HIAS-E-42

Asset Specificity, Human Capital Acquisition, andLabor Market Competition
Hodaka Morita
School of Economics, University of New South Wales
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

Cheng-Tao Tang
Graduate School of International Relations, International University of Japan

January 2017

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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http://hias.ad.hit-u.ac.jp/
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Asset Specificity, Human Capital Acquisition, andLabor Market Competition

Hodaka Morita⇤

Cheng-Tao Tang†

January 31, 2017⇤
School of Economics, University of New South Wales., and Hitotsubashi Institute for Advanced
Study, Hitotsubashi University. Email: hodakamorita@gmail.com†
Graduate School of International Relations, International University of Japan. Email: chengtao.tang@gmail.com

1Abstract
Firms let their employees operate assets to produce goods and services. Firm-specificity of asset
and human capital, key concepts of transaction cost economics and labor economics respectively, play
important roles in determining firms’ productivity and welfare consequences of their competition. How
are the degrees of firm-specificity of asset and human capital determined? We address this question
through exploring a new model that captures interconnections among asset specificity, human capitalacquisition, managerial capability, and labor mobility. We consider a two-period model with two firms,
where period 1 is the skill-acquisition period and period 2 is the output period. In the beginning of
period 1, each firm chooses a level of its asset specificity and employs a certain number of workers fromthe labor market. The level of asset specificity is interpreted as the extent to which the firm tailors
its asset to the unique features of the firm’s business strategies and products. A firm’s second-period
productivity is determined by its managerial capability, the extent to which its asset is tailored, andits workers’ familiarity with its asset specificity. Managerial capability here means the capability of afirm’s top management to develop an eﬀective strategy and create a unique competitive position.
We find that, as the importance of managerial capability increases, the labor mobility increases,
and both the level of asset specificity and firm size decrease. When a firm chooses the specificity of
its asset and the number of workers it employs in period 1, it estimates how many workers it willretain and how many workers it will hire from its rival in period 2. A higher importance of managerial
capability increases the diﬀerence of period 2 productivity between a high-capability and a low-capability
firm. Then, as the importance of managerial capability increases, each firm anticipates higher labormobility, because a larger number of workers will move from a low-capability to a high-capability firm.
Anticipation of higher labor mobility, in turn, reduces each firm’s incentives to hire more workers andincrease the level of asset specificity in period 1.
We discuss implications of our model in the contexts of cross-industry and cross-country comparisons. In a newly emerging industry or in a business undergoing revolutionary technological changes, abusiness’s success critically depends on the quality of its strategic decision making because these industries face a high level of uncertainty. Whereas in industries facing lower levels of uncertainty, strategic
decision making is less important. These arguments suggest that the importance of managerial capability is higher in the former types of industries, and the importance tends to be lower in the latter types ofindustries. Our model then predicts that labor mobility is higher, specificity of asset and human capitalis lower, and average firm size is smaller in industries of the former type and vice-versa in industries ofthe latter type. Also, as the economy makes a transition from industrial capitalism to post-industrial
capitalism, modern economies are becoming increasingly knowledge intensive which renders the disadvantage to the firms that heavily rely on physical assets. Our model yields new implications regarding
the consequences of the transition.
Keywords: Asset specificity, competition, firm size, firm specificity, human capital, managerial capability
JEL: J24, L20, M50

2

1Introduction

Firms let their employees operate their assets to produce goods and services. Firmspecificity of asset and human capital, key concepts of transaction cost economics(Williamson, 1975, 1979, 1985) and labor economics (Becker, 1962) respectively, play
important roles in determining firms’ productivity and welfare consequences of theircompetition. The firm’s productivity becomes higher when the firm has its asset utilized
by a worker who is familiar with that asset’s specific feature. Consequently, there exists
important connection between asset specificity and labor mobility, because the acquired
human capital of the workers who utilize the specificity of the asset loses its value
if workers switch employers. In reality, there also exists connection between a firm’s
managerial capability and labor mobility. A firm with high managerial capability iscapable of expanding its business by hiring more workers, whereas a firm with low
capability ends up shrinking its business by dismissing workers. This paper explores a
new model that captures interconnections among asset specificity, acquisition of firmspecific human capital, firm size, managerial capability, and labor mobility.
We consider a two-period model with two firms, where period 1 is the skill-acquisition
period and period 2 is the output period. Each firm i (= 1, 2) initially owns an asset, andin the beginning of period 1 it chooses a level of its asset specificity and employs a certain
number of workers from the labor market. Workers can switch their employers between
periods 1 and 2. The level of asset specificity is interpreted as the extent to which thefirm tailors its asset to the unique features of the firm’s business strategies and products.
As such, a firm incurs higher costs as it chooses a higher level of asset specificity. The
idea that a firm chooses a level of its asset specificity is consistent with the insight put
forth by Doeringer and Piore (1985), who discussed firm-specificity of technology where
technology refers to the entire set of tasks that comprises a work process. They asserted
that firm-specificity of technology can be chosen by a firm’s management.
In the current model, a firm’s second-period productivity is determined by its managerial capability, the extent to which its asset is tailored (i.e., asset specificity), and its
workers’ familiarity with its asset specificity. Managerial capability here means the capability of a firm’s top management to develop an eﬀective strategy and create a unique3

competitive position. Let ai denote the realization of firm i’s managerial capability. For
simplicity, ai is assumed to be a random draw from a symmetric binary distribution
satisfying Pr(ai = aH ) = Pr(ai = aL ) = 12 , aH > aL , and aH =aL = ↵. Assume

that ai is ex-ante unknown to all agents including firm i itself and becomes common
knowledge at the beginning of period 2 of firm i’s operation. A worker employed by firmi becomes familiar with the specificity of firm i’s asset. Hence, if the worker is retainedby the same firm i, the worker’s second-period productivity is increasing in the level ofthe firm’s asset specificity. In contrast, a worker employed by firm j (6= i) in period 1 is
not familiar with the specificity of firm i’s asset so that, if firm j’s first-period employees
move to firm i in period 2, their productivity is not improved by the level of firm i’s
asset specificity. We introduce diseconomy of firm size in the model by assuming that afirm must incur per-worker supervision cost, and that the cost increases as the numberof workers employed by the firm increases.
In our analysis of the model, we focus on the range of parameterizations in which,
if one firm turns out to have a high managerial capability and the other has a low
capability, some workers move from the low-capability firm to the high-capability one inequilibrium. A key comparative statics result is that, as the importance of managerial
capability increases, the labor mobility increases, and both the level of asset specificity
and firm size decrease.
The intuition behind this result can be explained as follows. When firm i chooses
the specificity of its asset and the number of workers it employs in period 1, it estimatesthe expected number of its period 1 workers that the firm will retain for its period 2
operation, and the expected number of workers that the firm hires from its rival firmj in period 2. A higher importance of managerial capability increases the diﬀerence of
period 2 productivity between a high-capability and a low-capability firm. Then, as theimportance of managerial capability increases, each firm anticipates that a larger numberof workers switch their employers in period 2 if the two firms’ managerial capabilities
turn out to be diﬀerent. Hence higher importance of managerial capability decreases the
expected number of retained workers, and it increases the expected number of workers
who switch their employers. Holding the initial number of workers constant, this reduceseach firm’s incentive to choose higher specificity of its asset, because the proportion of4retained workers decreases and the productivity gains over the investment cost for eachretained worker becomes smaller.
Meanwhile, anticipating that the number of retained workers decreases while thesupervision cost per worker in period 1 remains the same, each firm has a lower incentive
to hire more workers in period 1. Because a larger number of period 1 workers increases
a firm’s period 2 profit from its retained workers at the expense of increasing supervision
cost per worker in period 1. Also, the reduction of each firm’s first-period workers (i.e.,
average firm size) decreases the proportion of retained worker in period 2, and in turndecreases the marginal benefit of choosing a higher asset specificity. Hence, the force oflowering average firm size further reduces each firm’s level of asset specificity. The result,then, is that the equilibrium level of asset specificity and average firm size decrease as
the importance of managerial capability increases. And, higher labor mobility in period
2 and lower number of first-period workers both work in the direction of increasing theturnover rate. The model also yields similar comparative statics results concerning theimportance of a firm’s asset specificity.
The connection between labor mobility and the levels of firms’ asset specificity andtheir employment sizes, uniquely explored in our model, yields empirical implications
and predictions from a previously unexplored perspective, given that the importanceof managerial capability and firm-specificity of asset can diﬀer across industries andcountries. In a newly emerging industry or in a business undergoing revolutionary technological changes, a business’s success critically depends on the quality of its strategic
decision making because these industries face a high level of uncertainty. Whereas inindustries facing lower levels of uncertainty, strategic decision making is less important.
These arguments suggest that the importance of managerial capability is higher in theformer types of industries, and the importance tends to be lower in the latter types ofindustries. Our model then predicts that labor mobility is higher, specificity of asset
and human capital is lower, and average firm size is smaller in industries of the former type and vice-versa in industries of the latter type. Also, as the economy makes
a transition from industrial capitalism to post-industrial capitalism, modern economies
are becoming increasingly knowledge intensive which renders the disadvantage to thefirms that heavily rely on physical assets. Our model yields new implications regarding5

the consequences of the transition (see Section 4 for more discussions).

1.1

Related literatureThis paper is related to two strands of literature: one on transaction cost economics
and the other on human capital acquisition. The literature on the transaction cost economics, which centers on the view of Williamson (1975, 1979, 1985) and Klein, Crawford
and Alchian (1978), now spans a large body of research. Whinston (2003) clearly indicates two starting point observations of the transaction cost economics. First, thecontent of contract between contracting parties is incomplete in many exchange relationship. Second, the investment of asset for a specific purpose often creates a surplus
to be shared among contracting parties and makes those parties eﬀectively “locked into”
the transaction over the course of their relationship. The surplus, often referred to as“appropriable quasi-rents,” plays a critical role in the theory of the firm literature.1 The
incompleteness nature of the contract and the presence of appropriable quasi-rents together open up possibilities for ex-post opportunistic behavior. As Masten (1984) points
out, “The more specialized those assets, the larger will be the quasi-rents at stake overthat period, and hence the greater the incentive for agents to attempt to influence theterms and trade through bargaining or other rent-seeking activities once the investments
are in place.” This ex-post opportunistic behavior, regarded as socially ineﬃcient, canbe prevented by costly remedies such as vertical integration (Williamson, 1979, 1985;
Klein et al., 1978).
Along this line of transaction cost economics literature, the concept of asset specificity is an important—perhaps the most important—building block in organizing sometransactions one way and other transactions another.2 While the focus in this literature
centers on the relationship between the structure of the internal organization and the asset specificity where the specificity of the asset is often treated as an exogenous variable.
Attention has not been paid to the interlinkage among firms’ endogenous determinationof asset specificity, employees’ human capital acquisition, and inter-firm competition in
1

See Gibbons (2005) for a review.
Williamson (1985, p.52) points out that “[asset specificity] is the most important and most distinguishes transaction cost economics from other treatment of economic organization, ...”
26the labor market. The present study sheds new light on the transaction cost economics
by studying this interlinkage.
Concerning human capital acquisition, a significant amount of literature grew out
from Becker (1962)’s classical distinction between general and firm-specific human capital. One literature along this line focuses on general human capital and optimal investment decision (e.g. Ben-Porath, 1967; Wallace and Ihnen, 1975). Another literature thatgrew out of Becker’s work focuses on specific human capital. For example, Hashimoto(1981) formalizes Becker’s argument that the cost of and the return to specific humancapital are shared by the worker and the firm. The approach following Becker’s analysis
of human capital investment implicitly assumes that the pre- and post-training wages
and investment levels in general and specific human capital are contractible. As pointed
out by Gibbons and Waldman (1999), an equally useful approach to human capital
investment is to assume investment levels are not contractible and that post-training
wages are determined by bargaining. This approach introduces a number of results that
do not arise in Becker’s formulation.
For example, Chang and Wang (1996) presents an asymmetric learning model, where
human capital accumulation is not contractible. They show that the equilibrium training
level is negatively related to the turnover rate and positively related to the specificityof training. This result is in contrast to Becker’s standard result where firms provide
general training only if employees pay the cost (Becker, 1962).3 Acemoglu and Pischke(1999) generalize the argument where firms have incentives to invest in the generalhuman capital of their workers. They show that several labor market imperfections(including asymmetric learning) that generate ex-post rent from bilateral monopoly canyield this conclusion. There also exists supporting evidence that firms share some of
the costs and returns to general training (e.g. Acemoglu and Pischke, 1996). Along thisline of literature, a degree of firm-specificity of human capital is often introduced and istypically treated as an exogenous variable.4
3

Bai and Wang (2003) also show that, unlike the predictions from traditional human capital theory,the probability of separation has a positive relationship with the specific human capital investment if
the uncertainty in labor productivity is either very high or very low.
4
? introduce demand-side constraints in the labor-market behaviors where skilled and unskilledlabors are not perfect substitutes in production and a firm incurs hiring cost when filling out its managerial position from the outside. They show that their model can generate various familiar practices

7

This study follows the approach where the accumulation of human capital is notcontractible but makes a departure in the following aspect. The firm-specificity of human capital in our model is endogenously determined by the degree of asset specificity.
Because in reality there is important connection between firm-specificity of asset andhuman capital acquisition: it is the workers who utilize the asset to produce goods andservices.
There are some other papers which depart from the standard labor-theoretical model
but address diﬀerent issues. Important examples include Morita (2001), Wang (2002),
Mailath, Nocke and Postlewaite (2004), and Morita (2012). Morita (2001) studies theconnection between continuous process improvement and the firm-specificity of training
and explores its implications on the U.S.-Japanese diﬀerences. Wang (2002) analyzes
the connection between product market conditions and job design and explores its implications on explanations for heterogeneity of human resource management practices
across countries, industries, and firms. Mailath, Nocke and Postlewaite (2004) analyze
the interaction between a firm’s choice of business strategy and its manager’s incentive
for investing in “business-strategy-specific” human capital and explore its implications
on the organization of business activities. Morita (2012) studies the interconnection
between firm dynamics and specific human capital. He also shows that, through a different mechanism, as the importance of a firm’s managerial capability increases, thesurvival rate of firms decreases, and the level of firm-specific human capital investment
decreases.5 Although this paper has many useful points of contact with this body ofwork, to our knowledge, our combination of asset specificity, acquisition of firm-specifichuman capital, firm size, managerial capability, and labor mobility is new, as is theattempt to explain cross-industry and cross-country facts in an integrated model.
The rest of the paper is organized as follows. Section 2 presents a model that eachsuch as fast-track promotions, up-or-out rules, and promotion from within, which were previously inexplicable within a competitive, symmetric information, and complete-contracting framework.
5
In Morita (2012)’s two-period model, each first-period entrant has an advantage over second-period
entrants given that it has a worker who already accumulated a certain level of firm-specific humancapital. This advantage associated with firm-specific human capital becomes smaller as the importanceof managerial capability increases, resulting in a lower survival rate of the first-period entrant. Thechange of firm-specific human capital in his model directly comes from firm’s training investment; whilein our model, it is determined by firms’ asset specificity choice and whether or not workers switch
employers.

8firm consists of the top management, assets, and workers who operate assets to produce
goods and services. Section 3 analyzes the model, characterizes the equilibrium andpresents comparative statics results. Section 4 discusses the implications of the model
results. Section 5 concludes this paper.

2

The model

Consider an industry with two ex-ante identical firms in a two-period setting. Only one
homogenous good is produced in this industry and the price is normalized to one. Labor
is the only input, and a firm’s output is a summation of its employees’ outputs. Thereexists a large number of individuals, where each individual is of measure zero. In eachperiod, labor supply is perfectly inelastic and fixed at one unit for each individual. To
keep the analysis simple, firms and individuals do not discount the future, and they arerisk-neutral.
At the beginning of period 1, each individual has the same general human capital andlooks identical to the firms. Each firm i simultaneously makes a first-period wage oﬀer
wi,1 to ñi

0 individuals. Let ni denote the number of firm i’s first-period employees.

Individuals not employed by a firm become self-employed and remain so until the endof period 2, earning ! > 0 per period.
Each firm’s production eﬃciency is determined by its managerial capability and thelevel of its asset specificity, where managerial capability is interpreted as representing
the ability of a firm’s top management to develop an eﬀective strategy and a uniquecompetitive position. Let ai denote the realization of firm i’s managerial capability,
which is a random draw from a symmetric binary distribution satisfying Pr(ai = aH ) =Pr(ai = aL ) = 12 , aH > aL , and aH =aL = ↵. Assume that ai is ex-ante unknown

to all agents including firm i itself and becomes common knowledge at the beginningof period 2 of firm i’s operation. This specification is consistent with the widely heldview that ability of a firm’s top management is mostly innate, and diﬃcult to observeor assess ex-ante.
Also at the beginning of period 1, each firm i can invest zi
specificity at a cost of c(zi )0 in its level of asset

0, where c(·) is a convex function. The asset specificity
9refers to how a firm tailors its physical asset for a particular use. To obtain closed-form
solution in the analysis, let c(z) = 12 ✓z 2 and ✓ > 0. Each firm chooses its level of asset
specificity at the same time as making its first-period wage oﬀer to individuals, andcommits to the investment level. Each firm’s asset specificity aﬀects each first-period
worker’s production eﬃciency only in period 2. Formally, if a worker stays with thesame employer i in both periods, his production eﬃciency associated with asset zi is0) and f (zi ) for period 1 and 2 respectively, where f (0) > d and f 0 (zi ) > 0 ford(

all zi0. If a worker moves from firm i to the other firm j (j 6= i) in period 2, his

second-period production eﬃciency associated with new firm’s asset is g(> 0). That is,as the level of a firm’s asset specificity increases, the improved production eﬃciency only
comes from the retained workers. If firm i’s asset is utilized by outside workers who are
not familiar with the tailor-made feature of i’s asset, there is no production eﬃciency
gain. Without loss of generality, we assume that d = 0. To obtain closed-form solution,
let f (zi ) = 1 + szi where s > 0, and g = 1. Here s captures the importance of a firm’s
asset specificity on its retained workers.
To produce the good, each worker also requires employer supervision. Assume thatfirm i’s per worker supervision cost is bni in each period, where b > 0. That is, as
the number of workers increases, each worker’s net output declines because each worker
receives less supervision from the employer.6 Each firm i’s period 1 profit isni xai

bn2i

1 2
✓z
2 i

Wi,1 ,

where x > 0 captures the importance of managerial capability and Wi,1 denotes firm i’s
first-period total wage bill.
At the beginning of period 2, each firm i simultaneously makes its second-period
wage oﬀer wi,2 to all workers, including workers employed at the other firm. Given thewage oﬀers, each worker takes the highest one, and stays with his current employer if
oﬀers from both firms are the same. A firm expands in its firm size if, in addition toits first-period employees, it hires more workers from the other firm. A firm can also
contract in its firm size if its first-period employees move to the other firm. A firm stays
6

See Zábojník and Bernhardt (2001) and DeVaro and Morita (2013) for a similar specification.

10

unchanged in its size only if it does not expand nor contract. Let mi (>the amount of firm i’s newly hired workers if mini ) denote

0 in the second period, or dismissed

workers if mi < 0. Each firm i’s period 2 profit is8><(ni + mi ) (xai ) + ni (1 + szi ) + mi>:(n + m ) (xa + 1 + sz )
i

i

i

i

b (ni + mi )2b (ni + mi )

2

Wi,2

Wi,2

if mi

0

,

if mi < 0where Wi,2 denotes firm i ’s second-period total wage bill. Firm i expands in the second
period if mi > 0; firm i contracts if mi < 0.
The timing of the game is as follows.
Period 1[Stage 1] Each firm simultaneously makes first-period wage oﬀers wi to ñi individuals,
where i = 1, 2. At the same time, each firm chooses its level of asset specificity zi byincurring a cost. Individuals choose between working for a firm and self-employment.
When indiﬀerent, an individual chooses to work for a firm. When indiﬀerent between
oﬀers from both firms, an individual chooses randomly between them. Self-employedworkers remain so for both periods, earning a wage ! each period.
[Stage 2] Each firm i that employed ni workers at Stage 1 produces ni units of thegood.
Period 2[Stage 3] Firm i’s managerial capability ai is realized and becomes common knowledge.
Each firm i then makes its second-period wage oﬀer wi,2 to all workers, including workers
employed at the other firm. Given the second-period wage oﬀers, each worker takes thehighest one. A firm’s size expands if it hires workers from the rival firm, while a firm’s
size contracts if its first-period employees move to its rival firm. Otherwise, a firm stays
unchanged in its firm size.
[Stage 4] Each firm i that employed ni + mi workers produces ni + mi units of the
good.

11

3

Analysis

In this section, we consider the symmetric Subgame Perfect Nash Equilibria (SPNE) inpure strategy, in which ni = ñi in the equilibrium. That is, each firm i chooses the samelevel of asset specificity and the same number of first-period workers, and employs all
individuals to whom it makes a first-period wage oﬀer in equilibrium.
We focus on equilibria in which a strictly positive number (measure) of workers
move from one firm to the other at the beginning of period 2 whenever the realizationsof managerial capability are diﬀerent in both firms, given that in reality expansion
and contraction of firm size are common in most industries. Proposition 1 identifies
necessary and suﬃcient conditions for such an equilibrium to exist, and characterizes
the equilibrium. Proposition 2 and 3 then present comparative statics results on the levelof asset specificity, the firm size and the expected labor turnover rate in the equilibrium.
Note, all proofs are in the Appendix.
Suppose that there exists a symmetric equilibrium in which a strictly positive numberof workers switch employers in period 2 when firms’ managerial capabilities are diﬀerent.
Consider a Stage 3 subgame where the managerial capability is realized. There are
four cases, namely (ai , aj ) = (aH , aL ), (ai , aj ) = (aL , aH ), (ai , aj ) = (aH , aH ), and(ai , aj ) = (aL , aL ). First consider the case where (ai , aj ) = (aH , aL ). If firm i expands
in its firm size, firm i is willing to oﬀer wi,2 = xaH + 12b(ni + mi ) where mi > 0,

which is equal to the expected productivity of firm j’s first-period employee at themargin; while firm j contracts in its firm size and is willing to oﬀer wj,2 = xaL + 1 +szj

2b(nj + mj ) where mj < 0. In the equilibrium, both firms oﬀer the same second-

period wages wi,2 = wj,2 . This condition, plus the condition that the number of workers
from expansion equals that from contraction in equilibrium, we obtain mi =1
4b [2↵x

szj

2b(ni

becomes 12 [2 + szj

mj =nj )] ⌘ mEi (ni , zj , nj ). Then, the equilibrium second-period wage

E (n , z , n ).
2b(ni + nj )] ⌘ wi,2i j
j

On the other hand, if firm i contracts and firm j expands, the equilibrium secondperiod wage and the size of expansion/contraction can be solved analogously; they are1
2 [2 + szi

2b(ni

1
nj )] and 4b
[ 2↵x

szj

2b(ni

nj )], respectively. Note that thislatter case where firm i contracts and firm j expands can be ruled out since122↵x szj +2b(ninj ) < 0 for any symmetric equilibrium—a contradiction. As for the former casewhere firm i expands and firm j contracts in the equilibrium, the following condition
must hold:
0<

1
[2↵x
4b

szj

2b(ni

nj )] < nj .

(1)

The right-hand side of condition (1) holds with strictly inequality; otherwise, firm j will
shut down. Hence, in the SPNE outcome where both firms have diﬀerent managerial
capabilities, the firm with high managerial capability expands and the other contracts
in their firm sizes. Firm i’s second-period profit conditional on ai = aH is given byE⇡iE (zi , ni , zj , nj ) ⌘ (ni + mEi (ni , zj , nj ))xaH + ni (1 + szi ) + mi (ni , zj , nj )2b(ni + mEi (ni , zj , nj ))(2)E(ni + mEi (ni , zj , nj ))wi,2 (ni , zj , nj )C (z , n , n ) and mC (z , n , n ) be
Next consider the case (ai , aj ) = (aL , aH ). Let wi,2
i i j
i i j
i

defined analogously given firm i contracts in its firm size. In the similar vein as theanalysis in previous case, we obtain that firm i’s second-period profit conditional onai = aL is given by7⇡iC (zi , ni , nj ) ⌘ (nib(nimC
i (zi , ni , nj ))(xaL + 1 + szi )2mCi (zi , ni , nj ))

(ni

(3)

C
mC
i (zi , ni , nj ))wi,2 (zi , ni , nj ).

Now consider a Stage 3 subgame where both firms have the same managerial capability, ai = aj . In the equilibrium, each firm i oﬀers the same second-period wage
xai + 1S (a , n ), and workers stay with their first-period employer so that
2bni ⌘ wi,2i ithere is no labor mobility. Then firm i’s second-period profit conditional on ai = aj isgiven by⇡iS (zi , ni ) ⌘ ni (xai + 1 + szi )bn2i

Sni wi,2(ai , ni ).

(4)Note that ⇡iS (zi , ni ) is independent of managerial capability when it is the same for7
The condition 0 < mC
i (zi , ni , nj ) < ni is the same as condition (1) in the equilibrium given that we
focus on symmetric equilibrium.

13

both firms (see Claim 1 in the Appendix for details). Hence, given that the equilibrium
is symmetric and strictly positive number of workers move across firms whenever both
firms have diﬀerent managerial capabilities, each firm i’s expected profit in period 2 is⇡i,2 (zi , ni , zj , nj ) ⌘ Pr(aH , aL )⇡iE (zi , ni , zj , nj )(5)

+ Pr(aL , aH )⇡iC (zi , ni , nj ) + Pr(ai = aj )⇡iS (zi , ni ),where the probability of each outcome (ai , aj ) is 14 .
At Stage 1, each firm i oﬀers the same first-period wage w1⇤ to ñi = ñ⇤ individuals
and employs n⇤ = ñ⇤ workers in the equilibrium, given the level of asset specificity andfirst-period number of workers are symmetric. Also, every individual who receives anoﬀer from firm i is indiﬀerent between taking and not taking the oﬀer, anticipating thatE (n , z , n ) if (a , a ) = (a , a ), w C (z , n , n )his/her second-period wage will be wi,2
i j
j
i j
H L
i,2 i i j
S (a , n ) if a = a . Since 2! is the lifetime wage for selfif (ai , aj ) = (aL , aH ), and wi,2i i
ijemployed individuals, w1⇤ + E[wi,2 (zi , ni , zj , nj )] = 2! holds for a worker employed byfirm i whereEE[wi,2 (zi , ni , zj , nj )] = Pr(aH , aL )wi,2(ni , zj , nj )

(6)

C
S
+ Pr(aL , aH )wi,2(zi , ni , nj ) + Pr(ai = aj )wi,2(ai , ni ).

⇤ (z , n , z , n ) where
Hence w1⇤ = wi,1
i i j
j

⇤
wi,1
(zi , ni , zj , nj ) = 2!

Since firm i’s period 1 profit is ni xai

E[wi,2 (zi , ni , zj , nj )].
1
2
2 ✓zi

bn2i

(7)

Wi,1 , firm i’s expected overall profit

is ⇧i (zi , ni , zj , nj ) in equilibrium, where⇧i (zi , ni , zj , nj ) ⌘ ni [ bni⇤wi,1(zi , ni , zj , nj )]1 2✓z + ⇡i,2 (zi , ni , zj , nj ).
2 i

(8)

In equilibrium, firm i chooses (zi , ni ) = (z ⇤ , n⇤ ) maximizes the value of ⇧i (zi , ni , z ⇤ , n⇤ ),given (zj , nj ) = (z ⇤ , n⇤ ). Then the necessary first-order conditions that define the equilibrium level of asset specificity and the equilibrium number of first-period workers are14

(@/@zi )⇧i (z ⇤ , n⇤ , z ⇤ , n⇤ ) = 0 and (@/@ni )⇧i (z ⇤ , n⇤ , z ⇤ , n⇤ ) = 0 for each firm i. Given
the symmetric level of asset specificity and the number of first-period workers in theequilibrium, we find (see Claim 1 in the Appendix for details) that

n⇤ =

z⇤ =

8s(1 2!) 2↵xs.
32b✓ 9s2(9)(12!)(32b✓ s2 ) 2↵xs2.
4b(32b✓ 9s2 )(10)In addition, the necessary (and suﬃcient) second-order condition for the maximization
problem of ⇧i (zi , ni , z ⇤ , n⇤ ) implies 32b✓

9s2 > 0 (see Claim 2 in the Appendix for

details).
Given that z ⇤0 and the denominator of z ⇤ is positive, the following condition

must hold in the equilibrium:↵
implying n⇤

8(1

2!)
,
2x

⇤ ⇤ ⇤
0 holds. Also, in equilibrium the condition (1)—0 < mE
i (n , z , n ) <1⇤ ⇤ ⇤n⇤ —must hold, where mEi (n , z , n ) = 4b (2↵xcondition on ↵:(11)

sz ⇤ ). It further implies the followings2 (1 2!)
(1 2!)(32b✓ + 7s2 )<↵<.
22x(4b✓ s )2x(32b✓ 7s2 )(12)That is, conditions (11) and (12) are necessary for the existence of a unique and symmetric equilibrium, in which positive number of workers switch employers whenever firms’22

s (1 2!)
2!)(32b✓+7s )managerial capabilities are diﬀerent. Note that 0 < 2x(4b✓< min{ 8(12x2!) , (1 2x(32b✓
}
s2 )
7s2 )

holds such that the set of ↵ that satisfying conditions (11) and (12) is non-empty (seeClaim 3 in the Appendix for details). That is, the necessary conditions (11) and (12)can be represented by an intermediate range of ↵. If the diﬀerence between high andlow capability is too small, labor mobility will not occur since retained workers productivity is still high for each firm. If the diﬀerence between high and low capability is toolarge, a firm underwent contraction will completely shut down given that the productivity of its workers is relatively high in its rival firm. Also note that in equilibrium
E/C

⇤ ⇤ ⇤
C ⇤ ⇤ ⇤
⇤
E
⇤ ⇤ ⇤
C
⇤ ⇤ ⇤
mE
i (n , z , n ) = mi (z , n , n ) ⌘ m and wi,2 (n , z , n ) = wi,2 (z , n , n ) ⌘ w2.

Then, Proposition 1 below tells us that this condition is not only necessary but also15

suﬃcient for the existence of a unique symmetric equilibrium.
Proposition 1. There exists a unique symmetric equilibrium in which a strictly positive number of workers move from one firm to the other whenever the realizations of both
firms’ managerial capabilities are diﬀerent in period 2, if and only if ↵ 2 (↵, ↵¯ ] where 0 <↵<↵¯ . The equilibrium is characterized by each firm i’s level of asset specificity and num⌘⇣2!) 2↵xs (1 2!)(32b✓ s2 ) 2↵xs2
ber of first-period workers (zi , ni ) = (z ⇤ , n⇤ ) = 8s(132b✓
,
.
2
2
9s
4b(32b✓ 9s )In the equilibrium, each firm chooses the level of asset specificity z ⇤ , makes firstperiod wage oﬀer w1⇤ to n⇤ individuals, and employs n⇤ workers at Stage 1. At Stage 3
if both firms have diﬀerent managerial capabilities, firm i whose managerial capability
ai turns out to be higher retains all its first-period employees and expands in its firm
E/Csize by hiring m⇤ workers from firm j (j 6= i) at the second-period wage w2
firm j contracts in its firm size and retains n⇤
E/Cwage w2

whereasm⇤ workers at the same second-period. Hence if ai 6= aj , there exists a strictly positive number of workers, m⇤ ,moving from the low-capability firm to the high-capability firm in the equilibrium. At
Stage 3 if ai = aj , each firm i retains all its first-period workers at the second-period
wage wS (ai , n⇤ ) and stays unchanged in its firm size (neither expansion nor contraction);thus there is no labor mobility. Then, the expected number of workers who switch theiremployers at the beginning of period 2 is 14 (2m⇤ ), and each firm i’s expected laborturnover rate is 14

2m⇤n⇤

in the equilibrium.

We will now turn to comparative statics on the equilibrium level of the asset specificity z ⇤ and the firm size n⇤ (the period 1 number of workers), and the expected equilibrium turnover rate 142m⇤n⇤. Note that the period 1 number of workers n⇤ can beinterpreted as average firm size measured by employment since n⇤ determines each firm’s
expected number of workers over two periods.
Proposition 2. As the importance of managerial capability (captured by x) increases,
the level of asset specificity decreases, the average firm size decreases, and the expectedlabor turnover rate increases in the equilibrium.
The key result here is that, as the importance of managerial capability increases,
the expected number of workers who switch their employers (expected labor mobility)16

in period 2 becomes larger, and each firm’s asset specificity and average employment
size decreases. The logic behind this result can be explained as follows. When firm ichooses the level of its asset specificity and the number of workers it employs in period
1, it estimates the expected number of its first-period employees that the firm will retain
for second-period operation, and the expected number of workers that the firm hires
from its rival firm j in period 2. A higher importance of managerial capability increases
the diﬀerence of period 2 productivity between a high-capability and a low-capability
firm. Then, as the importance of managerial capability increases, each firm anticipatesa larger number of workers switch their employers in period 2 if both firms’ managerial
capabilities turn out to be diﬀerent. Hence a higher importance of managerial capability
decreases the expected number of retained workers, and it increases the expected number of workers who switch employers. Holding the initial number of workers constant,this reduces each firm’s incentive to choose higher specificity of its asset, because theproportion of retained workers decreases and the productivity gains over the investment
cost for each retained worker becomes smaller.
Meanwhile, anticipating that the number of retained workers decreases while thesupervision cost per worker in period 1 remains the same, each firm has a lower incentive to hire more workers in period 1. Because a larger number of period 1 workers
increases a firm’s period 2 profit from its retained workers at the expense of increasing
supervision cost per worker in period 1. Also, the reduction of each firm’s first-period
workers decreases the proportion of retained worker in period 2, and in turn decreases
the marginal benefit of choosing a higher asset specificity. Hence, the force of lowering
average firm size further reduces each firm’s level of asset specificity. Taken together, theresult is that, the equilibrium level of asset specificity and the average firm size decreaseas the importance of managerial capability increases. The last result of Proposition 2
naturally follows from the key result mentioned above. As the importance of managerial
capability increases, the expected labor turnover rate, measured by the ratio of expected
number of workers who switch their employers in period 2 to the number of first-period
workers, increases unambiguously in the equilibrium.
Proposition 3. As the importance of a firm’s asset specificity (captured by s) increases,
17

the level of asset specificity increases, the average firm size increases, and the expectedlabor turnover rate decreases in the equilibrium.
Recall that in period 1 firm i estimates the number of workers it will retain andhire from the rival firm in period 2, respectively. As the importance of firm i’s asset
specificity increases, firm i will retain more of its first-period employees during thecontraction phase, while hire fewer workers from its rival during the expansion phasesince new workers become relatively less productive in firm i. This implies that each firm
anticipates a smaller number of workers switch their employers in period 2 if managerial
capabilities are diﬀerent. Lower expected labor mobility, along with the higher returnfrom retained workers (captured by s), implies that a firm has higher incentive to choosea higher level of asset specificity. These two eﬀects are mutually reinforcing because
the higher level of asset specificity reduces the expected turnover rate by increasing
the retained workers’ productivity. The results of the average firm size and expected
turnover rate follow through the logic analogous to the one explained for Proposition 2.

4

Discussion

4.1

An application to cross-industry diﬀerences

The first application concerns the idea that the importance of managerial capability candiﬀer across industries, as discussed in Morita (2012).8 For instance, in a newly emerging
industry or in a business undergoing revolutionary technological changes, a business’s
success critically depends on the quality of its strategic decision making because theseindustries face a high level of uncertainty about the needs of customers, the products
and services that will prove to be the most desired, and the best configuration of activities and technologies to deliver them. Whereas in industries facing lower levels of
uncertainty, strategic decision making is less important. These arguments suggest thatthe importance of managerial capability is higher in the newly emerging industries withhigher level of uncertainty or in the new business undergoing revolutionary technological8
See also empirical studies by Ely (1991) and Hogan and Sigler (1998). They find that the sensitivityof CEO’s compensation to firm performance diﬀers significantly across industries. Assuming that thesensitivity captures the importance of managerial capability for firm performance, these findings suggestthat the importance of managerial capability diﬀers across industries.

18

changes, while the importance tends to be lower in mature industries or business withlower level of uncertainty and less technological advancement. Our model then predicts
that labor mobility is higher, specificity of asset and human capital is lower, and averagefirm size is smaller in industries of the former type and vice-versa in industries of thelatter type.9
These predictions are consistent with empirical and observational evidence. Concerning firm size, it has been documented that firm size tends to be smaller in theemerging industries compared with mature industries (Kohn, 1997). Also, Dinlersoz
and MacDonald (2009) show that the average firm size (measured by the employment)in U.S. manufacturing industries from 1963 until 1997 declines during the rapid entry
phase of the industry life-cycle, where the rapid entry phase occurs when there exists
greater technological improvement. Concerning labor mobility, Benner (2002) studies
labor markets in Silicon Valley and points out that, “The rapid turnover and volatility inemployment in Silicon Valley is integrally connected to the nature of competition in theregion’s high-technology industries. In these industries, markets and technology change
extremely rapidly and in unpredictable ways.”

4.2

An application to U.S.-Japanese diﬀerences

The U.S. and Japan have been considered representing two contrasting employment
systems, which attracts significant attention in the literature of international comparison
in how internal labor markets operate. By capturing the interconnections among asset
specificity, acquisition of firm-specific human capital, managerial capability, and labormobility, our model oﬀers new explanations for and predictions on the U.S.-Japanese
diﬀerences based on the cross-country diﬀerences in the importance of both managerial
capability and firm-specificity of asset.
Acemoglu, Aghion and Zilibotti (2006) argue in their analysis of technology frontiersand firm selection that managerial skill is more important for undertaking innovative activities than for adopting and imitating existing technologies from the world technology
frontier. They then point out, based on their analysis of the correlation between distance9

In an alternative approach, Rossi-Hansberg, Sarte and Owens III (2009) also show that the averagefirm size decreases due to firm fragmentation in the evolution of urban structure.

19

to the frontier and R&D intensity using data from the OECD sectoral database, that
innovation becomes more important as the economy approaches the world technology
frontier and there remains less room for adoption and imitation. Following their analysis
and argument, we argue that the importance of managerial capability was substantiallylower in Japan than in the U.S. when most Japanese industries were catching up with
the West in the postwar growth period (Okimoto, 1989).
On the other hand, as the economy makes the transition from industrial capitalism
to post-industrial capitalism, modern economies are becoming increasingly knowledge
intensive which renders the disadvantage to the firms that heavily rely on physical assets.
For example, Iwai (2002) points out that, in the new era of post-industrial capitalism, thephysical assets have surrendered their central position to the knowledge-based humanassets that money can no longer buy and control.10 The advent of knowledge economy
thus reduces the importance of firm-specificity of asset across the world. But diﬀerent
countries have diﬀerent paths in making their transition to the knowledge economy.
Based on the knowledge assessment methodology (KAM) developed by World Bank
Institute that traces the challenges and opportunities of major economies’ transition toknowledge-based economy from 1995, the U.S. has a better shape than Japan in theirdevelopment of knowledge economy (Shibata, 2006). Following this reasoning, we argue
that the importance of firm-specificity of asset is higher in Japan than in the U.S.
Combining the above argument where Japan has lower importance of managerial
capability and higher importance of firm-specificity of asset compared with U.S., thepredictions of our model point out to the same direction. That is, labor mobility was
lower, and specificity of asset and human capital was higher in Japan than in the U.S.,
especially in the catching up period.11
These predictions are consistent with empirical and observational evidence. Concern10
Also, Drucker (1993) for example, has argued that in the new economy, knowledge is not justanother resource alongside the traditional factors of production—labor, capital, and land—but the only
meaningful resource today.
11
In an alternative approach where promotion serves as a signal of the worker’s ability, Owan (2004)shows that late-promotion practice which commonly observed in Japanese firms results in low turnoverrate and long-term employment relationship; whereas early-promotion practice which is commonly observed in U.S. results in the opposite eﬀects. Moreover, Chang and Wang (1995) explain the lower
turnover rate and higher human capital accumulation in Japan than in the U.S. Their model is characterized by multiple equilibria and is analyzed in the framework where current employers have superior
information on the abilities of their employees.

20ing labor mobility, it was found that the labor turnover rate was much higher in U.S.
than in Japan (see Hashimoto and Raisian, 1985; Mincer and Higuchi, 1988). Hashimotoand Raisian (1985), for instance, found that the 15-year job retention rates of the malepopulation between the early 1960s and the late 1970s were much higher in Japan than in
the United States across all age groups. Concerning firm-specific human capital, Koike(1977, 1988) found, in his comparative analysis of Japanese and U.S. industrial relations, that Japanese workers acquired more firm-specific human capital through rotation
among related jobs (see also Dertouzos, Lester and Solow, 1989; Ito, 1992). Concerning
specificity of asset, we are unaware of direct observations or evidence. However, it iswell known that Japanese firms conducted continuous process improvement more than
U.S. firms did in the postwar growth period.12 As argued by Morita (2001), if a firm
conducts continuous process improvement, the technology is improved but a degree of
specificity is introduced. This is because, in general, continuous process improvement
involves a number of small changes and modifications, which lead to highly firm-specifictechnologies. This then suggests that specificity of asset was higher in Japanese firms
than in the U.S. firms in the postwar growth period.
The Japanese economy has already caught up with the West, and most Japaneseindustries have got much closer to the world technology frontier. This increases theimportance of managerial capability which is crucial for undertaking innovative activities. Our model then predicts that the degree of the U.S.-Japanese diﬀerences become
smaller. That is, labor mobility had increased, and specificity of asset and human capital has decreased in Japan. In reality, however, such transitions in Japan are likely
to take place rather slowly due to vested interests and institutional inertia embeddedin the Japanese economic system (Lincoln, 2004). Several empirical studies have been
undertaken recently to find out whether or not the Japanese employment system has
exhibited significant changes, but findings are not clear-cut thus far.13
12

See Morita (2001) for a review of evidence.
For example, Kambayashi and Kato (2011) study labor mobility and conduct cross-national analysisof micro data from Japan Employment Status Survey and U.S. Current Population Survey. They findthat, on the one hand, core employees (age of 30-44 with at least five years of tenure) in Japan continued
to enjoy much higher job stability than the U.S. counterparts consistently over the last twenty-five years.
On the other hand, job stability for mid-career hires and youth employees deteriorated in Japan over thelast twenty-five years, whereas there was no comparable decline in job stability in the U.S. counterparts.
13

21

5

Concluding remarks

This paper has explored a new model that in an environment of labor market competition, each firm consists of the top management, assets, and workers who operate assets to
produce goods and services. Elaborating on the connection between labor mobility andthe levels of firms’ asset specificity and their employment sizes, we have demonstrated
that as the importance of managerial capability increases or the importance of a firm’s
asset specificity decreases, the the level of asset specificity decreases, the average firm
size decreases, and the expected labor turnover rate increases in the equilibrium. Thishas yielded empirical implications and predictions from a previously unexplored perspective, given that the importances of both managerial capability and firm-specificity
of asset can diﬀer across industries and countries.
The current model can be enriched or extended to address several interesting questions. First, firms may want to collude in the labor market if it is profitable to reducelabor mobility and to increase asset specificity. Then there will be policy implication
for government’s restriction on firms’ employment practices (e.g. no-poach agreement).
Second, we can further assume that firm i’s choice of asset specificity aﬀects both its
first-period employees’s productivity in firm j and the productivity of firm j’s firstperiod worker in firm i. Specifically, if a worker moves from firm i to the other firm j(j 6= i) in period 2, his second-period production eﬃciency associated with new firm’s

asset is decreasing in both his first-period employer i’s and new employer j’s asset specificities. This reflects the idea that if firm i’s asset is utilized by outside workers who
are not familiar with the tailor-made feature of i’s asset, the output becomes lower.
The other is that if a worker already gains familiarity in utilizing firm i’s asset, his/her
productivity in firm j (j 6= i), holding j’s asset specificity level constant, becomes lower,
perhaps arising because a worker’s prior experience from a diﬀerent asset utilization caninterfere with his/her learning eﬃciency in the current one. There may exist interesting
implications from this distinction between the diﬀerent sources of reduction in workers’
human capital acquisition. Third, it will be interesting to introduce downward-sloping
demand schedule such that entry of firms is free. Then, there will be implications onfirm dynamics and government’s policy on firm’s entry. We plan to fully explore these22extension/enrichment in the future works.

23

Appendix
Proof of Proposition 1:
Suppose that there exists a symmetric equilibrium characterized by (zi , ni ) = (z ⇤ , n⇤ )for each i, in which a strictly positive number of workers switch employers in period 2
whenever the realizations of managerial capability are diﬀerent in both firms. In the text,
it has been shown that conditions (11) and (12) are necessary for such an equilibrium
to exist. Here, in Claim 1–3 below, we present proof of some computational details thathave not been presented in the text.
Claim 1: In a symmetric equilibrium where a strictly positive number of workers switch
employers in period 2 whenever the realizations of managerial capability are diﬀerent in2

22!) 2↵xss ) 2↵xs
both firms, each firm i chooses zi = 8s(132b✓⌘ z ⇤ and ni = (1 2!)(32b✓⌘9s2
4b(32b✓ 9s2 )n⇤ .

Proof of Claim 1: Suppose that there exists a symmetric equilibrium in which a strictly
positive number of workers switch employers in period 2 when firms’ managerial capabilities are diﬀerent. First note that in the Stage 3 subgame, firm i’s second-period profit
conditional on ai = aH and ai 6= aj is ⇡iE (zi , ni , zj , nj ) = (ni + mEi (zi , ni , zj , nj ))(xaH +E
2 w E (z , n , z , n )(n +1+szi )+ni (1+szi )+mEjii (zi , ni , zj , nj ) b(ni +mi (zi , ni , zj , nj ))i,2 i i jmE
i (zi , ni , zj , nj )) as analyzed in the text, implying that⇡iE (zi , ni , zj , nj ) =1{(2↵x szj )2 + 4b2 (ni + nj )216b+4b [2↵x(ni + nj ) + 4sni zi s(ni + nj )zj ]},1given mEi (ni , zj , nj ) = 4b [2↵x

szj

2b(ni

(13)

E (n , z , n ) = 1 [2 + sznj )] and wi,2
i j
j
j
2

2b(ni + nj )].
Consider the Stage 3 subgame where (ai , aj ) = (aL , aH ). If firm i contracts and firm
j expands in their firm sizes, firm i will oﬀer wi,2 = xaL + (1 + szi )2b(ni + mi ) wheremi < 0 and firm j will oﬀer wj,2 = xaH +1 2b(nj +mj ) where mj > 0, that equal to the
expected productivity of each firm’s first-period employee at the margin, respectively.
In the equilibrium, both firms oﬀer the same second-period wages wi,2 = wj,2 , and the24number of workers from contraction equals that from expansion,
1mC
i (zi , ni , nj ) = 4b [2↵xszini )]. Then, the equilibrium second-period wage

2b(njC (z , n , n ) = 1 [2 + sz
becomes wi,2
i i j
i
2

2b(ni + nj )]. If firm i expands and firm j contracts

1
in the equilibrium, the expansion/contraction size is 4b[ 2↵xequilibrium can be ruled out given

mi = mj . We obtain2↵xszi + 2b(njszi + 2b(njni )]. Thisni ) < 0 for any symmetricequilibrium—a contradiction. Also, 0 < mC
i (zi , ni , nj )  ni holds in the equilibrium,1
which suggests 0 < 4b
[2↵x

szi

ni )] < ni . This condition is the same as2b(njthe condition (1) in the equilibrium. Then, firm i’s second-period profit conditional onai = aL is ⇡iC (zi , ni , nj ) = (ni(nimCi (zi , ni , nj ))(xaL + 1 + szi )

b(ni

2
mC
i (zi , ni , nj ))CmCi (zi , ni , nj ))wi,2 (zi , ni , nj ), implying that⇡iC (zi , ni , nj ) =

1
[2↵x
16b

szi

2b(ni + nj )]2 .

(14)

Next consider the the Stage 3 subgame where both firms have the same managerial
capability, ai = aj . First note that it cannot be the case that one firm expands and theother contracts in this case given that either

szj +2b(ni nj ) ⇧ 0 or

szi +2b(nj ni ) ⇧0 (the managerial capability gap is now 0) under symmetric equilibrium. Then, both
firms must stay unchanged in their firm sizes in the equilibrium. That is, each firm i will
oﬀer the wage to the extent that the expected productivity from the outside workers are
the same at the level xai + 12b(ni + mi ) where mi = 0 holds in the equilibrium. That

S = w S (a , n ) = xa + 1
is, wi,2i
i,2 i i2bni . Then firm i’s second-period profit conditional onai = aj is ⇡iS (zi , ni ) = ni (xai + 1 + szi )bn2i

S (a , n ), implying thatni wi,2i i⇡iS (zi , ni ) = ni [szi + bni ].

(15)

Consequently, in the equilibrium each firm i’s expected profit in period 2 is ⇡i,2 (zi , ni , zj , nj ) =
1 E
1 C
1 S
1
4 ⇡i (zi , ni , zj , nj ) + 4 ⇡i (zi , ni , nj ) + 2 ⇡i (zi , ni ), where Pr(aH , aL ) = Pr(aL , aH ) = 4

and Pr(ai = aj ) = 12 . Also note that w1⇤ + E[wi,2 (zi , ni , zj , nj )] = 2! holds for a worker
employed by firm i in the equilibrium, where1
E[wi,2 (zi , ni , zj , nj )] = 1 + [s(zi + zj )825

4b(3ni + nj )],(16)

E (z , n , z , n ), w C (z , n , z , n ), and w S (a , z , n , z ). Then,the weighted average of wi,2
i i j
j
j
i,2 i i j
i,2 i i i j⇤ (z , n , z , n ) where
we find w1⇤ = wi,1
i i j
j

⇤
wi,1
(zi , ni , zj , nj ) = 2!

1

1
[s(zi + zj )8(17)

4b(3ni + nj )].

Hence, each firm i’s expected overall profit is ⇧i (zi , ni , zj , nj ) = ni xai

bn2i

1
2
2 ✓zi

Wi,1 + ⇡2 (zi , ni , zj , nj ), implying that⇧i (zi , ni , zj , nj ) = ni [ bniw1⇤ (zi , ni , zj , nj )]

1 2
✓z
2 i

(18)

1
1
1
+ ⇡iE (zi , ni , zj , nj ) + ⇡iC (zi , ni , nj ) + ⇡iS (zi , ni ),442
where ⇡iE (zi , ni , zj , nj ), ⇡iC (zi , ni , nj ), and ⇡iS (zi , ni ) are defined in the equations (13),(14), and (15) respectively. Let compute the partial derivatives of ⇧(zi , ni , zj , nj ). We
find1[ 2↵xs + s2 zi + 2b(15sni + snj 16✓zi )],32b1(@/@ni )⇧i (zi , ni , zj , nj ) = 1 2! + [15szi + szj 4b(15ni + nj )].
16(@/@zi )⇧i (zi , ni , zj , nj ) =Given that (zi , ni ) = (zj , nj ) = (z ⇤ , n⇤ ) in the equilibrium, the first-order conditions
imply

8
>
<

>
:1

1
32b

s(sz ⇤

2↵x) + (sn⇤

2! + sz ⇤4bn⇤ = 0✓z ⇤ ) = 0

.

(19)

2

2!) 2↵xss ) 2↵xs
Solving the simultaneous equations, we obtain z ⇤ = 8s(132b✓
and n⇤ = (1 2!)(32b✓9s2
4b(32b✓ 9s2 )in the equilibrium. Then, if such a symmetric equilibrium exists, it must be unique. ⇤Claim 2: There exists a unique symmetric equilibrium in which workers switch employers whenever firms managerial capabilities are diﬀerent, if and only if conditions (11)and (12) presented in the text hold.
Proof of Claim 2: Given Claim 1, if firm i’s expected overall profit at Stage 1 is⇧i (zi , ni , zj , nj ), it must be the case that in the subsequent subgame, firm i’s optimal de-26

2cision in period 2 is to expand in its firm size if (ai , aj ) = (aH , aL ), to contract in its firm
size if (ai , aj ) = (aL , aH ), and to stay unchanged in its firm size if ai = aj . That is, there
exists a symmetric equilibrium with positive labor mobility whenever firms’ managerial
capabilities are diﬀerent, if and only if each firm i’s expected overall profit at Stage 1
is ⇧i (zi , ni , zj , nj ). Then, given firm j (j 6= i) chooses (zj , nj ) = (z ⇤ , n⇤ ), let us checkthe second-order condition of firm i’s maximization problem of ⇧i (zi , ni , z ⇤ , n⇤ ). We
have (@ 2 /@zi2 )⇧i (zi , ni , z ⇤ , n⇤ ) = 16b1 (16b✓s2 ), (@ 2 /@n2i )⇧i (zi , ni , z ⇤ , n⇤ ) =7
2 b, and(@ 2 /@zi @ni )⇧i (zi , ni , z ⇤ , n⇤ ) = (@ 2 /@ni @zi )⇧i (zi , ni , z ⇤ , n⇤ ) = 78 s. Hence, the first-orderleading principal minor of D2 ⇧i (zi , ni , z ⇤ , n⇤ ) is negative and the second-order leading
7
principal minor of D2 ⇧i (zi , ni , z ⇤ , n⇤ ) is 64(32b✓

is negative definite over all zithat ↵  8(12x2!) , 32b✓0 and ni

9s2 > 0 given zi

9s2 ). That is , D2 ⇧i (zi , ni , z ⇤ , n⇤ )0 if and only if 32b✓

9s2 > 0. Note

0. Also note that the positive expan-

s2 (1 2!)
2!)(32b✓+7s2 )sion/contraction condition, 2x(4b✓< ↵ < (1 2x(32b✓, is also necessary and suﬃs2 )7s2 )cient for the he existence of such symmetric equilibrium. Hence, the necessary and suf2

2

s (1 2!)
2!)(32b✓+7s )ficient conditions are represented by ↵  8(12x2!) and 2x(4b✓< ↵ < (1 2x(32b✓
.
s2 )
7s2 )

⇤

This implies the result.

Claim 3: There exists a range of parametrization such that conditions (11) and (12)presented in the text hold.
22

s (1 2!)
2!)(32b✓+7s )Proof of Claim 3: We want to show that 0 < 2x(4b✓< (1 2x(32b✓< 8(12x2!) .
s2 )
7s2 )
2

2!)(32b✓+7s )First note that 8(12x2!) > (1 2x(32b✓, 32b✓(7s) > 9s2 (7s), which holds given
7s2 )
2

2

2!)(32b✓+7s )s (1 2!)32b✓ > 9s2 . Also, (1 2x(32b✓> 2x(4b✓, 128(b✓)2 4s2 b✓ 7s4 > 32b✓s2 7s4 ,
7s2 )
s2 )
2

s (1 2!)32b✓ > 9s2 , which holds given 32b✓ > 9s2 . Further note that 2x(4b✓> 0 given
s2 )

32b✓

8s2 > 0 and 1⇤

2! > 0. This implies the result.
222!)(32b✓+7s )s (1 2!)Let ↵¯ = (1 2x(32b✓and ↵ = 2x(4b✓. Then all the necessary and suﬃcient
7s2 )
s2 )

⇤

conditions boil down to ↵ 2 (↵, ↵¯ ]. This completes the proof.
Proof of Proposition 2 and 3:⇤

2!) 2↵xs2↵s(i) Note z ⇤ = 8s(132b✓, we find @z@x = 32b✓ 9s2 < 0. Also,
9s2[8(1 2!) 2↵x](32b✓ 9s2 )+[8s(1 2!) 2↵xs](18s)@z ⇤> 0 given 8s(1@s1 =(32b✓ 9s2 )2

27

2!)

2↵xs > 0.

2

2

⇤

⇤

s ) 2↵xs2↵s@n(ii) Note n⇤ = (1 2!)(32b✓, we find @n@x = 4b(32b✓ 9s2 ) < 0. Also, @s =4b(32b✓ 9s2 )32s[4(1 2!) ↵]> 0 given 8s(1(32b✓ 9s2 )21(iii) Note m⇤ = 4b(2↵x⇤given @n@x⇤
< 0 and @m@x⇤

Also, we find @m
@s =
⇤

2!)

2↵xs > 0.
⇤1
sz ⇤ ), we find @m@x = 4b (2↵

@z ⇤
@z ⇤
@x ) > 0 given @x < 0.

⇤
1
@m⇤
⇤
> 0, we find @( 14 2mn⇤ )/@x = (2n⇤ )2 [ @x (2n )1 @z ⇤@z ⇤4b @s < 0 given @s > 0.
⇤

1
@m
⇤
find @( 14 2mn⇤ )/@s1 = (2n⇤ )2 [ @s (2n )

⇤

⇤

@m
Then given @n@s > 0 and @s < 0, wem⇤ ( @n@s )] < 0.

28

⇤

Then

⇤
m⇤ ( @n@x )] > 0.

⇤

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