OSIPP Discussion Paper : DP-2008-E-003

The Persistence of Differences in Productivity, Wages, Skill Mixes and Profits
Between Firms in a Rapidly Changing Environment
February 20, 2008
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 persistent differences in productivity, wages, skill mixes and profits between firms
in a changing and uncertain environment. Large expected organization capital (firm-specificknowledge) attracts skilled workers, who help to accumulate organization capital. Accumulated
large organization capital, in turn, confirms high expectations. This positive feedback brings
about persistent differences in these variables in an uncertain environment. We estimateparameters and simulate the model. Our results show that a positive assignment mechanism
accounts for a large part of the observed persistence; the difficulty of estimating organization
capital plays only an auxiliary role.
This paper is the revised version of a previous working paper, Takii (2007b). I would like tothank R. Anton Braun, Julen Esteban-Pretel, Yuichi Fukuta, Ryo Horii, Shigeru Iwata, Charles
Yuji Horioka, Hidehiko Ichimura, Boyan Jovanovic, Dan Sasaki, Akihisa Shibata, Futoshi
Yamauchi and seminar participants at numerous seminars and conferences 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

Osaka School of International Public Policy, Osaka University, 1-31,
Machikaneyama, Toyonaka, Osaka, 560-0043, Japan. E-mail: takii@osipp.osaka-u.ac.jp
acknowledged.

1

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. (2007)]. 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.
As suggested by Haltiwanger et al. (2007), the assignment model provides apotential explanation for these observed persistent heterogeneities. If a quasi-fixedfirm-specific resource and workers’ skills are complementary to each other, a firm
endowed with the large resource is willing to pay high wages to attract skilled workers.
Such a firm achieves high productivity and earns large profits.
However, this seemingly plausible explanation does not provide a complete answer.
First, why do some firms succeed in investing and maintaining their specific resources
while others do not? Evidence shows that the pace of job creation and job destruction
is quite rapid and that idiosyncratic factors are the main source of the observed gross
job flows in the US economy [e.g., Davis and Haltiwanger (1999)].

This indicatesthat firms always confront idiosyncratic changes that may destroy some firm-specificresources. What is the mechanism that enables productive firms to maintain theircore resources and prevents unproductive firms from investing these resources in achanging environment?
Second, can the assignment model provide a reasonable explanation even if we2cannot observe firm-specific resources? Evidence shows that unobserved heterogeneity explains a large part of the variations in productivity [e.g., Bartelsman and Doms
(2000)]. This indicates that intangible assets are likely to be the main component of
firm-specific resources. Because intangible assets are, by definition, diﬃcult to estimate, assignment based on intangible assets must rely on perceived values. How do
speculative beliefs influence the persistence of variables? More importantly, to what
extent is the observed persistence influenced by the discrepancy between beliefs andfundamental values? Because researchers disagree about the productive importanceof intangible assets [e.g., Bond and Cummins (2000) and Hall (2001).], this question
is important for understanding persistent inequalities in the era of the knowledge
economy.
In this paper, we aim to answer these questions and provide a unified explanationof observed persistence in changing and uncertain environments.

We propose adynamic assignment model for the relationship between the skills of workers andunobserved firm-specific knowledge, which we term a firm’s organization capital.1
There are three key assumptions in our model.
are complementary inputs.

1) Skill and organization capitalThis means that skilled workers can better utilize theavailable knowledge in a firm. 2) Skill is an input for the accumulation of organization
capital.

In particular, we model the firm’s organization capital as a variant of its

vintage human capital. 3) Although we cannot directly observe the amount of firmspecific knowledge, we can infer it from the firm’s output.

Hence, assignment isbased on the beliefs about organization capital that are formed from the firm’s past
performance. This allows us to analyze how not only the assignment mechanism but
also the discrepancy between beliefs and fundamental values influences the persistenceof observed variables.
The main logic can be explained as follows. If a firm’s organization capital is be1

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lieved to be high, this belief attracts skilled workers. On the other hand, because skill
is an input for the accumulation of organization capital, the employment of skilledworkers promotes the accumulation of organization capital. A firm that accumulates
more organization capital can be expected to improve its performance, which generates the perception that the firm has a higher level of organization capital. Hence,
this persistence is induced by two positive feedback mechanisms: feedback between
the accumulation of organization capital and the employment of skilled workers, andfeedback between the fundamental capability of a firm and the beliefs about thecapability.
From these two feedback mechanisms, we identify two sources of persistence: theheterogeneity of skills and the diﬃculty of measuring organization capital. 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 mostadvantages because it can attract the best workers who can provide the firm withthe 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.

We show that if there is no stochastic disturbance, every firm’s

rank remains the same and firms’ relative advantages (and disadvantages) persist
indefinitely.
The theory also predicts that a rise in the noisiness of information increases thepersistence of organization capital. If the revealed information is noisy, managers
learn little from the new observations, and thus do not change their beliefs drastically. Because 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 powerfor organization capital, the belief never changes.

In that case, we show that thefirm’s rank remains the same on average forever and that the dynamics of actualorganization capital exhibit temporal deviations from the constant belief.

4

To examine the quantitative importance of the two sources, we estimate parameters and simulate the model.

Because it is perceived organization capital that

attracts skilled workers who help to accumulate organization capital, the eﬀect ofthe perceived value on persistence provides information about the role of assignment
in persistence in our framework.

Exploiting this information, 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. By using the estimated parameters, we simulate not only the
autocorrelations of relative productivity, relative wages and expected relative profits
per worker, but also the correlation between relative productivity and relative wages;
note that, in this paper, “relative” refers to the logarithm of each value relative toindustry and year averages.

All simulated autocorrelations replicate the observed

autocorrelations quite well.

The model is also able to explain the observed highcorrelations between relative productivity and relative wages.

That is, our model

can quantitatively account for the stylized facts of interest.
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 large partof 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 resources5[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
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 [e.g., Becker (1973)
and Sattinger (1979)]. More recently, Kremer (1993) demonstrates that the modelof positive assortative matching among workers can explain a variety of evidence.
Similarly to his model, our model incorporates positive assortative matching among
workers. In addition, although Kremer’s (1993) model is static,2 our model conveys
the sprit of Kremer’s (1993) idea in a dynamic framework; that is, current skilled
2

Most assignment models are static and the distribution of assigned variables is treated as given.

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.

6

workers attract skilled successors.
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 itself changes because oflearning by doing and because of idiosyncratic shocks that change the usefulness of
the accumulated knowledge. Hence, even mature firms must continue to learn about
their capability. We suggest that this modeling strategy mimics the nature of firms’
behavior in a changing and uncertain environment.
Although estimating organization capital is diﬃcult, the key assumptions madein this paper are broadly consistent with the evidence.

Evidence shows that pro-ductive organizational arrangement demands skill [e.g., Chandler (1977), Caroli and
Van Reenen (2001) and Bresnahan et al. (2002)].

This is consistent with our as-

sumption of complementarity between organization capital and skill.

In addition,evidence from Caroli and Van Reenen (2001) suggests that firms need the intangible
assets accumulated by skilled workers to make organizational changes productive.
Evidence also shows that the intangible assets accumulated by skilled workers are an
important determinant of technology adoption [e.g., Doms et al. (1997)]. Hence, theevidence consistently indicates that organization capital, as modeled in this paper,plays an important role in improving productivity by stimulating technological andorganizational changes.
The paper is organized as follows.

In the next section, we set up a dynamic

positive assortative assignment model.

We clarify the mechanism through whichskill diﬀerences and noisy information enhance persistence in the model. In Section
3, we use regression analysis to estimate the parameters of our model. In Section 4,
we simulate our model by using the estimated structural parameters. In Section 5,
we discuss extensions and conclude the paper. Proofs of propositions and technicalaspects of the derivation of equations are presented in Takii (2007b).

7

2

A Dynamic Assignment ModelIn this section we establish a positive assortative assignment equilibrium between
unobserved organization capital and skills.

Our static assignment model is basedon that of Sattinger (1979). We extend the model to incorporate dynamics, uncertainty and learning to describe the nature of assignment in a dynamically changing
environment.
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 function3 :yit = eut A (kto )α qitψ , α > 0, ψ > 0,(1)

where A, α and ψ are constant parameters and ut is a firm-specific productivity
2

shock and is normally distributed with a mean of − σ2u and a variance of σ 2u . We callthis shock, ut , noise because its only role is to make organization capital diﬃcult to
R1
observe. Because the total mass of jobs is 1, we interpret 0 yit di as both a firm’s
total output and its labor productivity.

Assume that kto cannot be directly observed, but can be inferred from the realizations of output. 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.
3

We assume that the prior

Alternatively, we can assume the following production function without changing our results:
utyit = eA (kto )α

∙Z 1
0

β
qijt
dj

¸ ψβ

, α > 0, ψ > 0, β < 1,

where qijt is the quality of the jth worker in the ith job at date t.
Kremer’s (1993) idea of team production.

8

This assumption capturesdistribution of ln kto 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(2)

All firms are assumed to have the same σ kt at date t. However, 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.
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 (µkt ) ≡ arg max {E [yit |µkt , ln qit ] − w (ln qit )} , ∀i, µkt .
ln qit(3)

Assume that ln qt is normally distributed with a mean of µq and a standard deviation of σ q at any date. It is assumed that the belief, µkt , is normally distributed witha mean of µekt and a standard deviation of σ µt . We examine a positive assortative
assignment equilibrium between a belief, µkt , and a skill, ln qt .
The positive assortative assignment equilibrium means that the top x percent ofµkt is assigned to the top x percent of ln qt for any x. Let Φ (·) denote the standard
normal distribution.

Given that

µkt −µekt
σµt

andln qt −µq
σq

are distributed as standard

normal variables, a positive assortative equilibrium implies that¶µ¶µ
χi (µkt ) − µqµkt − µekt=1−Φ, ∀i, µkt .
1−Φ
σ µt
σq

(4)

For simplicity, we assume that jobs and workers have reservation values of 0.
Because the number of jobs is the same as the number of workers, nobody chooses
the outside option and every agent can find a partner. Hence, equations (3) and (4)characterize a static market equilibrium.
9Definition: A static market equilibrium consists of χi (·) and w (·) that satisfy equations (3) and (4).

We aim to find a policy function and a wage function that are consistent with thisdefinition of equilibrium. Equation (4) states that the policy function must satisfy
χ (µkt ) ≡ χi (µkt ) =σq(µ − µekt ) + µq .
σ µt kt

Hence, all jobs in a firm are filled by workers of the same quality. This policy
function means that, in equilibrium, highly qualified workers must be assigned to a
firm that has a high level of perceived organization capital. For this policy function
to be consistent with the definition of equilibrium, the policy function must solve£¤
equation (3). Consider a firm that has µkt = σσµtq ln qt − µq + µekt ≡ 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 ) = ψE yt |χ−1 (ln qt ) , ln qt , ∀ ln qt .

Moreover, because the reservation value of workers is 0, w (−∞) = 0. The following
wage function is derived from the marginal condition and the boundary condition:w (ln qt ) =ψσ q
E [yt |χ−1 (ln qt ) , ln qt ]
ασ µt
.
ψσq
1 + ασµt(5)

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 µkt . Hence, a firm that has
high perceived organization capital pays high wages.
10The firm’s expected profits are strictly increasing in µkt .
Z 1
E [yt |µkt , χ (µkt )]e[E [yt |µkt , χ (µkt )] − w (χi (µkt ))] di =.
π (µkt ) =ψσq
1 + ασ0µt(6)In sum, the levels of skill and expected profits are strictly increasing functions ofµkt and that the wage function is a strictly increasing function of ln qt . Hence, thedynamics for skills, wages and expected profits follow the dynamics of µkt . On theother hand, labor productivity, ln yt , is strictly increasing in ln kto and ln qt . Hence,the dynamics of labor productivity are influenced by the dynamics of ln kto and µkt .
To understand the dynamics of productivity, wages, skills and profits, we analyze thedynamics of ln kto and µkt 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
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

11improves organization capital in the next period:4okt+1= B (kto )φ (qte )γ eεt , 0 ≤ φ < 1, γ > 0,

where qte =

R1
0

(7)

qit di, B, φ and γ are constant parameters and εt is a random vari2

able, which is normally distributed with a mean of − σ2ε and a standard deviation ofσ ε . Because of rapid changes in technology or demand, there is uncertainty about
the productive usefulness of the accumulated knowledge. The random variable, εt ,summarizes the shifts in the productivity of the accumulated knowledge because ofchanges in the environment. The parameter φ measures the technological persistenceof organization capital. Because some 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 (7).
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 orculture. Second, the learning-by-doing assumption simplifies the model, but conveys
the main logic of the paper5 . Hence, most of our analysis avoids the technical diﬃ4

Given that every job is filled by workers of the same quality, equation (7) generates dynamics

that are the same as those from the transition equation,okt+1= B1 (kto )φ1 (yt )γ 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.
5

In Takii (2007b), we assume that a firm solves a dynamic optimization problem by taking

12culties associated with dynamic optimization problems. Third, the wage and profit
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 learning-by-doing
assumption later.
Because all jobs in a firm are filled by workers of the same quality, in which case,ln qt = σσµtq (µkt − µekt ) + µq , in equilibrium, the dynamics of organization capital canbe written aso= ln B + φ ln kto + γ
ln kt+1∙¸
σqe(µ − µkt ) + µq + εt .
σ µt kt(8)

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

After the job 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

(9)
(10)

where
E [ln kto |st , µkt , σ kt ] = (1 − ht ) µkt + ht st = (1 − ht ) µkt + ht (ln kto + u∗t ) , (11)
³
´2
ασkt
σu

ht =

1+

³

ασkt
σu

´2 .

into account the dynamics of organization capital.

(12)

It constructs a recursive positive assortativeequilibrium and examines its properties. This shows that the dynamics of organization capital are
the same as those obtained when one assumes that there is learning by doing. Diﬀerences arise in
the wage and profit functions.

13Equation (11) 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 (12), ht is negatively related to σ u . If the varianceof ut 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.6
Because σ kt is the same in all firms, equation (10) shows that σ kt+1 is also thesame in all firms. Similarly, because µkt and st are normally distributed, equation (9)shows that µkt+1 is also normally distributed. Hence, the normality of the distribution
is preserved. The following mean and standard deviation of the belief in the next
period can be derived:
σ2µekt+1 = ln B + φµekt + γµq − ε ,
2
sµ
¶2
γσ q
σ µt+1 =φ+
σ 2µt + φ2 ht σ 2kt .
σ µt

(13)
(14)

Furthermore, by substituting equations (11) and (13) into equations (8) and (9), we
can also rewrite the dynamics of ln kto and µkt as follows:
γσ q(µ − µekt ) + ε∗t ,(15)
σ µt kt
¸
∙
γσ q
o
e
(µkt − µekt ) + φht u∗t ,(16)= 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 (15) shows the dynamics of ln kto .

The first term of equation (15) is

influenced by technological persistence, φ. That is, if organization capital is above
6

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,
2007a), as a tractable measure of prediction ability.

14

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
the firm accumulate further organization capital. Note that when the ratio of theqstandard deviation of skills to that of perceived organization capital, γσ, is large, theσµtois large. The firms with large µkt derive the high benefits fromeﬀect of µkt on ln kt+1
qlarge γσbecause these leading firms attract the most talented workers, who provide
σ µt

the firms with the best knowledge. Therefore, relative advantages persist longer.
Equation (16) 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, ut , or a highlevel of organization capital. Because managers put a weight ht on new information,
the fraction φht of current organization capital is believed to be translated into thenext period’s level.

New information incorporates noise.

Hence, the φht portionof u∗t also influences the posterior belief. This eﬀect is captured by the third term,
φht u∗t , in equation (16).
The second term of equation (16) 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
15

influences the posterior.

qOverall, the fraction φ (1 − ht ) + γσof the prior belief
σµt

influences the posterior.
The equations (15) and (16) provide some intuition about the dynamics of ln kto
and µkt . First, the interpretation of equation (15) 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 (16), 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. This meansthat a firm with large µkt can persistently attract high ln qt and maintain a large µkt .
That is, the noisier is the information, the more persistent is the belief.
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 1 The aggregate economy converges to a unique stationary distribution. Moreover, the dynamics of an individual firm in the stationary distribution are
described by the following vector autoregression (VAR):
kt+1 = Mkt + ξt ,
where

⎡

M=⎣

φ,

γσq
σ µ∞

q
φh∞ , φ (1 − h∞ ) + σγσ
µ∞

⎤

⎡

⎦ , kt = ⎣(17)

D ln kto
Dµkt

and D ln kto = ln kto − µek∞ and Dµkt = µkt − µek∞ .

⎤⎡⎦ , 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.

q
When σ µt is small, γσσ µt

is large and, therefore, firms with high perceived levels of organization capital havelarge relative advantages. This increases σ µt . Hence, provided that σ q is positive,the distribution is not degenerate.

We investigate the properties of equation (17)and discuss what influences the persistence of organization capital.
16

Extreme Cases : It is instructive to start with the extreme cases in which σ ε = 0and σ u = ∞. When σ ε = 0, there are no changes in the environment. Hence, firmsthat have relatively high levels of organization capital can maintain their relativeadvantages.

On the other hand, when σ u = ∞, information is too noisy and thefirm can learn nothing about the level of organization capital. Therefore, its belief
never changes. The following proposition shows that the level of organization capital
persists in both cases.
Proposition 2

1. Suppose σ 2ε = 0. Theno
= φ ln kto + (1 − φ) µkt , µkt+1 = µkt .
ln kt+1

2. Suppose that σ 2u = ∞. Theno
= φ ln kto + (1 − φ) µkt + ε∗t , µkt+1 = µkt .
ln kt+1

The first part of the proposition shows that if σ ε = 0, the belief does not change
and real organization capital eventually converges to the level implied by the constantbelief. Because there are no changes in the environment, the belief about organization
capital is accurate and there is no additional information from output. Hence, thebelief never changes. Because the constant belief determines the quality of workers,
real organization capital eventually converges to the level at which it is believed tobe.
Similarly to the case of σ ε = 0, if σ u = ∞, the belief never changes. However,the level of organization capital fluctuates around this constant belief. Because thefirm cannot learn about its own organization capital, the firm never changes its own
belief.

Hence, the belief is constant.

As actual organization capital is subjected

to shocks, the movement of organization capital temporally deviates from the firm’s

17own belief. However, the level of organization capital remains the same on average
because of the constant belief.

General Case:Let us examine a more general case.

q∈ (0, ∞) and γσ∈ (0, ∞).
σε

σu
ασ ε

Suppose that φ ∈ (0, 1),

First, we analyze the stability of equation (17).

Then, we analyze what influences persistence.
Let λ1 and λ2 denote the eigenvalues of the matrix M. Then, equation (17) isqcovariance stationary if λ1 = φ + σγσ< 1 and λ2 = φ (1 − h∞ ) < 1.
µ∞Note that

λ2 is less than unity. This means that stability is guaranteed if λ1 < 1. Because
λ1 incorporates important information about persistence, it is termed the persistenceqparameter. It consists of assumed persistence, φ, and the assignment eﬀect, σγσ.
µ∞qand h∞ are endogenous variables. Thus, there are more fundamental
Clearly, σγσµ∞conditions for stability. In order to find out the conditions, we need to understandq
the relationship between the endogenous variables, h∞ and σγσ, and the exogenousµ∞
q
σuvariables, ασ
and γσ. It is shown that there exist functions η (·) and Σ (·, ·)7 such
σε
ε

that
h∞ = η
where η 0

³

σu
ασ ε

´

σu
< 0, lim ασ
→0 η
ε

³

σu
ασε

´

µ

σu
ασ ε

¶

∈ (0, 1) ,σu
= 1 and lim ασ
→∞ η
ε

(18)
³

σu
ασε

´

= 0, and

¶
µ
γσ q
γσ q
(19)
=Σ
, h∞ ∈ (0, 1 − φ) ,
σ µ∞
σε
³
´
³
´
´
´
³
³
γσ q
γσ q
γσq
q
γσ q
γσq
>
0,
Σ
<
0,
lim
=
0,
lim
=
,
h
,
h
Σ
,
h
Σ
,
h
where Σ1 γσ
∞
2
∞
∞
∞
→0
→∞
σε
σε
σε
σε
³
³σε
´ σε
´
q
q
1 − φ, limh∞ →1 Σ γσ
, h∞ ∈ (0, 1 − φ) and limh∞ →0 Σ γσ
, h∞ = 1 − φ8 .
σεσεσuEquation (18) shows that h∞ and ασhave a one-to-one relationship.
ε7
8Hence,

Explicit solutions for η (·) and Σ (·, ·) can be found in the Takii (2007b).
γσ
Note that the properties of the function Σ (·, ·) imply that when h∞ converges to 0 or σεqconverges to infinite, the persistence parameter, λ1 , converges to 1. This means that the previous
extreme case can be seen as the limit of this general case.

18

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.
qqEquation (19) shows that for a given h∞ and φ, σγσand γσexhibit a one-to-one
σεµ∞relationship. This shows that not only the large variance in skills but also the smallqvariance of shocks to the accumulation of organization capital induces large σγσ.
µ∞qMore interestingly, σγσ
is decreasing in h∞ .
µ∞When information is more accurate,

rational agents rely more on new information to make inferences about the current
level of organization capital. Therefore, rational agents can change their posterior
beliefs based on reliable information. This makes the variance of µkt large and,q
therefore, makes σγσ
small.
µ∞
q
Note also that equation (19) implies that λ1 = φ + σγσ< 1. Hence, the followingµ∞

proposition can be stated.
q
σu
Proposition 3 Suppose that φ ∈ (0, 1), and that ασ
and γσare finite. Equationσεε(17) is covariance stationary.
The two eigenvalues, λ1 and λ2 , are important determinants of the persistenceof the stochastic process, too. Let ρln kj denote the autocorrelation between current
organization capital and organization capital j periods before. In addition, let ρµj
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 ]. We can derive the autocorrelations of ln kto and µkt are functions of
σ2µ∞λ1 and λ2
ρln kj = (1 − ω) λj1 + ωλj2 , ρµj = λj1 ,¢γσq ¡1 − λ21σ µ∞¡¢.
where ω =(λ1 − λ2 ) φ2 h∞ + 1 − λ2119The above equation states that the autocorrelation of organization capital can be
expressed as a weighted average of λj1 and λj2 ; the autocorrelation of the belief is λj1 .
Note that λ1 > λ2 . It means that the autocorrelation of the belief about a firm’sorganization capital exceeds that of its actual organization capital: ρµj > ρln kj , ∀j.
Because idiosyncratic shocks directly influence the realization of random variables,
the variance of the realized random variable is generally larger than the variance ofthe conditional expectation.

The same logic applies in this case.

Given that the

belief is less volatile than is actual organization capital, the autocorrelation of thebelief exceeds the actual value.
q
σu
Next, we show how the parameters γσand h∞ (or ασ) aﬀect the autocorrelations.
σε
ε

dρ

ln kj
Proposition 4 1) There exist j ∗ and j ∗∗ such that for all j ≥ j ∗ , d γσq > 0 and forall j ≥ j∗∗dρ, dhln∞kj < 0.

dρ2) For all j, d γσµjq
σεdρ
> 0 and dhµj
< 0.
∞

σε

q
This proposition implies that an increase in γσ
and a decrease in h∞ increase theσεautocorrelation about the belief. The same changes can increase the autocorrelationof 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.

3

Regression Analysis

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∞

20

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 (17), 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 ] ,

(20)
(21)

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∞
¡
¢
ψσ q
φh∞ + ασφh∞ , b4 = b1 + b2 − b3 and vt = α ε∗t − φu∗t + u∗t+1 .
µ∞In order to estimate equations (20) and (21), we must estimate E [D ln yt |µkt ] fromthe data. We propose two methods for doing this. Because each strategy has its own
strengths and weaknesses, it is hoped that the strategies complement each other.

Estimation Method 1: The first method is relatively simple and provides evidence
that is consistent with the assumption that skilled workers help firms to accumulate
assets and raise future productivity.

It applies the following proposition, which isproven by equations (5) and (6). Although the dynamics of organization capital are
not influenced by the assumption that organization capital is accumulated throughlearning by doing, equations (5) and (6) are aﬀected by this assumption. Therefore,
the following useful proposition represents a benefit of assuming that there is learningby doing.
Proposition 5 Perceived relative productivity is equal to relative wages and expected21relative profits per worker.
E [D ln yt |µkt ] = D ln w (χ (µkt )) = D ln π e (µkt ) ,where D ln w (χ (µkt )) = ln w (χ (µkt ))−E [ln w (χ (µkt ))] and D ln π e (µkt ) = ln πe (µkt )−E [ln π e (µkt )].
Proposition 5 states that perceived relative productivity can be estimated by using
relative wages. Hence, the following testable equation is derived from equations (20)and (21):D ln yt = θ1 D ln yt−1 + θ2 D ln yt−2 + θ3 D ln wt−2 + vt−1 ,(22)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 .
Equation (22) 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 5 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,
which is constructed from sequences of past relative productivity, influences future
relative productivity. Furthermore, to apply this alternative method, we need not
22

assume that there is learning by doing.

Hence, even if top managers’ skills areimportant elements 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 (21):D ln yt = b3

t−1
X
i=0

(b4 )i Dyt−1−i + bt4 E [D ln y0 |µk0 ] + $t ,(23)

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 ]. This contrasts with E [vt |D ln yt ] 6= 0 in equation(20).
Note that the parameters b1 and b3 diﬀer only if h∞ is less than 1 or, equivalently,
if σ u is positive.

In this case, E [u∗t |D ln yt ] 6= 0 and, therefore, E [vt |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, they eﬃciently extract this information from current productivity. Hence, b3 deviates fromthe fundamental parameter b1 . That is, the bias itself contains useful information
on h∞ .
To separate b1 from b3 , we apply ordinary least squares (OLS) to equation (23)and use an instrumental variables (IV) approach to estimate equation (20). 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. This helps to identify h∞ . We use thistechnique in 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 (23) is estimated under the constraint that the coeﬃcient of23

(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 (20) 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 .

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
US. It contains information from balance sheets, and information on incomes, cash
flows and financial variables.

The value added per worker and the average wage

rate are constructed for each firm and each year. Details of our data construction
procedure are given in Appendix.
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 1, 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. (The24

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 1: 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.
estimation method is described in Appendix.)

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 1 includes only companies that report labor and related expenses.
The “Large Sample” includes companies whose labor costs we have estimated.
All coeﬃcients in Table 1 are significant and positive, which is consistent with ourtheoretical 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 is 0.12
25

in the small sample and 0.04 in the large sample. The coeﬃcient is smaller in thelarge sample. However, the results from both samples are significant and demonstrate
the positive eﬀect. The results support the hypothesis that skilled workers improvea 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 derived
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 2 and 3.

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 2 reports the results from the regression equation(23). Table 3 reports the results from the regression equation (20).
Pi
Table 2 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

26

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
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 3 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.
Our regression results are consistent with the hypothesis that people learn about afirm’s capacity from its past performance and form a belief that influences future
performance.
Table 3 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, 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 usefulfor 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 the27

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 2: 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 3: 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 2 reports
OLS results, and Table 3 reports IV results. Standard errors are reported inparentheses. ∗ denotes significance at the 5 percent level.
the 0.5 percent level.
28∗∗

denotes significance atlarge 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.
In summary, the predictions of our model are broadly supported by the data.
In particular, the evidence is consistent with two important predictions: 1) skilledworkers help firms to accumulate assets and raise future productivity; and 2) people
learn about a firm’s capacity from its past performance and form beliefs that influence
its future performance.

4

Numerical Exercises

By using the estimated parameters of the previous section, we report our estimatesqof φ, h and σγσ.
µ∞Using these structural parameters, we simulate our model andexamine the extent to which assignment and the noisiness of information aﬀect thepersistence of relative productivity, relative wages and relative profits per worker.
ψσqq
Suppose that ασ
is known. The parameters φ, σγσand φh∞ can be identifiedµ∞µ∞from the following three equations:
φ = b1 −

ψσ q
γσ q
ψσ q
b3
φh∞ ,
= b2 +
φh∞ , φh∞ =
.
ψσ q
ασ µ∞
σ µ∞
ασ µ∞
1 + ασµ∞As discussed in the previous section, b3 contains useful information on h∞ . Given aqknowledge of h∞ , φ and σγσ
are primarily related to b1 and b2 , respectively. Becauseµ∞the fraction φ of current organization capital is transformed into next period’s organization capital, b1 contains information about φ. On the other hand, because belief
in a high level of organization capital attracts skilled workers, b2 contains information
about the eﬀect of assignment.99γσNote that we can separately identify φ and σµ∞q even if h∞ = 1. That is, our model canγσseparate σµ∞q from φ not because of noisy information, but because it distinguishes the eﬀects offirms’ decisions from those of firms’ capabilities.

29

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 4: 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.
ψσ q
To identify the parameters, we need to know the value of ασ.
µ∞Hence, weψσqcalibrate it from data. The wage function (5) implies that ασ
can be calibratedµ∞
t]t]from E[w, and we estimate E[wfromE[yt ]
E[yt ]

PI PT
i

t

" Pm

it w
f it
f
Pmit
yf it
f

IT

#

, where wf it and yf it are thewage 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 ofindustries and T is the number of years.
t]
Table 4 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
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 et al. (2002)30

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 5: The Estimated Structural Parametersψσqq
The parameters ασ, φ, h∞ and σγσmeasure the relative contribution of skills toµ∞µ∞current labor productivity, technological persistence, the accuracy of theinformation contained in realized labor productivity for predicting the level oforganization capital and the importance of assignment for persistence, respectively.
“Small Sample” includes only companies that report labor and related expenses.
“Large Sample” also includes companies for which labor costs are estimated.
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 5 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 partially31

qexplains 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.10

Output isuseful for 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.

In particular, the results inthe 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. It is shown later that λ1 is also the most importantparameter for the persistence of relative productivity, relative wages and expected
relative profits. The remarkable stability of the persistence parameter explains why
our simulation results are not particularly sensitive to variations in sample size andestimation method.

Persistence of Productivity, Wages and Profits per Workers: To understandthe 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 autocorrelations ofV ar(ln yt )V ar(E[ln yt |µ ])trelative productivity and expected relative productivity are derived as follows:
³
´
h
³
´
i
ψσq
ψσ q
j−1
2
φh∞ 1 + ασµ∞ λ1
φh∞ 1 + ασµ∞ λ1 + 1 − λ1
ρln yj =, ρE[ln y|µ]j = λj1 ,³´2
ψσq1 + ασ(φh∞ )2 + 1 − λ21µ∞q
and j ≥ 1.
where λ1 = φ + σγσµ∞Note that the predicted autocorrelations can be calculated by using the estimated10

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

for the purpose of simulation.

32

ψσ q
q
structural parameters, ασ, φ, h∞ and σγσ. These equations show 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 5 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 ones11 .
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 the11

Note that, in this model, autocorrelations are equivalent to correlations between current relativevalues and relative values from j periods previously. Hence, we estimate the observed correlations
by using
PT PI Pmit

∙

Pmit
f

ln xf it
mit

¸∙
¸
Pmit
ln xf i(t−j)
fln xf i(t−j) −mit

ln xf it −tifρln xj = s
∙
¸2 s
∙
¸2 ,
Pmit
Pmit
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

(24)

where xf it represents either the labor productivity, labor expenses per worker or the operating
income per worker of the f th firm in the ith industry in year t, mit is the number of firms operatingin the ith industry in year t, I is the number of industries and T is the number of years.

33

persistent 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 profits 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.
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∞
q= 0. Figures 4 and 5 report the results ofthat there is no assignment eﬀect, σγσµ∞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 wages12 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.
12

There is a caution for the interpretation of Figure 5. Equation (5) 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.

34

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:

35

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:

36

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:

37

8

9

10

We also ask “What would happen if h∞ = 1?”.

Table 5 shows that laborproductivity is quite informative in the large sample. This means that the noisinessof information is not the main source of persistence in the large sample.

Hence,

we conduct this exercise only for the small sample and investigate whether the small
sample confirms the findings from the large sample.
q
When h∞ changes, σγσchanges through equation (19), which in turn influences
µ∞

ψσq
ψσq
ψ γσ q. This is because ασ
= αγ
.
ασ µ∞
σ µ∞
µ∞

These combined eﬀects are reported in Figure

6. This shows that an improvement in information causes only slight changes in thepersistence of productivity, wages and expected profits per worker even in a small
sample.
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 correlation between

D ln yt and D ln wt can be simulated by:³´
ψσq
1 + ασφh∞µ∞ρln y ln w = r³
.
´2
ψσq
2 2
2
1 + ασµ∞ φ h∞ + 1 − λ1This equation states that the correlation can be predicted by using our estimatedparameters.

We compare the simulated correlations with the observed ones. The

observed correlation between relative productivity and relative wages is estimated byusing a method similar to that used to estimate equation (24).
Table 6 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 are
38

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:

39

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:

40

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:

41

8

9

10

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 6: 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.
still high. Hence, the high predicted correlation reasonably captures the feature of
the actual correlation.
Similarly to the previous argument, if we assume that σ q = 0 the correlation is
between 0.28 and 0.4.13 . This means that the observed positive correlation between
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.

5

Conclusion

In this paper, we developed a dynamic assignment model of the relationship between
the skills of workers and unobserved firm-specific knowledge, which we term a firm’s
13

When σ q = 0, the wage is 0.

Hence, the result obtained from the model that does notincorporate skill variations, in Table 6, can be interpreted as the correlation when σ q ≈ 0.

42

organization capital, to provide a unified explanation of observed persistence in achanging and uncertain economy. We posed two specific questions. 1) What is themechanism that enables productive firms to maintain their core resources, and what
prevents unproductive firms from investing these resources in a changing environment? 2) To what extent is the observed persistence influenced by the discrepancy
between beliefs and fundamental values? The answers based on our analysis are thattwo feedback mechanisms induced by assignment between unobserved organization
capital and skills prevent unproductive firms from investing in firm-specific knowledge and that the inaccuracy of beliefs about fundamental values explains a minor
proportion of the observed persistence.
Some points are worth discussing.

We defined organization capital as the in-tangible assets embodied in an organization and modeled this capital as a variant of
vintage human capital.

To develop the simplest possible model, we assumed thatthe labor market is perfectly competitive and that a firm receives all the benefits of
its organization capital. This is the standard approach used in modeling organization capital in a macroeconomic framework [see, e.g., Atkeson and Kehoe (2005) andSamaniego (2006)]. However, if all workers were to leave a firm at the same time,
one would not expect that firm to be able to maintain intangible assets.

Hence,

it is implicitly assumed that knowledge transferred from senior workers to younger
workers maintains organization capital.
It would be interesting to discuss how our results might change if we extended ourmodel to incorporate a long-term relationship in order to analyze explicit interaction
between senior workers and junior workers. As explained by Prescott and Visscher(1980), a source of organization capital is firm-specific or relation-specific humancapital.

Because firm-specific human capital is valuable only to a particular firm,

this discussion is particularly important if firm-specific human capital is the main
component of organization capital.
We would not expect the qualitative eﬀects of assignment on persistence to be43

aﬀected by explicitly incorporating a long-term relationship.

One could consider

a model in which senior workers develop the organization capital of the firm andin which there is assignment between organization capital and the skills of newly
employed workers.

Profits and wages would continue to depend on organization

capital, and the dynamics of organization capital would continue to be influenced byassignment. Of course, there are some diﬀerences. We expect that it would alter
the wage function and magnify the quantitative impacts of assignment on persistence.
Although these considerations would raise several interesting separate issues, because
incorporating an internal labor market would complicate the model, it would represent
an interesting extension of our model.
In general, assignment models are not suitable for addressing questions about the
dynamics of firm size. This is because they require that the number of workers be
fixed. However, it would be possible to extend our model in order to analyze firm size.
Assuming that assignment between top managers and organization capital determines
the total factor productivity (TFP) of a firm, other factors such as physical assets
and the number 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 togenerate theoretical predictions about the dynamics of firm size.

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 (4) 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 theoretical predictionof our model. Nonetheless, it would be interesting to examine how assignment aﬀects
entry and exit by firms. We plan to investigate this issue in future research.

44

6

Data Appendix14• 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
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
14(#X)f t implies COMPUSTAT number X of f th firm in year t and (#X)f it implies COMPUS-

TAT number X of f th firm in ith industry in year t. Summary statistics on the variables used for
estimation are in Takii (2007b).

45

expenses that includes employee benefits.

Second, if a firm does not reportlabor and related expenses, we estimate these expenses by
h
i⎤
⎡P
i
h
(#42)
/
(#41)
+
(#189)
f ∈Zit
f it
f it
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.

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