dρ

ln kj
Proposition 13 1) There exist j ∗ and j ∗∗ such that for all j ≥ j ∗ , d γσ
> 0 andq

for all j ≥ j ∗∗ ,dρln kj
< 0.
dh∞

dρ

dρ

σε

2) For all j, d γσµjq > 0 and dhµj
< 0.
∞
σε

q
and a decrease in h∞ increase theThis proposition implies that an increase in γσσε

autocorrelation about the belief. The same changes can increase the autocorrelation26

of organization capital after enough time has passed.

Because there is positive

assignment between the belief and skills, there are direct eﬀects on the autocorrelation
about the belief. However, both influence the autocorrelation of actual organization
capital because future actual organization capital is influenced by the firm’s current
belief. Actual organization capital can temporally deviate from the belief. However,as time passes, an increase in the persistence of the belief dominates the temporal
disturbance and increases the persistence of organization capital itself.

4

Empirical Examination

In this section, we derive empirically testable equations and examine the validityof our model.

We show that the predictions of our model are broadly supportedby the data.

The estimated parameters are used to identify the structure of ourqmodels: technological persistence, φ; the eﬀect of assignment on persistence, σγσ;µ∞and a measure of the accuracy of information, h∞ . These structural parameters are
inputs into the simulation exercises of the next section.
Because we cannot observe kto , we must translate the results from the previous
section into dynamics for observable variables.

One such variable is output, yt .

Given that the number of workers is assumed to be unity, we estimate yt by using
labor productivity.

The dynamics of a firm’s labor productivity relative to theindustry and year average and the expected relative productivity in the steady state
are derived from equation (25), as follows:D ln yt+1 = b1 D ln yt + b2 E [D ln yt |µkt ] + vt ,¤£E D ln yt+1 |µkt+1 = b3 D ln yt + b4 E [D ln yt |µkt ] ,

(30)
(31)

ψσ q
ψσq
q
where D ln yt = ln yt − E [ln y], b1 = φ + ασ
φh∞ , b2 = σγσ− ασ
φh∞ , b3 =
µ∞
µ∞
µ∞
¡
¢
ψσq
φh∞ + ασφh∞ , b4 = b1 + b2 − b3 and vt = α ε∗t − φu∗t + u∗t+1 . The derivations ofµ∞equations (30) and (31) are given in Appendix 1.

27

Before showing how to estimate these parameters, it is instructive to discuss how
ψσqq
the estimated parameters are related to φ, σγσand h∞ . Suppose that ασ
is known.
µ∞µ∞q
The parameters φ, σγσand φh∞ can be identified from the following three equations:µ∞φ = b1 −

ψσ q
γσ q
ψσ q
b3
φh∞ ,
= b2 +
φh∞ , φh∞ =
.
ψσ q
ασ µ∞
σ µ∞
ασ µ∞
1 + ασµ∞These equations imply that φ,

γσq
σµ∞

and h∞ are primarily related to b1 , b2 and b3 ,respectively.
To clarify the intuition behind these relationships, note that equation (30) can be
written as
D ln yt+1 = φD ln yt +

γσ q
ψσ q
E [D ln yt |µkt ] +φh∞ [D ln yt − E [D ln yt |µkt ]] + vt .
σ µ∞
ασ µ∞

As in equation (23), the coeﬃcient on D ln yt represents technological persistence andthe coeﬃcient on E [D ln yt |µkt ] represents the eﬀect of assignment on persistence.
That is, because the fraction φ of current organization capital is transformed into
next period’s organization capital, the fraction φ of current labor productivity islinked to labor productivity in the next period. On the other hand, because belief
in a high level of organization capital attracts skillful workers, the coeﬃcient on£¤
E D ln yt−1 |µkt−1 represents the eﬀect of assignment on organization capital in thenext period. The third term does not appear in equation (23). It arises because ofa prediction error. When realized output exceeds expected output, people update
their beliefs. The updated belief attracts better workers in the next period, whichraises output. Hence, if the eﬀects of prediction error are removed, the coeﬃcientsqon D ln yt and E [D ln yt |µkt ] in equation (30) separately identify φ and σγσ.
µ∞qNote that we can separately identify φ and σγσeven if h∞ = 1.
µ∞1, a firm knows the exact level of current organization capital.

When h∞ =However, futureorganization capital remains unknown because it is influenced by shocks. Hence, afirm’s employment decision must be based on its beliefs, rather than on real values.
qThat is, our model can separate σγσfrom φ not because of noisy information, butµ∞

because it distinguishes the eﬀects of firms’ decisions from those of firms’ capabilities.
28

The prediction error is clearly influenced by the accuracy of information. In our
regression analysis, the parameter b3 provides useful information about h∞ .

Note

that

b1 − b3 = φ (1 − h∞ ) .

(32)

This expression shows that the diﬀerence between b1 and b3 provides information
about h∞ . In equation (30), vt and D ln yt are correlated because E [u∗t |D ln yt ] 6= 0.
Because measured productivity is influenced not only by the level of organization
capital, but also by current temporal shocks, observed productivity contains information about current shocks. When rational agents predict future productivity, theyeﬃciently extract this information from current productivity.

Hence, b3 deviatesfrom the fundamental parameter b1 . When the variation of u∗t is a large componentof the variation in measured productivity, measured productivity is influenced moreby u∗t and less by ln kto . Hence, measured productivity is useful for predicting u∗t , but
not for predicting ln kto . Therefore, h∞ is small and the diﬀerence between b1 and b3
is large.

Estimation Method 1: To implement the procedure discussed above, we must
estimate E [D ln yt |µkt ] from the data.

We propose two methods for doing this.

Because each strategy has its own strengths and weaknesses, it is hoped that thestrategies complement each other.
The first method applies the following proposition, which is proven by equations(14) and (15). Although the dynamics of organization capital are not influenced by
the assumption that organization capital is accumulated through learning by doing,
equations (14) and (15) are aﬀected by this assumption.

Therefore, the following

useful proposition represents a benefit of assuming that there is learning by doing.

29

Proposition 14 Perceived relative productivity is equal to relative wages and expected relative profits per worker.
E [D ln yt |µkt ] = D ln ŵ (χ̂ (µkt )) = D ln π̂ e (µkt ) ,where D ln ŵ (χ̂ (µkt )) = ln ŵ (χ̂ (µkt ))−E [ln ŵ (χ̂ (µkt ))] and D ln π̂ e (µkt ) = ln π̂e (µkt )−E [ln π̂ e (µkt )].
Proposition 14 states that perceived relative productivity can be estimated byusing relative wages. Hence, the following testable equation is derived from equations(30) and (31):
D ln yt = θ1 D ln yt−1 + θ2 D ln yt−2 + θ3 D ln wt−2 + vt−1 ,(33)where θ1 = b1 , θ2 = b2 b3 and θ3 = b2 (b1 + b2 − b3 ) > 0. By using the estimated valueof θ1 , θ2 and θ3 , we can identify b1 , b2 and b3 , which allows us to estimate φ, h∞ and
γσ q
.
σµ∞

Equation (33) shows that after controlling for the first and second lags of relative
productivity, the second lag of relative wages must have a positive impact on current
relative productivity. Given that skilled workers equip firms with better firm-specificknowledge, the theory predicts that there is a positive association between past wages
and current productivity.
One econometric issue exists.

Because vt−1 contains u∗t−1 , it is correlated withD ln yt−1 . Hence, we need an instrument for this variable. Proposition 14 provides asuitable instrument. Because the firm makes employment decisions without observing
realized output, relative wages, D ln wt−1 ≡ ln wt−1 − E [ln w], are not influenced by
the realization of the noise term, u∗t−1 , but are correlated with D ln yt−1 because of

positive assignment. Hence, D ln wt−1 can be used as the instrument.

Estimation Method 2: The next estimation method is more complex. However, it
allows us to examine a diﬀerent prediction of our theory. This is that the belief, which30

is constructed from sequences of past relative productivity, influences future relative
productivity.

Furthermore, to apply this alternative method, we need not assume

that there is learning by doing. Hence, even if top managers’ skills are importantelements of organization capital, this estimation method can be used to identify
parameters.

To construct E [D ln yt |µkt ] from the data, we derive the following

regression equation from equation (31):
D ln yt = b3

t−1
X
i=0

(b4 )i Dyt−1−i + bt4 E [D ln y0 |µk0 ] + $t ,(34)

where $t = D ln yt − E [D ln yt |µk ]. Note that $t is not correlated with D ln yt−1−ifor all i ≥ 0 and E [D ln y0 |µk0 ].
As already discussed, the parameters b1 and b3 diﬀer because E [vt |D ln yt ] 6= 0.
Hence, to separate b1 from b3 , we apply ordinary least squares (OLS) to equation (34)and use an instrumental variables (IV) approach to estimate equation (30). The IV
estimate provides a consistent estimator of the parameter b1 and the OLS estimate
provides a biased estimator of b1 , which is b3 .

Hence, the diﬀerence between the

IV estimates and the OLS estimates indicates the extent to which labor productivity
provides information about the error term and identifies h∞ . We use this techniquein applying the second estimation procedure discussed below.
Assume that there is a proxy for E [D ln y0 |µk0 ]. First, we choose an arbiPit
trary value of b4 , and construct t−1
i=0 (b4 ) Dyt−1−i and (b4 ) E [D ln y0 |µk0 ] from thedata. Second, equation (34) is estimated under the constraint that the coeﬃcient of(b4 )t E [D ln y0 |µk0 ] is 1. This yields b̂3 , where b̂3 is the estimated value of b3 . Third,Pitusing b̂3 and b4 , we estimate E [D ln yt |µkt ] by b̂3 t−1
i=0 (b4 ) Dyt−1−i +b4 E [D ln y0 |µk0 ].

Fourth, using the estimated value of E [D ln yt |µkt ], we estimate equation (30) by us-

ing the IV regression. We use D ln yt−1 and D ln wt as instruments for D ln yt andE [D ln yt |µkt ]. We need an additional instrument for E [D ln yt |µkt ] because b̂3 contains a measurement error. This IV estimation procedure yields b̂1 and b̂2 , where b̂1
and b̂2 are the estimated values of b1 and b2 .
31

Fifth, because there is a regulatoryrelationship, according to which b4 = b1 + b2 − b3 , we replace b4 by b̂1 + b̂2 − b̂3 andrepeat the same procedure until the estimated b4 converges to the assumed b4 .

Data: We use COMPUSTAT industry annual data from 1970 to 2004 for estimation.
COMPUSTAT provides data on an unbalanced panel of publicly traded firms in
the U.S. It contains information from balance sheets, and information on incomes,
cash flows and financial variables.

The variables value added per worker and theaverage wage rate are constructed for each firm and each year. Details of our data
construction procedure and summary statistics on the constructed variables are given
in Appendix 1.
We estimate D ln yf t and D ln wf t by ln yf t −Pmtjln yf tmt

and ln wf t −Pmtjln wf t,mt

where yf t is value added divided by the number of workers and labor expenses per
worker in the f th firm in year t, respectively, and mt is the number of firms in thecorresponding four-digit industry in year t.

We estimate each firm’s initial priorbelief, E [D ln y0 |µk0 ], from the average value of D ln yf t over the five consecutive
years following the firm’s initial appearance in COMPUSTAT after 1970. Therefore,
the following regression is estimated by using data for 1975—2004.

Results: First, in Table 2, we report the regression results from the first estimation
method.
Because only few companies report labor and related expenses in COMPUSTAT,
we estimate labor costs for companies that do not report this information. (Theestimation method is described in Appendix 1.)

To investigate the potential bias

arising from the use of this estimation method, we also report regression results basedon the sample of companies that report labor and related expenses.

The “Small

Sample” in Table 2 includes only companies that report labor and related expenses.
The “Large Sample” includes companies whose labor costs we have estimated.
All coeﬃcients in Table 2 are significant and positive, which is consistent with our

32

The dependent variable is D ln yt .
Small Sample Small Sample Large Sample Large Sample
D ln yt−1

D ln yt−2

D ln wt−2

0.708∗∗

0.696∗∗

0.708∗∗

0.671∗∗(0.027)(0.027)(0.008)(0.008)0.120∗∗

0.132∗∗

0.158∗∗

0.084∗∗(0.026)(0.026)(0.009)(0.008)0.116∗∗

0.101∗∗

0.040∗∗

0.087∗∗(0.018)(0.018)(0.006)(0.006)

D ln kt# of observations 3113

0.038∗∗

0.106∗∗(0.007)(0.002)

3113

30135

20119

Table 2: Estimation Method 1
The variables D ln yt , D ln wt and D ln kt are relative labor productivity, relative
wage payments and the relative capital—labor ratio, respectively. The “Small
Sample” includes only companies that report labor and related expenses. The“Large Sample” also includes companies whose labor costs we have estimated. The
variable D ln wt−1 is used as the instrument for this regression. Standard errors arereported in parentheses. ∗∗ denotes significance at the 0.5 percent level.
theoretical predictions. Moreover, the results do not depend on the sample size.
More interestingly, two-year lagged relative wage payments have a positive impact
on current relative productivity even after conditioning the first and second lags ofrelative productivity.

The elasticity of two-year lagged relative wage payments is0.12 in the small sample and 0.04 in the large sample.
in the large sample.

The coeﬃcient is smaller

However, the results from both samples are significant anddemonstrate the quantitative eﬀect. The results support the hypothesis that skilledworkers improve a firm’s assets.
In this regression, we implicitly assume that there are no adjustment costs ofinvestment in physical capital. Given this assumption, physical capital can be derived33

as a function of organization capital. Organization capital not only directly increases
labor productivity, it also increases the physical capital stock, which in turn raises
labor productivity.

Because we are interested in the total eﬀect of organization

capital on labor productivity, we ignore the physical capital stock.
However, if adjustment costs of investment in physical capital are important,a high current level of labor productivity can partially be explained by the initialphysical capital stock per worker.

The omission of physical capital might havebiased our estimates. To investigate this possibility, we add relative physical capitalper worker, D ln kt . We estimate D ln kt by using ln kf t −Pmt
fln kf t, where kf t is themtinitial capital stock per worker in the f th firm in year t.
The inclusion of D ln kt hardly changes the coeﬃcients in the small sample, butraises the elasticity of D ln wt−2 and lowers that of D ln yt−2 in the large sample. This
indicates that adjustment costs of investment might have biased our results in thelarge sample. This is a potential problem. However, this minimal bias is unlikely
to aﬀect our simulation results. We discuss this point later.
We report the regression results obtained by using the second estimation method
in Tables 3 and 4.

The initial value of b4 is chosen to be 0.5. The result is notsensitive to this choice. The results in these tables are based on the estimated b4
matching the assumed b4 . Table 3 reports the results from the regression equation(34). Table 4 reports the results from the regression equation (30).
PiTable 3 shows that b3 (the coeﬃcient on t−1
i=0 (b4 ) D ln yt−1−i ) is 0.64 in the small

sample and 0.72 in the large sample.

The large sample produces a slightly larger

value of b3 . To check whether the constrained regression produces a bias, we also
ran an unconstrained regression. This regression yields a similar value of b3 . Thissuggests that our estimates are not sensitive to the constraint.
The unconstrained regression also reveals an interesting feature of the data: theweighted initial prior has a persistent eﬀect on labor productivity. This means thatthe eﬀect of initial values declines over time, but does not fade out altogether. The

34

theory predicts a coeﬃcient on the weighted initial prior of 1, but this is not supported
by data. However, the coeﬃcients are not far from 1. In particular, the coeﬃcientin the small sample is close to 1, 0.93. These results indicate that the model is a
useful first-order approximation of the data.
As already discussed, if adjustment costs of investment in physical capital are
important, our results might be biased. Hence, we also include D ln kt in our regressions. This does not materially change the coeﬃcients of the regressions. Hence,
our results are robust in this respect.
Table 4 shows that, after controlling for current relative productivity, the constructed belief about relative productivity continues to influence relative productivity
in the next year. Note that E [D ln yt |µkt ] is constructed from past observations. Ourregression results are consistent with the hypothesis that people learn about a firm’s
capacity from its past performance and form a belief that influences the employmentof skilled workers and future performance.
Table 4 shows that b1 (the coeﬃcient on D ln yt ) is 0.84 in the small sample and0.72 in the large sample. Given that b3 is 0.64 in the small sample and 0.72 in thelarge sample, b1 exceeds b3 in the small sample, but both are similar in the large
sample. Hence, equation (32) implies that h∞ < 1 in the small sample, while h∞= 1 in the large sample.

That is, according to the results from the large sample,

labor productivity is useful for predicting organization capital.
Adding relative physical capital stock per worker hardly changes the coeﬃcientsin the small sample, but causes the coeﬃcient of E [D ln yt |µkt ] to decrease in thelarge sample. This indicates that the large-sample regression results might overstate
the eﬀects of assignment if adjustment costs of investment in physical capital are
important. However, as is discussed later, this potential problem is unlikely to aﬀect
our simulation results.

35

The dependent variable is D ln yt .

Pt−1ii=0 (b4 ) D ln yt−1−i

Small

Small

Small

Large

Large

Const

Unconst Const

Const

Unconst Const

0.637∗∗

0.639∗∗

0.715∗∗

0.719∗∗

0.667∗∗(0.009) (0.003) (0.003)(0.003)(0.009) (0.009)

bt4 E [D ln y0 |µk0 ]10.931∗∗

0.627∗∗

1

0.847∗∗

1(0.026)
D ln kt# of observations

3645

3645

Large

1

(0.006)0.074∗∗

0.082∗∗(0.007)(0.002)

3638

32211

32211

32114

Table 3: Estimation Method 2 — the First Stage
The dependent variable is D ln yt+1 .
D ln yt

Small

Small

Large

Large

0.838∗∗

0.798∗∗

0.716∗∗

0.757∗∗(0.041) (0.040) (0.013) (0.013)
E [D ln yt |µkt ]0.090∗

0.118∗∗

0.211∗∗

0.103∗∗(0.043) (0.042) (0.014) (0.013)
D ln kt+1# of observations 2772

0.049∗∗

0.085∗∗(0.008)(0.003)

2771

23019

23012

Table 4: Estimation Method 2 — the Second StageWe report regression results in which the estimated b4 matches the assumed b4 in
two tables. “Small” refers to the small sample, which includes only companies thatreport labor and related expenses. “Large” refers to the large sample that includes
companies whose labor costs we have estimated. “Const” denotes the constrained
regression and “Unconst” denotes the unconstrained regression. Table 3 reports
OLS results, and Table 4 reports IV results. Standard errors are reported inparentheses. ∗ denotes significance at the 5 percent level.
the 0.5 percent level.
36∗∗

denotes significance atSmall Sample Large Sample
E[wt ]
E[yt ]

E[wt ]=
E[yt ]

PI PT
i

t

" Pm

ii w
f it
f
Pmii
yf it
f

IT

#

0.429

0.608

Table 5: Labor Share, where wf it and yf it are the wage payments and laborproductivity of the f th firm in the ith industry in year t, mit is the number of firms
operating in the ith industry in year t, I is the number of industries and T is thenumber of years. “Small Sample” includes only the companies that report labor
and related expenses and “Large Sample” includes also companies for which wehave estimated labor costs.

5

Simulation Resultsq
In this section, we report our estimates of φ, h and σγσ.
µ∞Using the estimatedparameters, we simulate our model and examine the extent to which assignment andthe noisiness of information aﬀect the persistence of relative productivity, relative
wages and relative profits per worker.

ψσq
ψσ q
Calibration of ασ: To identify the parameters, we need to know the value of ασ.
µ∞µ∞For this purpose, we use a calibration technique.

The wage function (14)implies
"
#

ψσ q
t]
t]
that ασcan be estimated from E[w, and we estimate E[wfromE[yt ]
E[yt ]
µ∞

PI PT
i

t

Pmit
wf it
f
Pmit
yf it
f

IT

,

where wf it and yf it are the wage payments and labor productivity of the f th firm in
the ith industry in year t, mit is the number of firms operating in the ith industry
in year t, I is the number of industries and T is the number of years.
t]
Table 5 reports our estimate of E[w. This is 0.43 in the small sample and 0.61 inE[yt ]t]the large sample. Because E[wapproximates the labor share, the estimate of 0.43 isE[yt ]

fairly small. This may be because COMPUSTAT only includes publicly traded firms,
which are relatively capital intensive and above average in size. In particular, because
37only relatively large firms in COMPUSTAT report labor and related expenses, “Small
Sample” contains only fairly large firms. In 2000, the average capital stock of firms
not reporting labor and related expenses was 421 million dollars, whereas that offirms reporting labor and related expenses was 787 million dollars.
Note that our estimated values are similar to those obtained by previous studies
based on COMPUSTAT data. Dhawan and Gerdes (1997) report an estimated labor
share from COMPUSTAT of 0.3.

Summary statistics in Bresnahan, Brynjolfssonand Hitt (2002) indicate a labor share for their selected sample from COMPUSTAT
of 0.53.
t]
Although our estimated E[wvaries between samples, fortunately our simulation
E[yt ]

results are not particularly sensitive to these variations. Below, we use our estimatesq
to estimate φ, h∞ and σγσ.
µ∞Estimated Structural Parameters: The results from our regression analysis yield
the following parameters of interest:ψσqq, φ, h∞ and σγσ
.
ασ µ∞
µ∞

Table 6 reports theresults.
Our estimates generally diﬀer between the small and large samples. The large
ψσqt]
diﬀerence in ασarises because of diﬀerent estimated values of E[w. The value ofE[yt ]
µ∞
ψσq
ασ µ∞
ψσq
ασ µ∞

measures the relative importance of worker quality to production.

Hence, ifis large, a firm’s productivity is aﬀected more by assignment. This partiallyqexplains why σγσ
is relatively large and φ is relatively small in the large sample.
µ∞

Similarly, our estimates of h∞ are also sensitive to the sample size.5 Output is usefulfor predicting organization capital in the large sample, but not in the small sample.
While diﬀerent sample sizes yield large diﬀerences in the estimated parameters,
diﬀerent estimation methods produce similar results.
5

In particular, the results in

Because h∞ cannot exceed unity, if the estimated value of h∞ is greater than 1 we set h∞ = 1

for the purpose of simulation.

38

Small Sample Small Sample Large Sample Large Sample
Estimation 1

Estimation 2

Estimation 1

Estimation 2

ψσq
ασµ∞

0.751

0.751

1.549

1.549

φ

0.499

0.565

0.279

0.282

h∞

0.556

0.644

1

0.993

γσ q
σµ∞

0.456

0.363

0.646

0.645

q
λ1 = φ + σγσ
µ∞

0.955

0.928

0.925

0.927

Table 6: The Estimated Structural Parametersψσ qmeasures the relative contribution of skills to current labor productivity.
ασµ∞φmeasures technological persistence. h∞ measures the accuracy of the information
contained in realized labor productivity for predicting the level of organization
capital.

γσ qmeasures the importance of assignment for persistence.
σ µ∞“SmallSample” includes only companies that report labor and related expenses. “Large
Sample” also includes companies for which labor costs are estimated.
the large sample are almost identical. The robustness of the estimates to diﬀerent
estimation methods suggests that the results are reliable.
More importantly, although regressions from diﬀerent samples produce diﬀerentqvalues, the estimated persistence parameters, λ1 = φ + σγσ, are remarkably stable.
µ∞They range from 0.93 to 0.96.

Note that Proposition 11 suggests that λ1 is themost important parameter for the persistence of organization capital.

It is shown

later that λ1 is also the most important parameter for the persistence of relative
productivity, relative wages and expected relative profits. The remarkable stabilityof the persistence parameter explains why our simulation results are not particularly
sensitive to variations in sample size and estimation method.

Persistence of Productivity, Wages and Profits per Workers: To understand39the eﬀects of assignment on persistence, we calculate autocorrelations for productivity and expected productivity. Let us define the autocorrelations as ρln yj ≡E [E[D ln yt |µt ]E [D ln yt−j |µt−j ]]E[D ln yt D ln yt−j ]and ρE[ln y|µ]j ≡. The following proposiV ar(ln yt )V ar(E[ln yt |µ ])ttion, which is proved in Appendix 1, generates the theoretical prediction.
Proposition 15 The autocorrelations of relative productivity and expected relative
productivity are
³
´
h
³
´
i
ψσ q
ψσq
j−1
2
φh∞ 1 + ασ
λ
φh
1
+
λ
+
1
−
λ
∞
1
1
1
ασµ∞
µ∞
ρln yj =, ρE[ln y|µ]j = λj1 ,
³
´2
ψσ q
(φh∞ )2 + 1 − λ21
1 + ασµ∞q
and j ≥ 1.
where λ1 = φ + σγσµ∞Note that the predicted autocorrelations can be calculated by using the estimatedψσqq
structural parameters, ασ, φ, h∞ and σγσ. Proposition 15 shows that λ1 is theµ∞µ∞most important determinant of these autocorrelations. Given that there is a stableestimate of λ1 , we do not expect the predicted autocorrelations to depend greatly on
the sample size and estimation method. This expectation is confirmed below.
We compare the simulated correlations with the correlations observed in the data.
Proposition 14 states that perceived relative productivity is equal to the relative wage,
which is also equal to expected relative profits per worker.

Hence, for wages andexpected profits per worker, we can use the autocorrelations of expected relative
productivity to compare the simulated correlations with the observed ones.

Note

that, in this model, autocorrelations are equivalent to correlations between current
relative values and relative values from j periods previously. Hence, we estimate theobserved correlations by using

40

Pmit
Pmit
ih
i
PT PI Pmit hln xf itln xf i(t−j)
ffln xf it −ln xf i(t−j) −
t
i
f
mit
mit
r
ρln xj = r
Pmit
Pmit
h
i
h
i2 ,
2
PT PI Pmit
PT PI Pmit
ln xf itln xf i(t−j)
ffln xf it −ln xf i(t−j) −
t
i
t
i
f
f
mit
mit

(35)

where xf it represents either the labor productivity, labor expenses per worker or theoperating income per worker of the f th firm in the ith industry in year t, mit is thenumber of firms operating in the ith industry in year t, I is the number of industries
and T is the number of years.
Figures 1, 2 and 3 summarize the results of our simulations. Figure 1 compares thesimulated correlation for relative productivity with the one estimated from the data.
As already discussed, the results are similar despite diﬀerences in sample size andestimation method. All predicted correlations fit the data quite well. In particular,
the simulation results in the large sample are remarkably good. All results indicatethat the model can quantitatively account for the observed persistence of productivity
diﬀerences.
Figure 2 conducts the same exercises for the relative wage. All results suggestthat the model’s predictions are consistent with the data.

Hence, the results in

Figure 1 are unlikely to be the result of coincidence. Our model can also explain thepersistent diﬀerences in wage payments.
Figure 3 compares the predicted correlation for expected relative profits per worker
with the correlation of relative profits per worker. Although the predicted correlation
is much larger than the observed correlation, this result is expected. Our theory explains the correlation for expected relative profits per worker, but not the one for real
relative profit per worker. Real profits per worker are aﬀected by unpredictable idiosyncratic shocks. Hence, this correlation is expected to be smaller. This reasoning
is consistent with the pattern observed in Figure 3.

41

The correlation between current relative productivity and past relative productivity(small sample)
1.2
correlation coefficient

1
0.8
data
model (estimation 1)
model (estimation 2)

0.6
0.4
0.2
0
0

1

2

3

4

5
year

6

7

8

9

10

The correlation between current relative productivity and past relative productivity(large sample)
correlation coefficient

1.2
1

0.8

data
model (estimation 1)
model (estimation 2)

0.6
0.4
0.2
0
0

1

2

3

4

5
year

6

Figure 1:

42

7

8

9

10

The correlation between current relative wages and past relative wages(small sample)
1.2

correlation coefficient

1
0.8

data
model (estimation 1)
model (estimation 2)

0.6
0.4
0.2
0
0

1

2

3

4

5
year

6

7

8

9

10

The correlation between current relative wages and past relative wages(large sample)

correlation coefficient

1.2
1
0.8

data
model (estimation 1)
model (estimation 2)

0.6
0.4
0.2
0
0

1

2

3

4

5
6
year

7

Figure 2:

43

8

9

10

The correlation between current relative profits per worker and past relative profits per worker(small sample)

correlation coefficient1.2
1
data: relative profits per worker

0.8

model (estimation 1): expected
relative profits per worker
model (estimation 2): expected
relative profits per worker

0.6
0.4
0.2
0
0

1

2

3

4

5
year

6

7

8

9

10

The correlation between current relative profits per worker and past relative profits per worker(large sample)

correlation coefficient

1.2
1data : relative profits per worker

0.8

model (estimation 1) : expected
relative profits per worker
model (estimation 2) : expected
relative profits per worker

0.6
0.4
0.2
0
0

1

2

3

4

5

6

7

year

Figure 3:

44

8

9

10

Let us conduct a counterfactual experiment. We first ask “What would happen if
people were homogenous and, therefore, there were no assignments in the economy?”
This experiment can be done by assuming that σ q = 0 and the other parameters areψσ qconstant. The assumption of σ q = 0 implies a zero labor share, ασ= 0, and impliesµ∞qthat there is no assignment eﬀect, σγσ= 0. Figures 4 and 5 report the results ofµ∞this experiment.
Figure 4 shows that, if σ q = 0, the autocorrelations for relative productivity
diminish to about 0 after five years.

This result does not depend on either the

sample sizes or the estimation method.

All results show that relative temporaladvantages disappear quickly if there are no benefits from positive assignment.
This point is confirmed by Figure 5. It shows that if σ q = 0, the autocorrelationsof perceived relative productivity (which is equivalent to relative wages6 and expected
relative profits per worker) become 0 after between four and six years. This result is
not aﬀected by either sample size or estimation methods. This means that positive
assignment accounts for much of the observed persistence in wage payments andprofits.
We also ask “What would happen if h∞ = 1?”. Recently, many economists haverecognized the importance of intangible assets to firms and have tried to estimate
them [e.g., Hall (2001)].

This exercise may provide valuable information on how

better estimation of organization capital might aﬀect the persistence of variables.
Table 6 shows that labor productivity is quite informative in the large sample.
This means that the noisiness of information is not the main source of persistencein the large sample. Hence, we conduct this exercise only for the small sample andinvestigate whether the small sample confirms the findings from the large sample.
6

There is a caution for the interpretation of Figure 5. Equation (14) says that if σ q = 0, wages

must be 0 for all firms. Hence, the relative wage is always 0. In order to maintain a link between
the relative wage and expected relative productivity, σ q has to be slightly larger than 0.

Hence,

when we discuss the persistence of the relative wages, the results in Figure 5 have to be interpreted
as an approximation.

45

The autocorrelation without skill variation : productivity(small sample)
1.2

correlation coefficient

1

model (estimation 1)

0.8

model (estimation 2)

0.6

model without skill variation(estimation 1)
model without skill variation(estimation 2)

0.4
0.2
0
0

1

2

3

4

5
year

6

7

8

9

10

The autocorrelation without skill variation: productivity(large sample)

correlation coefficient

1.2

model (estimation 1)

1

model (estimation 2)

0.8
0.6

model without skill variation(estimation 1)
model without skill variation(estimation 2)

0.4
0.2
0
0

1

2

3

4

5 6
year

7

Figure 4:

46

8

9

10

The autocorrelation without skill variation : wages and expected profits per worker(small sample)
1.2

correlation coefficient

1
model (estimation 1)
0.8

model (estimation 2)

0.6

model without skill variation(estimation 1)
model without skill variation(estimation 2)

0.4
0.2
0
0

1

2

3

4

5
year

6

7

8

9

10

The autocorrelation without skill variation: wages and expected profits per worker(large sample)

Correlation Coefficient

1.2
1model (estimation 1)

0.8

model (estimation 2)

0.6

model without skill variation(estimation 1)
model without skill variation(estimation 2)

0.4
0.2
0
0

1

2

3

4

5 6
Year

7

8

Figure 5:

47

9

10

The autocorrelation without noise: productivity(small sample)
1.2

correlation coefficient

1

model (estimation 1)

0.8

model (estimation 2)

0.6

model without noise(estimation 1)
model without noise(estimation 2)

0.4
0.2
0
0

1

2

3

4

5
year

6

7

8

9

10

The autocorrelation without noise: wages and expected profits per worker(small sample)
1.2

correlation coefficient

1

model (estimation 1)

0.8

model (estimation 2)

0.6

model without noise(estimation 1)
model without noise(estimation 2)

0.4
0.2
0
0

1

2

3

4

5
year

6

7

Figure 6:

48

8

9

10

q
When h∞ changes, Lemma 9 shows that σγσchanges through equation (29), which
µ∞

ψσq
ψσ q
ψ γσq
in turn influences ασ. This is because ασ= αγ. These combined eﬀects are
σµ∞
µ∞
µ∞

reported in Figure 6. This shows that an improvement in information causes only
slight changes in the persistence of productivity, even in a small sample.
Figure 6 also illustrates an interesting finding. When h∞ = 1, the autocorrelations of relative wages and expected relative profits are smaller than those from thebenchmark model; that of relative productivity is larger in the short run and smallerin the long run. When h∞ = 1, because beliefs are frequently updated the autocorrelations of relative wages and expected relative profits per worker are small. Although
the same eﬀect influences the autocorrelation of relative productivity, there is an additional eﬀect. When h∞ = 1, σ u = 0. Hence, labor productivity is not aﬀected bytemporal shocks and the autocorrelation may be larger. Although temporal shocks
negatively aﬀect the persistence of relative productivity in the short run, their eﬀect
is dominated by that of beliefs in the long run.
In summary, these exercises consistently suggest that positive assortative assignment accounts for much of the observed persistence of a firm’s relative advantages(disadvantages), whereas the noisiness of information plays a relatively minor role.

Positive Correlation Between Relative Productivity and Relative Wages:
Our model can predict the correlation between relative productivity and relative
wages, ρln y ln w , where ρln y ln w ≡ √E[D ln yt D ln wt ]V ar(D ln yt )V ar(D ln wt ). The following proposition isproved in Appendix 1.
Proposition 16 The correlation between D ln yt and D ln wt can be simulated by
´
³
ψσq
φh∞
1 + ασµ∞ρln y ln w = r³
.
´2
ψσq
2 2
2
1 + ασµ∞ φ h∞ + 1 − λ1This proposition states that the correlation can be predicted by using our estimated parameters. We compare the simulated correlations with the observed ones.
49

Small

Small

Large

Large

Estimation 1 Estimation 2 Estimation 1 Estimation 2
Data

0.774

0.774

0.842

0.842

Model

0.853

0.863

0.879

0.886

Model without Skill Variation 0.305

0.404

0.278

0.280

Model without Noise

0.919

0.879

0.886

0.923

Table 7: The Correlation Between Relative Productivity and Relative Wages“Small” includes only companies that report labor and related expenses. “Large”
also includes companies for which we have estimated labor costs.
The observed correlation between relative productivity and relative wages is estimatedby using a method similar to that used to estimate equation (35).
Table 7 reports the results. The model predicts a slightly higher correlation thanthe observed one: the model predicts a correlation of between 0.85 and 0.89, whereas
the one recorded by the data is between 0.77 and 0.84. However, 0.77 and 0.84 arestill high. Hence, the high predicted correlation reasonably captures the feature of
the actual correlation.
There is an explanation of why the correlations predicted by the model exceed
the observed correlations.

To prove Proposition 16, we assume that organization

capital is accumulated through learning by doing. However, some workers may beintentionally employed to develop organization capital. As Appendix 2 suggests, if
a firm’s maximization problem takes into account the accumulation of organization
capital, the wage is related not only to current expected productivity, but also to
the market value of a firm. Hence, the predicted correlations would be lower thanthose reported in Table 7.
Similarly to the previous argument, if we assume that σ q = 0 the correlation is
between 0.28 and 0.4.7 . This means that the observed positive correlation between
7

When σ q = 0, the wage is 0.

Hence, the result obtained from the model that does not50

relative productivity and the relative wage largely arises because of positive assortative assignment between organization capital and the quality of workers. On theother hand, if h∞ = 1 the correlation is slightly larger.

This indicates that thenoisiness of information contributes little to reducing the correlation.
In summary, according to the theory, a large assignment eﬀect and the generationof fairly accurate information for inferring organization capital from output combine
to explain the high observed correlation between labor productivity and wages.

6

Conclusion

In this paper, we developed a theory that accounts for persistent diﬀerences in productivity, wages, skill mixes and profits between firms. Firms with a high level oforganization capital attract skilled workers, who increase organization capital further.
This positive feedback brings about persistent diﬀerences in these variables.

It isalso shown that history can be important for predicting future persistence because
organization capital is unobservable.

Our model is consistent with empirical evi-

dence. Our simulation results show that positive assortative assignment explains alarge proportion of observed persistence.
Some points are worth discussing. We defined organization capital as all the intangible assets embodied in an organization. As explained by Prescott and Visscher(1980), a source of organization capital is firm-specific human capital. Firm-specifichuman capital may raise an issue not explicitly considered in this paper.

Follow-

ing Atkeson and Kehoe (2005) and Samaniego (2006), in this paper, we assumed
that firms receive benefits from organization capital. However, because firm-specifichuman capital is valuable only to a particular firm, it is diﬃcult to determine who
benefits from it.
Fortunately, the dynamics of organization capital are not aﬀected by the explicit
incorporate skill variations, in Table 7, can be interpreted as the correlation when σ q ≈ 0.

51

incorporation of firm-specific human capital. In the context of this paper, organization capital is essentially vintage human capital, which is influenced by the historyof skilled workers in a firm. Hence, one could consider a model in which a sequenceof senior workers develops the organization capital of the firm and in which thereis assignment between organization capital and worker skills. Although this would
alter the wage function derived in the paper, it would not materially aﬀect persistence. This is because profits and wages would continue to depend on organization
capital, and the dynamics of organization capital would continue to be influenced byassignment. Because incorporating an internal labor market raises several separate
issues and complicates the model, it would represent an interesting extension of ourmodel.
We were unable to address questions about the dynamics of firm size because
our assignment model requires that the number of workers is fixed.

However, it

would be possible to extend the model in order to analyze firm size.

Assumingthat assignment between top managers and organization capital determines the totalfactor productivity (TFP) of a firm, other factors such as physical assets and thenumber of workers can be derived as functions of TFP. That is, the larger is TFP,
the higher are the levels of capital and labor. This approach can be used to generatetheoretical predictions about the dynamics of firm size. In this case, assuming that
there is learning by doing is questionable and, thus, one would rely on the results inAppendix 2. This interesting extension is left for future research.
Finally, it would be interesting to extend the model to incorporate entry and exit
by firms. We ignored entry and exit by firms to focus on eﬀects on the persistence ofvariables. Incorporating entry and exit would inevitably introduce nonlinearity andmake it diﬃcult to find an analytical solution. Hence, one would use computational
exercises for this analysis. Because equation (12) implies that a firm’s position, relative to the top, is important in an assignment model, one would expect that the
cut-oﬀ points at the bottom of distribution would not greatly aﬀect the theoretical52

prediction of our model.

Nonetheless, it would be interesting to examine how as-signment aﬀects entry and exit by firms. We plan to investigate this issue in future
research.

7

Appendix 1

The Derivation of Equation (22): Because E [µkt E [(ln kto − µkt ) |µkt ]] = 0, thecovariance of µkt and ln kto − µkt is 0. Hence,
¶2
¶2
µ
µ
γσ q
φht
2
2
2
o
σ µt+1 =φ+
σ µt + (φht ) V ar (ln kt − µkt ) +
σ 2u
σ µt
α
¶2
¶2
µ
µ
£ £
¤¤
γσ q
φht
2
2
2
o
=
φ+
σ µt + (φht ) E E (ln kt − µkt ) |µkt +
σ 2u
σ µt
α
¶2
¶2
µ
µ
γσ q
φht
=
φ+
σ 2µt + (φht )2 σ 2kt +
σ 2u
σ µt
α
¶2
µ
µ
¶2 2 2α σ kt
γσ q
φht
2
=
φ+
σ µt +
σ µt
α
ht
¶2
µ
γσ q
=
φ+
σ 2µt + φ2 ht σ 2kt .
σ µt

The Derivation of Equations (28), (26) and (27): The derivations of equations(26) and (27) follow from Proposition 6. Hence, we only discuss the derivation ofequation (28). From Proposition 6, it follows that¢¡1 − φ2
γσ q
r
=
³
´2 .
σ µ∞
¡
¢ 2
2
σk∞
φ + 1 + 1 − φ φ h∞ γσq(36)

Note that equation (18) implies σ 2k∞ = φ2 (1 − ht ) σ 2k∞ + σ 2ε in the steady state.
Hence, it follows thatσ 2k∞ =σ 2ε.
1 − φ2 (1 − h∞ )
53

(37)

Substituting equation (37) into equation (36) yields equation (28).

The Proof of Proposition 11: Because equation (25) implies that
kt =

j
X

Mi−1 ξt−i + Mj kt−j ,i=1

hence, we can show that
#
#
"" j
X
£
£ 0 ¤
¤
Mi−1 ξt−i + Mj kt−j k0t−j = Mj E kt−j k0t−j .
E kt kt−j = E

(38)

i=1

q
Note that Lemma 8 shows that the two eigenvalues of the matrix M are λ1 = φ + σγσµ∞and λ2 = φ (1 − h∞ ).

Hence, we can easily calculate the following corresponding

eigenvectors:
⎡
⎣

1
1

⎡

⎤

⎦ z1 , ∀z1 ∈ R f or λ1 , ⎣

⎤

q
− σγσ
µ∞

⎦ z2 , ∀z2 ∈ R f or λ2 .

φh∞Define two matrices, Λ and Z:
⎤
⎤
⎡
⎡
γσq
λ1 , 0z , − σµ∞ z2⎦, Z =⎣ 1⎦ , ∀z1 , z2 ∈ R.
Λ =⎣
0, λ2
z1 , φh∞ z2

Then, the matrix M can be decomposed into Λ and Z: M = ZΛZ−1 .

(39)

Hence, itfollows from equation (38) that£¤¤£
E kt k0t−j = ZΛj Z−1 E kt−j k0t−j ,⎡q
λj ,φh∞ λj1 + σγσ
1
j −1
µ∞ 2
⎣
where ZΛ Z=¡¢
q
φh∞ + σγσφh∞ λj − λj ,µ∞1

2Because we can easily calculate⎡¤£
E kt−j k0t−j = ⎣σ 2µ∞ + σ 2k∞ ,σ 2µ∞

σ 2µ∞ ,

σ 2µ∞

54

⎤

γσq ¡ j
j¢
−
λ
λ
1
2
σ µ∞
⎦.
γσq j
j
λ + φh∞ λ2
σ µ∞ 1

⎤

⎦,

(40)

an autocovariance matrix can be calculated as follows:
¸
∙
⎡
⎤
γσ
φh∞ λj1 + σ q λj2
j 2
j 2
µ∞
2
σ k∞ ,
λ1 σ µ∞γσ£¤ ⎢ λ1 σ µ∞ +φh∞ + σ q⎥µ∞E kt k0t−j = ⎣
⎦.
¡ j
j¢ 2
j 2
j 2
φh∞
λ1 − λ2 σ k∞ + λ1 σ µ∞ , λ1 σ µ∞γσφh∞ + qσ µ∞£¤
By using the first row and first column element E kt k0t−j and the second row and£¤second column element of E kt k0t−j , the following autocorrelations of ln kto and µkt
are derived:ρkj = (1 − ω) λj1 + ωλj2 , ρµj = λj1 ,
γσ q
σµ∞

µ
´2 ¶ .
³σ µ∞(λ1 − λ2 ) 1 + σk∞ω ≡Note that equation (22) implies σ 2µ∞ = (φσ µ∞ + γσ q )2 +φ2 h∞ σ 2k∞ in the steady state.
This means that
µ

σ µ∞
γσ q

¶2

= λ21

µ

σ µ∞
γσ q

¶2

2

+ φ h∞

µ

σ k∞
σ µ∞

¶2 µ

σ µ∞
γσ q

¶2

.

Hence, it follows from this equation that
¶2
µ
φ2 h∞
σ µ∞
=
.
σ k∞
1 − λ21(41)

Substituting this equation into the definition of ω, we have¢γσq ¡1 − λ21
σ µ∞
¡
¢.
ω=
(λ1 − λ2 ) 1 − λ21 + φ2 h∞
The Proof of Proposition 13: The impacts on ρµj are clear.

Hence, we only

derive the eﬀects on ρln kj .
"Ã# µµ ¶j !
¶
dρln kj
dω
λ
γσ q
2
j−1
³ ´ = λ1
j (1 − ω) − 1 −λ1 γσq Σ1
, h∞
q
λ1
d σµ∞
σε
d γσ
σε
¯# µ
¯
"
¶
¯
dω ¯¯
γσ q
¯
j−1
j (1 − ω) − ¯λ1 γσq ¯ Σ1
> λ1
, h∞
¯ d σµ∞ ¯
σε
55

dρ

ln kj
Hence, there exists a j ∗ such that d γσ
> 0, ∀j ≥ j ∗ . Similarly( σεq )
"
Ã
!#
µ
¶
µ
¶
dρ lnkj
γσ q
γσ q
dω
dω
j (1 − ω) Σ2= λj−1, h∞ − λ1, h∞ +
γσq Σ2
1
dh∞
σε
d σµ∞
σε
dh∞
(Ã
!
)
µ
¶
dω
γσ q
dω
λ2 − jωφ .
+λj−1, h∞ +
γσ q Σ2
2
d σµ∞
σε
dh∞

dρ

Hence, there exists a j ∗∗ such that dhkj
< 0, ∀j ≥ j ∗∗ .
∞The Derivation of Equations (30) and (31): Substituting equation (13) into
equation (9) yields, in the steady state,
ψσ q
σ2D ln yt = α (ln kto − µekt ) +Dµkt + ut + u ,
σ µ∞
2
¶
µ
ψσ q
Dµkt .
α+E [D ln yt |µkt ] =
σ µ∞

(42)
(43)

We define the vectors yt and ζ t and a matrix G such that
⎤
⎤
⎤
⎡
⎡
⎡
ψσq
σ 2u
D ln yt
u + 2
α,
σµ∞
⎦, G =⎣⎦.
⎦ , ζt = ⎣ tyt = ⎣q
E [D ln yt |µkt ]0, α + σψσ
0
µ∞

(44)

Then,

yt = Gkt + ζ t .
Combining this equation and equation (25) yields
yt+1 = GMG−1 yt + Gξt + ζ t+1 − GMG−1 ζ t .

(45)

The results follow immediately because it is easy to derive
⎡

⎤

ψ σq
⎦,
φh∞ J, where J= ⎣(46)
α σ µ∞
1 −1
⎡ ³
⎤
´
³
´
2
2
2
α εt + σ2ε − φ ut + σ2u + ut+1 + σ2u
⎦.
= ⎣
0

GMG−1 = M +Gξ t + ζ t+1 − GMG−1 ζ t

1 −1

56

The Proof of Proposition 15: We define vectors yt and ζ t and a matrix G in thesame way as we defined equation (44). Using equation (45), we find
yt =

j−1
X
¡
¡
¢i−1
¢j−1
vt−i + GMG−1
yt−(j−1) ,
GMG−1
i=1

where vt−i = Gξ t + ζ t+1 − GMG−1 ζ t . Hence the covariance matrix is¡ T ¢ ¡
¢j−1 ¡¢ ¡¢j−1 ¡¢
T
E yt yt−j
E yt−(j−1) yt−j
E yt+1 ytT .
= GMG−1= GMG−1

We define matrices Λ and Z in the same way as we defined equation (39). Then, as
stated in the proof of Proposition 11, M = ZΛZ−1 . Hence,´¡¢j−1 ³−1 j−1
GMG−1= GZΛZ−1 G= Zy Λj−1 Z−1
y ,
where

⎤
z2∗
³
´ ⎦ ∀z1∗ , z2∗ ∈ R.
Zy = GZ = ⎣ψσq∗z1 , − φh∞ + ασµ∞ φh∞ z2∗
⎡

z1∗ ,

³

γσ q
ψσq
− ασ
φh∞
σ µ∞
µ∞

´

Hence, we can show that

¢
¡ j−1
¡
¢
λ1 − λj−1
ψσ q
2−1 j−1j−1 −1
GMG= ZΛ Z +φh∞ J,
γσq
φh∞ + σµ∞ ασ µ∞(47)where ZΛj Z−1 is defined in equation (40) and J is defined in equation (46). Next,££¤¤
we calculate E yt+1 ytT . Because yt+1 = GMG−1 yt + vt , E yt+1 ytT can be de¢¡¢¡composed into E yt ytT and E vt ytT .
¡
¡
¢
¢
¡
¢
E yt+1 ytT = GMG−1 E yt ytT + E vt ytT¡¢¡¢
Because E yt ytT and E vt ytT are calculated as
´2
´2
⎤
³
⎡ ³
ψσq
ψσq
2 2
2 2
2
2 2
1 + ασµ∞ α σ µ∞ + α σ k∞ + σ u , 1 + ασµ∞ α σ µ∞¢¡⎦ ,(48)³´2´2³
E yt ytT = ⎣
ψσ q
ψσq
2 2
2 2
1 + ασµ∞ α σ µ∞ ,1 + ασµ∞ α σ µ∞⎡⎤¡ T¢
−φσ 2u , 0⎣⎦,=
E vt yt
(49)
0,
0
57

we can derive
³
´
´
⎤
⎡ ³
µ
¶
ψσq
ψσq
2
2
2
¡
¢
ψσ q ⎣ 1 + ασµ∞ λ1 σ µ∞ + φσ k∞ , 1 + ασµ∞ λ1 σ µ∞ ⎦³´´³
E yt+1 ytT = α2 1 +
.
ψσq
ψσq
2
2
2
ασ µ∞
1 + ασµ∞ λ1 σ µ∞ + φσ k∞ , 1 + ασµ∞ λ1 σ µ∞(50)Hence, equations (47) and (50) can be used to show that
⎡ ³
´ ³
´2
´ ³
´2 ⎤
³
µ
¶
ψσq
σµ∞
ψσ q
σ µ∞
1
+
λ
λ
+
φ,
1
+
¡ T ¢
1 σ
1 σ
ψσ q
ασµ∞
ασ µ∞
⎣ ³
´ ³ k∞ ´2´ ³ k∞ ´2 ⎦ .
³
λj−1= α2 σ 2k∞ 1 +E yt yt−j
1
ψσq
ψσ q
ασ µ∞
1 + ασλ1 σσµ∞λ1 σσµ∞+ φ, 1 + ασ
µ∞
µ∞
k∞
k∞
Using the first row and first column of this covariance matrix, we can derive theautocorrelation between ln yt and ln yt−j ,
∙³
¸
³
´
´ ³
´2
ψσq
ψσq
σµ∞
j−1
1 + ασµ∞ λ1
1 + ασµ∞ λ1 σk∞ + φρln yj =
.
³
´2 ³
´2
´2
³
ψσq
σµ∞
σu
1 + ασµ∞+ 1 + ασk∞
σ k∞

³
´2
2
h∞
Note that equation (41) implies σσµ∞= φ1−λ2 and that equation (26) implies
k∞
1
´2
³
σu
1−h∞
= h∞ . Hence, ρln yj can be rewritten as
ασ k∞
³
´
h
³
´
i
ψσq
ψσ q
j−1
2
φh∞ 1 + ασµ∞ λ1
φh∞ 1 + ασµ∞ λ1 + 1 − λ1
ρln yj =
.
³
´2
ψσq
2
2
1 + ασ
(φh
)
+
1
−
λ
∞
1
µ∞

Similarly, using the second row and second column of the covariance matrix, we can¤£derive the autocorrelation between E [ln yt |µkt ] and E ln yt−j |µkt−j ,ρE[ln y|µk ]j = λj1 .

The Proof of Proposition 16: Proposition 14 implies ln ŵ (χ̂ (µkt ))−E [ln ŵ (χ̂ (µkt ))] =E [ln yt |µkt ] − E [ln yt ]. Hence,ρln y ln w =Cov [[ln yt − E [ln yt ]] E [ln yt |µkt ] − E [ln yt ]].
σ y σ E[y|s]58

Using equation (48), it is easy to show that³´
ψσq1 + ασµ∞ σσµ∞k∞
ρln y ln w = r³
´2 ³
´2
´2 .
³
ψσq
σµ∞
σu
1 + ασµ∞+ 1 + ασk∞
σ k∞

³
´2
φ2 h∞
Note that equation (41) implies σσµ∞=and that equation (26) implies
1−λ21
k∞
³
´2
σu
∞
= 1−h
. Hence, ρyw can be rewritten as
ασ k∞
h∞
´
³
ψσq
1 + ασµ∞ φh∞ρln y ln w = r³
.
´2
ψσq
2 2
2
1 + ασµ∞ φ h∞ + 1 − λ1

Data Appendix: ((#X)f t implies COMPUSTAT number X of f th firm in year tand (#X)f it implies COMPUSTAT number X of f th firm in ith industry in year t.)• Selection of data: We used industry annual data from 1970 to 2004 from COMPUSTAT. However, because we constructed initial priors for each firm by using
the initial five annual observations in COMPUSTAT, our regression is based ondata for 1975—2004.

We deleted observations for which either the estimatedwage or value added was negative and deleted those for which the labor shareexceeded unity. This was because such observations are not consistent with themodel’s assumptions. Because we are interested in deviations from the industry average, we retain industries that have at least five firms throughout theyears for which data are available. Industries are classified based on four-digit
industry codes.
• Total expenses are defined as (#41)f t + (#189)f t , where (#41)f t is the cost of
goods sold and (#189)f t measures administrative, selling and general expenses.
• Labor expenses: If a firm reports labor and related expenses, (#42)f t , that
includes employee benefits, we use this as our measure of labor expenses. The
59

small sample comprises these firms.
as follows.

Otherwise, we estimate labor expenses

First, if a firm reports labor and related expenses that exclude

employee benefits, we replace labor expenses by⎡P⎤⎣Pf ∈Yt (#42)f t /(#29)f t

nY t

f ∈Xt (#42)f t /(#29)f t

nXt⎦ (#42)f t , ∀t,where (#29)f t is the number of workers in the f th firm in year t and Yt is theset of firms that includes employee benefits for year t, Xt is the set of firms that
exclude employee benefits for year t, nY t is the number of firms in set Yt and
nXt is the number of firms in set Xt . This is an estimate of labor and related
expenses that includes employee benefits.

Second, if a firm does not reportlabor and related expenses, we estimate these expenses by
i⎤
h
⎡P
i
h
f ∈Zit (#42)f it / (#41)f it + (#189)f it⎣⎦ (#41)f it + (#189)f it , ∀t, i,nZitwhere Zit is the set of firms that report labor and related expenses in the
ith industry in year t and nZit is the number of firms in set Zit .

Note that(#41)f it + (#189)f it is defined as total expenses. This is our estimate of labor
expenses for firms in the large sample.
• yf t : Value added divided by the number of employees (#29)f t . Value added ismeasured as sales (#12)f t minus the value of materials, which is total expenses
minus labor expenses.
• wf t : Labor expenses divided by the number of employees (#29)f t .
• πf t : Operating income (#13)f t divided by the number of employees (#29)f t .
• kf t : Total net value of property and plant and equipment at the end of theprevious year (#8)f t−1 divided by the number of employees (#29)f t . Hence,
we approximate the initial capital stock by using the value at the end of theprevious year.
60

yit1990 Mean

wit

π it

kit

82.9 34.8 48.1 32.9

Std. Dev. 120

14.0 110

29.9

Median67.6 31.4 36.9 27.9

# of Obs

142

142

145

52.2 92.8 47.1

2000 Mean

142

141

Std. Dev. 39.8 14.0 31.6 22.7
Median139

49.1 84.2 41.5

# of Obs

86

86

86

85

Table 8: Summary Statistics for the Small Sample
yit

wit

π it

kit

77.2

33.8

43.4

103

Std. Dev. 366

45.7

360

338

Median

48.0

28.4

14.4

25.4

# of Obs

1371 1371 1371 1353

1990 Mean

2000 Mean

153

59.1

94.0

221

Std. Dev. 533

169

488

1141

Median

75.3

41.2

26.5

32.7

# of Obs

1561 1561 1558 1524

Table 9: Summary Statistics for the Large Sample
The nominal value is reported. Dollar figures are in thousands.

61

8

Appendix 2

In this appendix, we show that even if a firm takes into account the dynamics of organization capital for its maximization problem, the dynamics of organization capital
do not change. However, the wage function and the value function are aﬀected.
We define a recursive positive assortative equilibrium with unobserved organization capital. To formally define the equilibrium, we distinguish prevailing uncertainty
about ln kto , σ akt from a firm’s uncertainty about ln kto , σ kt , which are, however, equalin equilibrium. Let xt = (µekt , σ µt , σ akt )T denote the vector of these aggregate statevariables. The firm’s problem can be rewritten as
⎫
⎧
2
∗
⎬
⎨
|µ
,
σ
,
ln
q
]
−
w
(ln
q
:
x
)
E
[y
t kt
t
t
t
kt
∗
V (µkt , σ kt : xt ) = max,¡¢R
ln qt ⎩
+β V ∗ µkt+1 , σ kt+1 : xt+1 dΓs (st |µkt , σ kt ) ⎭

(51)

¶
µ
¤
£
α2 σ 2kt
2+ ψ ln qt ,s.t.E yt |µkt , σ kt , ln qt = exp ln A + αµkt +2

σ2µkt+1 = ln B + φ [(1 − ht ) µkt + ht st ] + γ ln qt − ε ,2qφ2 (1 − ht ) σ 2kt + σ 2ε ,σ kt+1 =µekt+1 = f (xt ) , σ µt+1 = g (xt ) , σ akt+1 = m (xt ) ,where Γs (st |µkt , σ kt ) is a conditional distribution function of a signal st given µkt
and σ kt , and functions f (·), g (·) and m (·) represent firms’ expectations about thetransition of the aggregate state variables. Our candidate equilibrium is a positive
assortative equilibrium. Hence, similarly to the static problem, the policy function
must satisfyχ∗ (µkt , σ kt : xt ) =σq[µkt − µekt ] + µq .
σ µtGiven this policy function, the dynamics of µekt+1 and σ µt+1 areσ2µekt+1 = ln B + φµekt + γµq − ε ,2q2(φσ µt + γσ q ) + φ2 ht (σ akt )2 .
σ µt+1 =62

Hence, a recursive positive assortative equilibrium with unobserved organization
capital is defined as follows.
Definition 17 A recursive positive assortative equilibrium with unobserved organization capital comprises values of χ∗ (µkt , σ kt : xt ), V ∗ (µkt , σ kt : xt ), w∗ (qt : xt ), f (xt ),g (xt ) and m (xt ) that satisfy the following conditions.
1. An individual firm solves its maximization problem (51).
2. The labor market is cleared:χ∗ (µkt , σ kt : xt ) =σq[µkt − µekt ] + µq .
σ µt

3. Expectations are rational:
σ2
f (xt ) = ln B + φµekt + γµq − ε ,2q22(φσ µt + γσ q ) + φ ht (σ akt )2 ,g (xt ) =qm (xt ) =φ2 (1 − ht ) (σ akt )2 + σ 2ε ,
where
ht =

³

1+

ασa
kt
σu

³

´2

ασa
kt
σu

a

(52)
(53)
(54)

2

´2 , σ kt = σ kt .

Because σ akt = σ kt in equilibrium, σ kt is used subsequently to denote the aggregate
state variable. Note that if this equilibrium exists, the dynamics of kto and µkt are thesame as those presented in the main text. Hence, our main results are unaﬀected.
Because the labor- market clearing condition and rational expectations assumption
determine the functions χ∗ (µkt , σ kt : xt ), f (xt ), g (xt ) and m (xt ), we need to find avalue function and a wage function that are consistent with the definition of equilibrium. The next theorem derives the value function and the wage function.

63

Theorem 18 Suppose that φ ∈ [0, 1), α ∈ (0, ∞), ψ ∈ (0, ∞), γ ∈ (0, ∞), σ ε ∈[0, ∞), σ q ∈ [0, ∞) and σ u ∈ [0, ∞].

There exists a unique recursive positive as-sortative equilibrium with unobserved organization capital. In equilibrium, the valuefunction and the wage function are∞X∗

V (µkt , σ kt : xt ) =

Πis=1

i=0

γσq

φ + σµt+s−1

E [yt+i |µkt , σ kt ]q
1 + ασψσµt+i

,

(55)

ψσ q
E [yt |µkt , σ kt ]ασµt|µkt = σµt (ln qt −µq )+µe
(56)
ψσ q
kt
σq
1 + ασ
µt
γσ q
Z
β σµt
¢
¡
+
V µkt+1 , σ kt+1 : xt+1 dΓs (st |µkt , σ kt ) |µkt = σµt (ln qt −µq )+µe ,
γσ q
kt
σq
φ + σµt∗

w (ln qt : xt ) =where Π0s=1 φ+ βφγσq

βφ= 1 andσ µt+s−1⎡³´ α2 σ2
σqln A + ψ µq − σµt+iµekt+i + 2kt+i +

⎤

⎢
⎥
µ
¶2
E [yt+i |µkt , σ kt ] = exp ⎣ ³
,
ψσ q
´ £
α+
£
¤
¤ ⎦
σ µt+i
ψσqα + σµt+i E µkt+i |µkt , σ kt +V ar µkt+i |µkt , σ kt
2
µ
¶
£
¤
γσ q
E µkt+i |µkt , σ 2kt = µekt+i + Πiτ =1 φ +(µkt − µekt ) ,σ µt+i−τ
µ
¶2
i
X
£
¤
γσ q
2
τ −1
Πs=1 φ +φ2 ht+i−τ σ 2kt+i−τ ,V ar µkt+i |µkt , σ kt =σµt+i−sτ =1andµekt+iσ µt+iσ 2kt+i
and

µ
¶
σ 2ε
ln B + γµq −+ φi−1 µekt ,
2
³¡
¢T ´
,
= g i (xt ) ≡ g µekt+i−1 , gi−1 (xt ) , mi−1 (xt )
´
³¡
¢T
,
= mi (xt ) ≡ m µekt+i−1 , g i−1 (xt ) , mi−1 (xt )1 − φi=1−φq
g (xt ) = g (xt ) = (φσ µt + γσ q )2 + φ2 ht (σ akt )2 ,q
1m (xt ) = m (xt ) = φ2 (1 − hat ) (σ akt )2 + σ 2ε .
164

Proof. Consider a mapping, T
⎧
⎫
∗
⎨
⎬
|µ
,
σ
,
ln
q
]
−
w
(ln
q
:
x
)
E
[y
t kt
kt
t
t
t
∗
,
T V = max¡¢R
ln qt ⎩
+β V ∗ µkt+1 , σ kt+1 : xt+1 dΓs (st |µkt , σ kt ) ⎭¶µα2 σ 2kt
E [yt |µkt , σ kt , ln qt ] = exp log A + ψ ln qt + αµkt +,2
σ2µkt+1 = ln B + φ [(1 − ht ) µkt + ht st ] + γ ln qt − ε .
2Suppose that the value function and the wage function are represented by equations(55) and (56). DefinedE [yt |µkt , σ kt , ln qt ]MP Q (µkt , σ kt , ln qt ) =+βd ln qt

Z¡¢
dV ∗ µkt+1 , σ kt+1 : xt+1dΓs (st |µkt , σ kt ) .
d ln qt

Then, we can derive¶¢
σ µt ¡ew (ln qt : xt ) = MP Q
ln qt − µq + µkt , σ kt , ln qt .
σq¡¢Hence, when a firm is endowed with µkt = χ−1 (ln qt , σ kt : xt ) = σσµtq ln qt − µq + µekt ,∗0µit can equate the marginal cost of ln qt , w0 (ln qt : xt ) to the marginal benefit of ln qt ,MP Q (µkt , σ kt , ln qt ) by choosing ln qt . It is also easy to check thatw∗00 (ln qt : xt ) >∂MP Q (µkt , σ kt , ln qt )|µkt = σµt (ln qt −µq )+µe .
kt
σq∂ ln qt

Hence, the objective function of the firm for which µkt = χ−1 (ln qt , σ kt : xt ) is strictly
concave and ln qt is a unique optimal decision.

65

This decision is consistent with apolicy function of χ (µkt , σ kt : xt ). Hence,
µ
¶
σq
∗
∗
e
[µ − µkt ] + µq : xt
T V = E [yt |µkt , σ kt ] − w
σ µt ktZ¡¢+β V ∗ µkt+1 , σ kt+1 : xt+1 dΓs (st |µkt , σ kt )Z¡¢βφαE [yt |µkt , σ kt ]V ∗ µkt+1 , σ kt+1 : xt+1 dΓs (st |µkt , σ kt )
+
=
γσ q
ψσ q
φ + σµt
α+ σ
µt

=
=
=

∞
βφ
αE [yt+1+i |µkt , σ kt ]βφ X i
+
Πs=1
γσ q
γσ q
φ + σµt i=0
φ + σµt+s
α + σ ψσqαE [yt |µkt , σ kt ]q
α + ψσσµtµt+1+iαE [yt |µkt , σ kt ]+q
α + ψσ
σµt

∞
X

Πju=1

j=0

∞
X
j=1

βφ
γσq

φ + σµt+u−1

Hence, T V ∗ = V ∗ .
Finally, we show that

Πju=1

βφ

γσ q

φ + σµt+u−1

αE [yt+j |µkt , σ kt ]P∞qα + σψσµt+j

βφi
i=0 Πs=1 φ+ γσq

σ µt+s−1

αE [yt+j |µkt , σ kt ]q
α + σψσµt+j.

αE[yt+i |µkt ,σ kt ]
ψσ q

α+ σ

is bounded. Because φ+ βφγσqµt+i

σ µt+u−1

1 and Proposition 6 show that µekt+i , σ kt+i and σ µt+j converge to finite values, it is£¤£¤enough to show that E µkt+i |µkt , σ kt and V ar µkt+i |µkt , σ kt converge to finite val-ues. We define

Di = Πi−1
x=0

µ
¶
γσ q
φ+
.
σ µt+xSuppose that φ ∈ (0, 1), σ u < ∞ and that σ ε ∈ (0, ∞). Then, Proposition 8 shows³´
qqthat limx→∞ φ + σγσ< 1. Hence, limi→∞ Di → 0. This means that= φ + σγσ
µ∞
µt+x
£
¤
£
¤
E µkt+i |µkt , σ kt and V ar µkt+i |µkt , σ kt converge to finite values. Now suppose that£¤φ = 0, σ u = ∞ or σ ε = 0. Then, V ar µkt+i |µkt , σ 2kt = 0. Note that if limi→∞ Di£¤
is bounded, limi→∞ E µkt+i |µkt , σ kt is bounded. We need to prove that limi→∞ Di
is bounded.

Suppose that φ = 0; then, σ µt+1 = γσ q for all t.

This means thatq
φ + σγσ= 1 for all t. Hence, Di = 1. Suppose that σ 2u = ∞ or σ 2ε = 0. Becauseµt+xq
σ µt = φσ µt + σ q γ, 1 = φ + σγσ. Note thatµ∞σ µt+x =1 − φxσ q γ + φx σ µt .
1−φ
66

<

Hence,
Di

(

)
γσ
q
φ + 1−φx= Πi−1x=0
σ γ + φx σ µt
1−φ q
⎧
⎡
⎤⎫
⎨
⎬
γσ q
i−1
⎣
⎦
h
³
´i
= Πx=0 1 + (φ − 1) 1 −⎩σ q γ + 1 − φ + σσqµtγ σ µt φx ⎭
h
³
´i ⎫
⎧
⎨
(1 − φ) 1 − φ + σσqµtγ ⎬³´= Πi−1.
1−x=0⎩
1 − φ + σσqµtγ φ1x − 1 ⎭Note that for both 1 ≥ φ + σσqµtγ and 1 < φ + σσqµtγ , the following condition is satisfied:
h
³
h
³
´i ⎫
´i ⎫
⎧
⎧
⎨
⎨
(1 − φ) 1 − φ + σσqµtγ ⎬(1 − φ) 1 − φ + σσqµtγ ⎬
´
³
´ .
³
≤ Πi−1
1−
1−
Πi−1
x=0
x=0
σq γ
1
1
⎭
⎭
⎩
⎩
1−φ+
−
1
1
−
φ
+
(1
−
φ)
−
1
x
x
σ µt

φ

φ

Hence, it can be shown that
Di

½
∙
µ
¶¸¾
∙µ
¶
¸¾
½
i−1
X
σq γ
σq γ
x
x
1−φ 1− φ+
= exp
φ+
−1
≤
log 1 + φ
σ µt
σ µt
x=0
∙µ
¶
¸
¶
¸
∙µ
i−1
X
σq γ
1 − φiσq γxφ+− 1 = exp
−1 .
φ+
φ
≤ exp
σ µt
1−φ
σ µt
x=0
Πi−1
x=0

The third inequality uses the fact that log (1 + g) is concave in g. This means that³´σq γ
φ + σµt − 1
lim Di ≤ exp
.
i→∞
1−φ
Hence, Di is bounded.

References[1] Abowd, John M., John Haltiwanger, Ron Jarmin, Julia Lane, Paul Lengermann,
Kristin McCue, Kevin McKinney and Kristin Sandusky (2004), “The Relation
among Human Capital, Productivity and Market Value: Building Up from MicroEvidence,” mimeo.
67

[2] Acemoglu, Daron (1997), “Matching, Heterogeneity and the Evolution of Income
Distribution,” Journal of Economic Growth, Vol. 2, No. 1, 61—92.
[3] Arrow, Kenneth J. (1962), “The Economic Implication of Learning by Doing,”
Review of Economic Studies, Vol. 29, June, 155—173.
[4] Atkeson, Andrew and Patrick J. Kehoe (2005), “Modeling and Measuring Organization Capital,” Journal of Political Economy, Vol. 113, No. 5, 1026—1053.
[5] Bahk, Byong-Hyong and Michael Gort (1993), “Decomposing Learning by Doingin New Plants,” Journal of Political Economy, Vol. 101, No. 4, 561—583.
[6] Baily, Martin, Neil C. Hulten and David Campbell (1992), “Productivity Dynamics in Manufacturing Plants,” Brookings Papers on Economic Activity. Microeconomics, 187—249.
[7] Becker, Gary S. (1973), “A Theory of Marriage: Part 1,” Journal of Political
Economy, 81, 813—846.
[8] Bresnahan, Timothy F., Erik Brynjolfsson and Lorin M. Hitt (2002), “Information Technology, Workplace Organization, and the Demand for Skilled Labor:Firm-Level Evidence.” Quarterly Journal of Economics, Vol. 117, 339—376.
[9] Burdett, Kenneth and Dale T. Mortensen (1998), “Wage Diﬀerentials, Employer
Size, and Unemployment,” International Economic Review, Vol. 39, No. 2, May,
257—273.
[10] Chandler, A. (1977), The Visible Hand, the Managerial Revolution in American
Business, The Belknap Press of Harvard University Press, Cambridge, Massachusetts.
[11] Chari, V. V. and Hugo Hopenhayn (1991), “Vintage Human Capital, Growth,and the Diﬀusion of New Technology,” Journal of Political Economy, Vol. 99,
No. 6. December, 1142—1165.
68

[12] Costrell, Robert M. and Glenn C. Loury (2004), “Distribution of Ability andEarnings in a Hierarchical Job Assignment Model,” Journal of Political Economy,
Vol. 112, No. 6, December, 1322—1363.
[13] Dhawan, Rajeev and Geoﬀrey Gerdes (1997), “Estimating Technological Change
Using a Stochastic Frontier Production Function Framework: Evidence from U.S.
Firm-Level Data,” Journal of Productivity Analysis, Vol. 8, 431—446.
[14] Eeckhout, Jan and Boyan Jovanovic (2002), “Knowledge Spillovers and Inequality,” American Economic Review, Vol. 92, No. 5, 1290—1307.
[15] Ericson, Richard and Ariel Pakes (1995), “Markov-Perfect Industry Dynamics:
A Framework for Empirical Work,” Review of Economic Studies, Vol. 62, 53—82.
[16] Faria André L. (2003), “Mergers and the Market for Organization Capital,”
Mimeo.
[17] Hall, Robert D. (2001), “The Stock Market and Capital Accumulation,” American Economic Review, Vol. 91, No. 5, December, 1185—1202.
[18] Haltiwanger, John C., Julia I. Lane and James R. Spletzer (1999), “Productivity
Diﬀerences across Employers: The Roles of Employer Size, Age, and Human
Capital,” American Economic Review, Vol. 89, No. 2, May, 94—98.
[19] Haltiwanger, John C., Julia I. Lane and James R. Spletzer (2000), “Wages,
Productivity, and the Dynamic Interaction of Businesses and Workers,” NBER
Working Paper 7994.
[20] Jovanovic, Boyan (1982), “Selection and the Evolution of Industry,” Econometrica, Vol. 50, No. 3, May, 649—670.
[21] Jovanovic, Boyan (1998), “Vintage Capital and Inequality,” Review of Economic
Dynamics, Vol. 1, 497—530.

69

[22] Jovanovic, Boyan and Peter L. Rousseau (2001), “Vintage Organization Capital,” NBER Working Paper 8166.
[23] Kaldor, Nicholas (1934), “The Equilibrium of the Firm,” Economic Journal, Vol.
44, 60—76.
[24] Kremer, Michael (1993), “The O-Ring Theory of Economic Development,” Quarterly Journal of Economics, Vol. 108, August, 551—575.
[25] Lucas, Robert E. (1978),“On the Size Distribution of Business Firms,” Bell Journal of Economics, Vol. 9, 508—523.
[26] McGahan, Anita M. (1999), “The Performance of US Corporations: 1981—1994,”
Journal of Industrial Economics, Vol. 47, No. 4, December, 373—398.
[27] Nelson, Richard R. and Sidney G. Winter (1982), An Evolutionary Theory of
Economic Change, The Belknap Press of Harvard University Press, Cambridge,
Massachusetts.
[28] Penrose, E. (1959), The Theory of the Growth of the Firm, Oxford University
Press, Oxford, U.K.
[29] Prescott, E. and M. Visscher (1980), “Organization Capital,” Journal of Political
Economy, Vol. 88, 446—461.
[30] Robinson, Austin (1934), “The Problem of Management and the Size of Firms,”
Economic Journal, Vol. 44, 242—257.
[31] Rosen, Sherwin (1972), “Learning by Experience as Joint Production,” Quarterly
Journal of Economics, Vol. 86, August, 366—382.
[32] Samaniego, Roberto M. (2006), “Organizational Capital, Technology Adoption
and the Productivity Slowdown,” Journal of Monetary Economics, Vol. 53, October, 1555—1569.
70

[33] Sattinger, Michael (1979), “Diﬀerential Rents and the Distribution of Earnings,”
Oxford Economic Papers, Vol. 31, March, 60—71.
[34] Shimer, Robert (2005), “The Assignment of Workers to Jobs in an Economy
with Coordination Frictions,” Journal of Political Economy, Vol. 113, No. 5,
996—1025.
[35] Simon, Herbert A.(1997), Administrative Behavior, The Free Press, New York.
[36] Takii, Katsuya (2003), “Prediction Ability,” Review of Economic Dynamics, Vol.
6, 80—98.
[37] Takii, Katsuya (2007), “The Value of Adaptability - Through the Analysis ofa Firm’s Prediction Ability,” Journal of Economics and Business, March-April
2007, vol.59, 144-162.

71