OSIPP Discussion Paper : DP-2003-E-010

Entrepreneurial Efficiency
October 1, 2003
Katsuya Takii
Associate Professor, Osaka School of International Public Policy (OSIPP)【キーワード】E13,O40.
【要約】This paper examines a particular aspect of entrepreneurship, namely
firms' ability to respond appropriately to unexpected changes in theenvironment (i.e., their adaptability). An increase in firms' adaptability
improves allocative efficiency in a competitive economy, but can reduce it
when opportunities are distorted. It is shown that adaptability canaggravate distortions in the presence of political risk. Because efficiency
affects the total factor productivity (TFP) of an economy, the model can
explain how entrepreneurship influences TFP. The quantitative effect offirms' adaptability on TFP is investigated using the Census of
Manufacturing in Japan.

I would like to thank Carmen Li, Boyan Jovanovic and seminar participantsat the University of Essex, the Kansai Macro Workshop, Osaka Prefecture
University and OSIPP for helpful comments. I also thank Kenjiro Hirata
for his excellent assistance. Osaka School of International Public Policy,

Osaka University, 1-31, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan.
Tel: 81-6-685-5631, E-mail: takii@osipp.osaka-u.ac.jp1

1

Introduction

It is widely known that entrepreneurship plays an essential role in several economic
activities. When new technology is discovered, demographic change occurs or a government changes its policy, it is entrepreneurs who react to the changes and reallocate
resources to seek opportunities. However, there have been only a few attempts todevelop a formal model of entrepreneurship in macroeconomics. The absence of entrepreneurship from macroeconomics can be explained in part by the extensive useof an exogenously given aggregate production function.

As the relation between

output and inputs is given by the aggregate production function, there is no need foran economic agent who Þnds a productive use for inputs.
This paper develops a tractable macroeconomic model that enables a theoreticaland empirical examination of a particular aspect of entrepreneurship, namely theability to react appropriately to unexpected changes in the environment (i.e., adaptability).

In this model, the aggregate production function is derived from Þrms’reactions to these changes.

Therefore, we can analyze how Þrms’ adaptability af-

fects the macroeconomy, while retaining the advantages of the aggregate production
function.
We assume that Þrms’ adaptability is determined by entrepreneurs’ ability torecognize changes in the environment. When an unexpected change in productivity
occurs, the marginal products of inputs deviate from input prices, and this generates
opportunities for entrepreneurs to exploit. If entrepreneurs recognize the change andreact to it, the deviation of the marginal products of the inputs from input prices
would be small.

Hence, the improvement in Þrms’ adaptability raises allocativeeﬃciency, and therefore increases productivity in the economy.

It is shown thatthe increased adaptability of Þrms raises the total factor productivity (TFP) of theaggregate production function in a competitive economy.
2This result does not apply if opportunities are distorted. As the social marginal
products of resources diﬀer from the private marginal products, even if a talented entrepreneur can equate private marginal products to input prices, this may not improve
allocative eﬃciency. This intuition is analyzed formally in the context of entrepreneurs seeking rent. Because the results of political negotiations are uncertain, entrepreneurs must respond to changes in the political environment. However, because
rent-seeking activities simply redistribute incomes, adapting to political changes generates negative externalities: when an entrepreneur is good at taking opportunities,
there are fewer opportunities for others. It is shown that increased Þrms’ adaptability
can lower TFP.
The eﬀect of Þrms’ adaptability on TFP is investigated empirically. This paper
shows that the adaptability of Þrms can be estimated by the squared correlation between an unexpected shock and Þrms’ reactions to the shock. Using data from the
Census of Manufacturing in Japan, 1985—1999, we estimate Þrms’ adaptability andexamine the eﬀect of adaptability on TFP by prefecture and industry. Our model
predicts that, if political shocks are negligible, full recognition of previously unpredicted changes increases TFP by at least 14% in Japan. However, such recognition
lowers TFP by the same amount if all changes are due to political shocks.
A similar view of entrepreneurs is emphasized by Kirzner (1973) and Schultz(1975). Kirzner (1973) emphasizes the essential role of entrepreneurial discovery inmarket processes. Schultz (1975) deÞnes entrepreneurial ability as the ability to interpret new information and allocate resources to proÞtable opportunities. Althoughboth researchers insist that equilibrium analysis is not suitable for understanding theimportance of entrepreneurship, we suggest that there are beneÞts of using an equilibrium model. Hence, we represent similar aspects of entrepreneurship in a generalequilibrium model, and it is hoped that this approach complements theirs.

3This strategy has been adopted by Holmes and Schmitz (1990).

Holmes andSchmitz (1990) incorporate a Schultzian entrepreneur into an equilibrium analysis,
and emphasize the importance of the division of labor between entrepreneurs andmanagers.1 This paper diﬀers from theirs in three respects: (1) it assigns entrepreneurship a diﬀerent social role, namely that of inßuencing allocative eﬃciency; (2)it incorporates this role into the neoclassical growth model; and (3) it examines theeﬀect of entrepreneurship on the macroeconomy quantitatively.
This paper also contributes to a controversy in the literature on pressure groups.
Becker (1983) shows that competition between pressure groups can lead to eﬃcient
allocation.

On the other hand, Tullock (1967) argues that rent-seeking activities

waste resources. The detrimental eﬀect of entrepreneurs’ rent-seeking activities has
recently been re-emphasized in the context of economic growth [e.g., by Baumol(1990), Murphy et al. (1991) and Holmes and Schmitz (2001)]. In contrast to theprevious literature, this paper shows that, even if rent-seeking activities are costless,competition between pressure groups may not lead to eﬃcient allocation in the presence of political risk. Hence, the social welfare loss due to rent-seeking activities may
have been underestimated by the existing literature.
Evidence obtained in this paper is related to that obtained from plant-level data.
Much evidence from plant-level data suggests that the reallocation of resources towards more productive uses is an important component of productivity growth [seereviews by, e.g., Davis and Haltiwanger (1999) and Bartelsman and Doms (2000)].
This paper argues that the reallocation of resources might be aﬀected by Þrms’ adaptability.
This paper is organized as follows. The next section describes the model. Sec1

The diﬀerent roles of entrepreneurs are incorporated into equilibrium analysis by Kihlstrom andLaﬀont (1979) with respect to risk bearing, by Schmitz (1989) with respect to imitation, and by
Aghion and Howitt (1992) with respect to innovation.

4

tion 3 deÞnes the measure of adaptability and analyzes the macroeconomic eﬀect of
Þrms’ adaptability.

Section 4 extends the model to analyze the eﬀect of politicaluncertainty. Section 5 shows how the theory is implemented empirically. Section 6
reports the empirical results. Section 7 concludes by summarizing the main results
and discussing possible extensions.

2

The Model

We present a simple general equilibrium model, which is based on that of Lucas(1978). It lays the foundations for analyzing the macroeconomic eﬀect of entrepreneurship in the next section.
An agent can be an entrepreneur or a worker. Each Þrm needs one entrepreneur
to organize the Þrm. In this model, for simplicity, there is no distinction between
entrepreneurs, managers and Þrms. This simpliÞcation is made to develop a tractablemodel that focuses on Þrms’ adaptability.
Firms are continuously distributed on [0, mN ], where m ∈ (0, 1) is the proportionof entrepreneurs in the total population, N. Although the variables m and N canchange over time, time subscripts are omitted throughout the paper unless they arenecessary for clarity.

The representative entrepreneur’s problem is described Þrst,and then the market equilibrium is deÞned.

The entrepreneur’s problem: An entrepreneur establishes a Þrm, employs capital stock and workers, and produces output. The entrepreneur faces the following
production function:
Yi = zi A [F (Ki , T Li )]α , 0 < α < 1
where zi is a Þrm-speciÞc productivity shock for the ith Þrm, and Yi , Ki and Li5

are the amounts of the ith Þrm’s output, capital stock and labor input, respectively.
T measures the eﬀectiveness of labor, which is increased by general factors such as
the level of educational attainment and economy-wide technological progress.
is assumed that F exhibits constant returns to scale in K and L.

It

By deÞning

f (k) = F (k, 1), where k = TKL , we can express F (K, T L) as a function of capitalper unit of eﬀective labor in production: F (K, T L) = f (k) T L.

We assume thatf 0 (.) > 0, f 00 (.) < 0, limk→0 f (k) = 0, limk→0 f 0 (k) = ∞ and limk→∞ f 0 (k) = 0.
The parameters α and A measure the span of control and the productivity of management, respectively.

Because α ∈ (0, 1), the production function has decreasing

returns to scale in Ki and Li .
It is assumed that the productivity of management is a function of the eﬀectiveness of the entrepreneur. Because agents are homogeneous, the eﬀectiveness of theentrepreneur is the same as that of workers, which is given by T .

Assuming that

A = T 1−α , the production function can be written as
Yi = zi [f (ki ) Li ]α T.
There are three advantages to assuming that A = T 1−α . First, this assumption implies that the production function has constant returns to scale in capital stock, laborand managerial input.

Hence, it can be shown that the Þrm’s proÞt is equivalent

to the returns to managerial input in a competitive environment [Mas-Colell et al.
(1995), p.135].

Secondly, this production function, which has constant returns toscale, results from maximizing total output in a hierarchical organization, as Rosen(1982) has shown.

Given Rosen’s model, managerial input, T , is required to su-

pervise diﬀerent plants. Thirdly, when T grows at a constant rate, this assumption
guarantees the existence of a balanced growth path, which is shown subsequently.
An entrepreneur has an important task other than the supervisory one.

Be-

cause the movement of zi is unpredictable ex ante, when zi changes the entrepreneur
6

must recognize the direction and magnitude of this change to respond appropriately.
When the entrepreneur makes production decisions, she does not observe zi , but does
observe a noisy signal, si , from which the realization of zi can be inferred.

It isassumed that the entrepreneur’s inference is based on a conditional distribution function, Qh (z|s), where h measures the entrepreneur’s ability to recognize changes in z.
The conditional distribution function is the same for all entrepreneurs.

It implies

that all entrepreneurs share the same knowledge about the relation between the productivity shock and the observable signal. A more detailed information structure is
speciÞed subsequently.
Note that z is assumed to be an idiosyncratic shock.

Hence, the information

required to infer z must be local information. That is, entrepreneurs must process
their local information.

However, as both z and s are idiosyncratic, prices in thismodel do not depend on them. Hence, prices are predictable without knowing what
others observe. That is, entrepreneurs do not need to know all the local informationin an economy because the price system summarizes the information they need. Thisis essentially the view of Hayek (1945). In this sense, this paper incorporates Hayek’s(1945) arguments into the neoclassical growth model, and examines the social eﬀectof local information.
It is assumed that the Þnancial market is complete. Therefore, entrepreneurs canhedge against any idiosyncratic risk. Entrepreneurs maximize their Þrm’s expectedproÞts:π (si ) = max
k,L

½Z

α

h

¾

zi [f (ki ) Li ] T dQ (zi |si ) − wT Li − rki T Li ,where w is the wage rate for eﬀective labor and r is the rental price of capital. The
Þrst-order conditions are:
w = α

Z

zi dQh (zi |si ) [f (k (si )) L (si )]α−1 f (k (si )) − rk (si ) ,7(1)

r = αZzi dQh (zi |si ) [f (k (si )) L (si )]α−1 f 0 (k (si )) ,for any s, where k (s) and L (s) are the optimal levels of k and L.

(2)
Because theproduction function is strictly concave in k and L and satisÞes the Inada conditions,
there exists a unique interior solution, and the Þrst-order conditions are necessary
and suﬃcient for the maximization problem.
Note that the two Þrst-order conditions imply that entrepreneurs equate the wage
rate (rental price) to the expected marginal product of labor (capital), not to theactual marginal product of labor (capital). Unexpected idiosyncratic shocks cause
marginal products to deviate from marginal costs, and these deviations provide opportunities for entrepreneurs to exploit. If entrepreneurs recognize the changes clearly,
they can take the opportunities.

This is the aspect of entrepreneurship that weemphasize in this model, and is also stressed by Kirzner (1973) and Schultz (1975).
The following is derived from the two Þrst-order conditions:wf (k (si )) − f 0 (k (si )) k (si )=rf 0 (k (si ))(3)This equation implies that the capital stock per unit of eﬀective labor in production
k (si ) does not depend on the realization of the signal si . Hence, we denote this byk.

As the right-hand side of equation (3) is strictly increasing in k, k is uniquelydetermined by wr .
Expected proÞts are derived by substituting the two Þrst-order conditions into
π (si ):

Z

(1 − α)π (s) dQhs (s) =(w + rk) TαZ

L (s) dQhs (s) .

(4)Equation (4) shows that expected proÞts are proportional to the costs of production.

The arbitrage condition: Because entrepreneurs can completely hedge their risks
in the Þnancial market, they do not bear risk. As agents are identical and can be8

entrepreneurs or workers, expected proÞts must be equal to the opportunity cost of
being an entrepreneur, which is the wage rate in the labor market.
Z
π (s) dQhs (s) = wT.

(5)

Hence, equations (4) and (5) imply(1 − α)(w + rk)w=
αZ

L (s) dQhs (s) .

(6)

Resource constraints: To close the model, the labor and capital markets mustclear:
Z= mN kT L (s) dQhs (s) ,Z(1 − m) N = mN L (s) dQhs (s) ,
K

a

(7)
(8)

where K a is the aggregate capital stock. Equation (7) is the capital market clearing
condition. The left-hand side is the supply of capital and the right-hand side is theRdemand for capital: mN is the number of Þrms and kT L (st ) dQ (st ) is the average
Þrm’s demand for capital. Equation (8) is the labor market clearing condition. The
left-hand side is the supply of labor and the right-hand side is the demand for labor.

Market equilibrium: Market equilibrium can be formally deÞned as follows:DeÞnition 1 A market equilibrium is {L (·) , k, w, r, m} that satisÞes the following
conditions.
1. The Þrm’s proÞt maximization conditions: equations (1) and ( 3).
2. The arbitrage condition: equation (6).
3. The resource constraints: equations (7) and (8).
90Let us deÞne θ (k) = ff(k)k.
(k)

The following theorem proves the existence anduniqueness of the equilibrium. The proof is given in the Appendix.
a

Theorem 2 Suppose that limk→∞ θ (k) < 1. Then, for any k a ≡ TKN ∈ (0, ∞) , a
unique market equilibrium exists.
When many agents become entrepreneurs, few are employees.

This increases

the demand for employees and reduces the supply of employees.

The wage rateis determined so that demand equals supply, which guarantees the existence of an
equilibrium.

The assumption limk→∞ θ (k) < 1 is a technical one.

agents become employees, k is larger.

When fewer

This condition implies that as k becomesinÞnite, the labor share does not converge to 0. That is, the wage rate must increaseat a faster rate than employment falls.

This guarantees that m has a solution in(0, 1).

The aggregate production function: Now we derive the aggregate production
function.

Let Y a and y a denote aggregate output and aggregate output per unitaof eﬀective labor in an economy, y a = TYN , respectively. The following propositionshows that entrepreneurs’ ability to recognize change can inßuence the TFP of theaggregate production function. The proof is given in the Appendix.
Proposition 3 Suppose that limk→∞ θ (k) < 1.

Then for any k a ∈ (0, ∞), there

exists an aggregate production function:y a = z (h) φ (k a )"Z ·Z

where z (h) ≡zdQh (z|s)
10

1
¸ 1−α

dQhs (s)#(1−α)(9),aφ (k ) ≡ fµ

ka
1 − m (k a )¶αm (k a )(1−α) [1 − m (k a )]α ,and m (k a ) ∈ (0, 1) is a solution of·µ a ¶¸
α
k
1−m
1−θ
=
.
1−α
1−m
m

This proposition shows that h aﬀects the TFP of the aggregate production function, but says nothing about the direction of the eﬀect. Before examining the eﬀectof h on TFP, it is useful to show the properties of the aggregate production function. The following proposition shows that the derived aggregate production function
satisÞes the traditional assumptions of macroeconomics. The proof is given in the
Appendix.
Proposition 4 Suppose that limk→∞ θ (k) < 1 and limk→0 θ (k) < 1.

Then thederived aggregate production function is increasing and concave in k a ∈ (0, ∞), andsatisÞes the Inada conditions:
φ0 (k a ) > 0, φ00 (k a ) < 0.
φ0 (k a ) = ∞lim φ (k a ) = 0, alim φ0 (k a ) = 0 and lim
a

ka →0

k →∞

k →0

Both concavity and satisfaction of the Inada conditions are essential to the existence of a globally stable unique steady state in the neoclassical growth model. Thatis, the derived aggregate production function satisÞes all the important assumptionsof the aggregate production function in the neoclassical growth model.

3

The Macroeconomic Eﬀects of Adaptability

In this section, we specify the information structure that entrepreneurs can access andexamine the macroeconomic eﬀects of entrepreneurs’ ability to recognize unexpected11changes in the environment.

TFP, z (h), is shown to be an increasing function ofentrepreneurial ability.

The components of z (h): Assume that log z comprises a predictable componentµ and an unpredictable component u:
log z = µ + u
where u is normally distributed with mean 0 and variance σ 2u . It is assumed thatthe unpredictable component u summarizes an unexpected change in productivity.
The entrepreneur cannot observe u before making production decisions, but she canobserve the signal s:s= u+εwhere ε is normally distributed with mean 0 and variance σ 2ε (h). This paper assumes
that a Þrm’s adaptability is determined by the entrepreneur’s ability to recognize theunexpected change.

This ability can be represented by the ability to predict u

after observing s.

Hence, we can apply Takii’s (2003) notion of prediction ability

in this context. Let Qhu (u|s) denote the conditional distribution of u given s. The
measure of an entrepreneur’s ability to recognize the unexpected change, u, is deÞned
as follows.
DeÞnition 5 The measure of an entrepreneur’s ability to recognize the unexpectedchange u (the measure of the Þrm’s adaptability) is deÞned by:R
V ar (u|s) dQhs (s)
,
h=1−
σ 2u
¢2
R¡
R
where V ar (u|s) =u − udQhu (u|s) dQhu (u|s).

This measure implies that the entrepreneur accurately recognizes u when she re-duces on average the conditional variance having observed s.
12

To compare ability

in diﬀerent environments,
ditional variance of u.

R

V ar (u|s) dQhs (s) is divided by σ 2u , which is the uncon-

The measure h ranges from 0 to 1.

If the entrepreneur

perfectly recognizes the change, h = 1, whereas if the entrepreneur does not recognize the change at all, h = 0.
Using the deÞnition of h, the variance of the noise term is endogenously determined
as follows:

σ 2ε (h) =(1 − h) σ 2u
.
h

(10)

As expected, when the entrepreneur more accurately recognizes an unexpected change,
the variance of the noise term is smaller. When h = 1, the variance is 0, and whenh = 0, the variance is inÞnite.

Macroeconomic eﬀects of entrepreneurship: Using this measure, z (h) canbe decomposed into productivity, risk and adaptability.

The following theorem

summarizes one of the main results in this paper. The proof is given in the Appendix.
Theorem 6 For any k a ∈ (0, ∞), a rise in h increases GDP per unit of eﬀective
labor in an economy, y a :·¸
ασ 2u hy = z φ (k ) exp.
2 (1 − α)
a

e

a

(11)

When an unexpected change in productivity occurs, if entrepreneurs accurately
recognize the change, the deviations of actual marginal productivities of inputs from
input prices would be small. Hence, an improvement in a Þrm’s adaptability raises
allocative eﬃciency, and therefore increases the productivity of the economy. Theorem 6 shows that an increase in productivity is represented by a rise in the TFP of thederived aggregate production function and that an increase in the Þrms’ adaptability
raises GDP per unit of eﬀective labor in an economy.
13

Steady state: It is easy to apply the derived aggregate production function to
the neoclassical growth model. Assume that the productivity measure, T , and the
population, N , grow at the constant rates gT and n, respectively. Assume also thatR ((cat Tt )1−θ −1)Nt −ρtthe utility function of the household is
e dt, where cat is consumption1−θ

per unit of eﬀective labor in an economy. The steady-state values of cat and kta in

the neoclassical growth model must satisfy the following:z (h) φ0 (k a ) = δ + ρ + θgT ,ca = z (h) φ (k a ) − (n + δ + gT ) k a ,2

ασ u hwhere z (h) = z e exp 2(1−α)
and δ is the depreciation rate of the capital stock. These

two equations yield the steady-state values of ca and k a . In the steady state, consumption per capita, GDP per capita and capital stock per capita grow at the constantrate gT . Hence, an increase in h raises the level of k a and ca , but does not change
the growth rate.
The lack of a growth eﬀect may be perceived as a weakness of the model, as
entrepreneurs are thought to develop new products and aﬀect economic growth. Of
course, if innovation were formally modeled, it would be possible to construct a modelin which entrepreneurial ability aﬀects the long-run growth rate. However, this paper
separates adaptability from innovative ability. If only adaptability is considered, it
is conceivable that entrepreneurial ability does not aﬀect the long-run growth rate,
although it would aﬀect temporal changes in the growth rate.

4

Political Risk and Entrepreneurship

In the previous section, it was argued that entrepreneurship can improve allocative efÞciency and the TFP of an economy, because when entrepreneurs recognize a change
14correctly, they can equate the marginal products of inputs to their prices.

How-

ever, in this section, we use an example to show that if opportunities are distorted,
adaptability can lower the TFP of an economy.
Assume that subsidies increase Þrm-speciÞc productivity and that all subsidies
are Þnanced by income tax:
zi = (1 − τ ) (1 + Gi )where τ is the constant average and marginal income tax rate and Gi is the subsidy forthe ith Þrm. It is assumed that in the absence of taxes and subsidies, the productivityof each Þrm is unity. The subsidies have two components: a predictable componentg (Ri ) and an unpredictable component ui .
log (1 + Gi ) = g (Ri ) + ui
where Ri is the rent-seeking activity of the ith Þrm and ui is normally distributedwith mean 0 and variance σ 2u .

The random variable ui can be interpreted as apolitical shock. Because political outcomes depend on the opinions, political tactics
and negotiations of politicians, the results are diﬃcult to predict. The random factor,
ui , represents this uncertainty. It is assumed that Ri is chosen before entrepreneurs
observe signals. Then, because entrepreneurs are identical, all choose the same levelof rent-seeking activity, Ri = R and g (Ri ) = g (R). If we set µ = log (1 − τ ) + g (R),the analysis of the previous section is applicable in this context.
As rent-seeking activities do not change aggregate income, µ must be chosen
endogenously to satisfy the following resource constraint:0=Z(1 − zi ) [f (k) L (si )]α T dQhzs (zi , si ) m(12)

where L (s), k and m are the market equilibrium solutions to the previous problem.
R
R
Note that [f (k) L (si )]α T dQhzs (zi , si ) m and zi [f (k) L (si )]α T dQhzs (zi , si ) m are15

the values of aggregate output before and after the transfer of income, respectively.
Hence, equation (12) requires that the income transfer does not change aggregate
output.
Note that L (s) is chosen when entrepreneurs expect an income transfer. Hence,the value of aggregate output before the income transfer takes place is aﬀected byentrepreneurs’ predictions of the realization of the political shock.

The following

theorem shows that an increase in Þrms’ adaptability lowers TFP when there ispolitical risk. The proof is given in the Appendix.
Theorem 7 Suppose that µ satisÞes equation (12 ). Then, for any k a ∈ (0, ∞), a
rise in h reduces GDP per unit of eﬀective labor in an economy, y a :¸−ασ 2u hy = φ (k ) exp.
2 (1 − α)
a

·

a

(13)

Political risk reduces TFP because political risk generates a negative externality:
when an entrepreneur is good at taking opportunities, there are fewer opportunities
for others. In fact, it is shown that when equation (12) is satisÞed, µ is chosen tosatisfyµ=−ασ 2u h σ 2u
− .
(1 − α)2Although individual entrepreneurs react to the political shock given µ, since thesereactions do not produce new value in the economy, adaptability lowers µ to satisfy
equation (12). This generates a negative externality.
Two comments are warranted.

First, equation (13) shows that, if there is nopolitical risk, competition between pressure groups leads to eﬃcient allocation, as
suggested by Becker (1983). More importantly, equation (13) shows that if there ispolitical risk, even if rent-seeking activities are costless, political uncertainty can confuse entrepreneurs and thereby lower productivity. That is, entrepreneurship might
have a detrimental eﬀect on the economy in the presence of political risk.
16

Hence,

the social welfare loss due to rent-seeking activities may have been underestimatedby the existing literature.

5

A Framework for an Empirical Study

The objective is to investigate the eﬀect of Þrms’ adaptability on TFP. The growth
0

a

a

(k )k
rate of TFP, gT F P , is usually deÞned as gT F P ≡ g Y − φ φ(ka ) g K , where g Y and g K
N

N

N

N

are the growth rates of GDP per capita and capital stock per capita, respectively.
Theorem 6 implies thatµφ0 (k a ) k aegT F P ≈ gz + 1 −φ (k a )

¶

gT +

£
¤
α
hdσ 2u + σ 2u dh .
2 (1 − α)(14)

To test the implication of equation (14), the following empirical equation is examined:∆ log T F P = ψ 0 + ψ z ∆ log z e + ψ T ∆ log T + ψ σ ht ∆σ 2u + ψ h σ 2ut−1 ∆h + ε,(15)

where ψ 0 , ψ T , ψ σ and φh are constant parameters. The growth rate of T , ∆ log T ,represents aggregate productivity growth. The growth rate of z e , ∆ log z e , can beinterpreted as Þrm-speciÞc productivity growth.

After controlling for these two

eﬀects, our theory predicts that a change in both risk and adaptability has a positiveeﬀect on the growth rate of TFP. Note that even if political risk is important, once
we control for z e , adaptability should have a positive eﬀect on TFP.
The estimation of this equation requires estimates of the variables, ∆ log T F P ,∆ log z e , ∆ log T , σ 2u and h, which are described below.

Data description: Proxies for Y , K, wT L, wT and r were constructed mainly fromthe Census of Manufacturing in Japan for 1985—1999, which was provided by I-N
Information Systems, LTD. The census covers all establishments in which four or17

more persons work as employers or employees, and the data are aggregated by cityand industry for each year. This enables panel data analysis of the behavior of thetypical establishment by city and industry. Details of the data and the constructionof variables are given in the Appendix.
The data are split into two periods: 1985—1991 and 1992—1999. These periods
roughly correspond to before and after the bursting of the bubble in Japan.

Weestimate, σ 2u and h using the constructed Y , wT L, wT , r and K, by prefecture,industry and period. Then, ∆ log T F P , ∆ log z e , ∆ log T , ∆σ 2u and ∆h are estimatedby prefecture and industry.

We treated 1988 and 1996 as representative years foreach period. That is, we estimated the aggregate production function’s TFP growthby prefecture and industry between 1988 and 1996. These estimated variables areused in our regression analyses. The estimation method is explained below.

Estimation of ∆ log T F P , ∆ log z e and ∆ log T : First, we derive the equations
that relate ∆ log T F P , ∆ log z e and ∆ log T to observable variables, and then provide
an interpretation of each equation.

Details of the estimation method are given in

the Appendix.
The following proposition explains the estimation of ∆ log T F P , ∆ log z e and ∆ log T .
The proof is given in the Appendix.
Proposition 8 If w is constant, the growth rate of TFP, ∆ log T F P , aggregate productivity growth, ∆ log T , and Þrm-speciÞc productivity growth, ∆ log z e , can be estimated as follows:Y
φ0 (k a ) k a
K
−
∆ log
(16)
a
N
φ (k )
N∆ log T = ∆ log wT,
(17)
¸
Z ·
Y (z, s)K(, 18)
Qhzs (z, s) − αθ (k) ∆ log∆ log z e = ∆ log
α(1−α)(wT L (s)) wTwT L∆ log T F P = ∆ log

18

0

a

a

(k )k
where φ φ(ka ) , αθ (k) and α are estimated by

φ0 (k a ) k a
1i,= αθ (k) = R ha
Y (z,s)φ (k )dQh (z, s)rK(s)α = Rh1

Y (z,s)rK(s)+wT L(s)
α

i

(19)

zs

,

(20)

dQhzs (z, s)

Y (z, s) ≡ z [f (k) L (s)] T, K (s) ≡ kT L (s) .
Equation (16) is the usual deÞnition of TFP growth, except that the methodof estimating the elasticity of output with respect to capital is unusual. Equation(19) shows that this can be estimated by the average capital share. Note that thedeÞnition of the average capital share corresponds to the usual one when there is no
random component.
Equation (17) shows that aggregate productivity growth can be estimated by
the growth rate of the average wage. When productivity growth is economy-wide,competition in the labor market pushes up workers’ wage rates.

Equation (17)reßects this intuition.
To interpret equation (18), we consider the case in which z is predictable. If z ispredictable, equation (18) can be re-written as:∆ log z = ∆ log Y − α [1 − θ (k)] ∆ log L − αθ (k) ∆ log K − [1 − αθ (k)] ∆ log wT.
This equation shows that ∆ log z represents the growth rate of value added thatcannot be explained by the growth rate of labor input, capital input or wage rates.
Hence, this is TFP growth excluding the contribution represented by the growthrate of wage rates.

That is, the estimated ∆ log z excludes aggregate shocks in

productivity changes.

An aggregate shock raises wage rates and causes the eﬀect

to deviate from the components of ∆ log z. Firm-speciÞc productivity growth is thecomponent of ∆ log z.

19

It may be objected that a constant w is not consistent with our theory, as shifts
in z e , h and σ 2u change w. However, as we are using regional data, workers can move
between regions. Hence, if the labor market is competitive, w is approximately thesame in all regions and industries. To the extent that z e , h and σ 2u do not change
on average, a constant w is justiÞed. In fact, the data support this assumption, as
is shown subsequently. Hence, this assumption is innocuous.

Estimation of h and σ 2u : Next, we explain the estimation of h and σ 2u . It is shown
subsequently that h can be estimated by the correlation between the unexpected
shock and the reaction to the shock. If a Þrm’s response to the shock is appropriate,
this correlation must be high. To conÞrm this intuition, we must deÞne the reaction
to the shock.
DeÞnition 9 The Þrm’s reaction to the shock R (L (s)), is deÞned as the logarithmof the deviation of actual labor input, L (s), from predicted labor input, L∗ :R (L (s)) = log L (s) − log L∗ ,where L∗ is estimated from the input level in the absence of an unexpected shock:
1
½ e
¾ 1−α
z
α [f (k a ) − f 0 (k a ) k a ]f (k a ) .
L∗ =
w

(21)

Equation (21) is derived by substituting z e into the Þrst-order conditions (1) andR(2) for zi dQh (zi |si ). Using the deÞnition of the Þrm’s reaction to the shock, thefollowing theorem is proved in the Appendix.

Theorem 10 The entrepreneurs’ adaptability, h, can be estimated by the correlation
between u and R (L (s)):¤2£h = ρuR(L(s)) , ρuR(L(s)) > 0,20

where¢¡Ru R (L (s)) − R (L (s)) dQhs (s) dQhus (u, s)ρuR(L(s)) = qR
.
R¡
R
¢2
u2 dQu (u) R (L (s)) − R (L (s)) dQhs (s) dQhs (s)R

Theorem 10 shows that h can be estimated by the squared correlation between
the unexpected shock and the Þrms’ reaction to the shock. The proof is based onthe Þrst-order condition (1).

The entrepreneur employs more than the predictedlevel of labor input when she believes that a positive productivity shock has beenrealized, and employs less than the predicted level when she believes that a negative
one has occurred. When the entrepreneur’s belief is accurate, the correlation mustbe larger.
To implement this idea, we need to estimate L∗ . This involves the estimation of an
unknown function f (·). However, if the economy is in the steady state, the correlation
between the unexpected shock and the reaction to the shock can be estimated without
using the function f (·).

In the steady state, k and w are constant.

Because thecorrelation coeﬃcient is invariant to an aﬃne transformation of a variable ( ρXY =ρX(ηY +ι) , where η and ι are constant ), the following corollary can be easily provenfrom the deÞnition of u and R (L (s)).
Corollary 11 In the steady state, the correlation between the unexpected shock andthe reaction to the shock can be estimated by the correlation between log Y −α log wT L−(1 − α) log wT and log wT L − α log wT L − (1 − α) log wT .
Corollary 11 shows that the main factor aﬀecting h is the correlation between
value added and labor expenses. Obviously, this is a fairly crude measure of adaptability. If we could explicitly model the information that entrepreneurs observe, amore accurate measure might be obtained. However, observable data are less likely
to reßect the ideas of Hayek (1945) and Kirzner (1973), who emphasize the importance of unobservable local information.

Moreover, Davis and Haltiwanger (1999)
21

insist that unobserved idiosyncratic factors play a dominant role in explaining theredistribution of workers. Genda (1998) conÞrms this for Japan.
The correlation measure reßects the value of local information. To understandthis, it is helpful to modify the equations in corollary 11.
log Y − α log wT L − (1 − α) log wT = log Y − α log L − log wT,
log wT L − α log wT L − (1 − α) log wT = (1 − α) log L.

(22)
(23)

Equation (22) measures Þrm-speciÞc productivity, since an aggregate productivity
shock must also increase wT . The two equations show that h can be measured by thecorrelation between a Þrm-speciÞc shock and labor input.2 Hence, we suggest that,
despite its potential problems, the correlation measure contains useful information
about the ability of Þrms to process local information.
The correlation measure can be aﬀected by various factors, including talent levels
in management groups, education, personal networks, population density, regionaltransportation costs, and communication costs within organizations. In the absenceof a theory of what determines adaptability, identifying the factors that enhance
entrepreneurship is beyond the scope of this paper.

However, we doubt that thecorrelation measure would be greatly aﬀected by factors aﬀecting adjustment costs,
as adjustment costs lower not only the covariance of value added and labor expenses,
but also their variances.
Corollary 11 uses labor expenses as a proxy for labor input for the estimation of h.
There are two justiÞcations for this proxy. First, the data set used for the empirical
study does not include data on the number of employees, but does include data on
the sum of employees and employers. Hence, labor expenses are the best available
measure of labor input. Second, many unobserved inputs cannot be represented by
2

In fact, we can show that the correlation between the unexpected shock and the reaction to theshock is equivalent to the correlation between log z and log L in the steady state.

22the number of employees.

Because it is diﬃcult to Þre employees, Þrms are likely

to react to unexpected changes by varying working hours or workers’ eﬀort levels.
Expenses best reßect these unobserved inputs.
If the economy is in the steady state, σ 2u can be estimated as follows:
σ 2u = V ar [log Y (z, s) − α log wT L (s) − (1 − α) log wT ] ,where V ar (x) is the variance of x. As discussed above, equation (22) implies that
σ 2u can be interpreted as a measure of changes in Þrm-speciÞc productivity.
Strictly speaking, the steady-state assumption is not consistent with regression
´
³
0 a a
(k )k
gT .
equation (2), because if the economy is in the steady state, gT F P ≈ 1 − φ φ(ka)

The economy is assumed to be in the steady state between 1985 and 1991, andbetween 1992 and 1999.

A large shock is assumed to occur around 1991, whichcaused the economy to move from one steady state to the other. The values of h andσ 2u during the transition period are approximated by the steady-state values of h andσ 2u .

Because the Japanese bubble burst in 1991, it is not unreasonable to assume

that the steady state changed around this time.

Alternative method of estimation: If the steady-state assumption is restrictive,
an alternative method of estimating h is to specify the production function. Assume
that f (k) = Bk β . Then the following corollary can be proven in the same way ascorollary 11 was proven.
Corollary 12 Suppose that f (k) = Bk β and w are constant. Then, the unexpected
shock and the reaction to the shock can be estimated by the correlation between log Y −α (1 − β) log wT L − αβ log K − (1 − α) log wT and log wT L − α (1 − β) log wT L −αβ log K − (1 − α) log wT . Moreover, the measure of risk, σ 2u , can be estimated by
the variance of log Y −α (1 − β) log wT L−αβ log K −(1 − α) log wT , where α (1 − β)23and αβ can be estimated respectively by1i,α (1 − β) = R h
Y (z,s)h (z, s)dQ
zs
wT L(s)i
αβ = R h
Y (z,s)
rK(s)

1

.

dQhzs (z, s)

When f (k) = Bk β and w are constant, the estimation of ∆ log z e is modiÞed asfollows:e∆ log z = ∆ log

Z

Y (z, s)α(1−β)[wT L (s)]

αβ

K (s)

wT (1−α)dQhzs (z, s) .

The robustness of the results that follow can be checked by using the two alternative
measures.

6

Empirical Results

This section reports the empirical results.

First, summary statistics are reported.

Then the regression results are reported.

Summary statistics: Table 1 reports the summary statistics of our estimates. The
annual TFP growth rate is about 2 and aggregate productivity growth is 1.7%. Thismeans that aggregate productivity growth accounts for most of the growth in TFP,
which is broadly consistent with the steady-state assumption.
On average, Þrm-speciÞc productivity declined slightly (-0.39% ∼ -0.45%), andthe level of idiosyncratic risk remained constant.

Note that we split the sample

at approximately the point when Japan’s bubble burst.

Barseghyan (2003) andNishimura et al (2003) argue that many unproductive Þrms survived in Japan during
the 1990s. The fall in average Þrm-speciÞc productivity might reßect the survival of
weak Þrms in economy.
24

Variable

Obs. Mean (1988-1996) Std. Dev. Mean (annual)∆ log T F P¡¢∆ log z e f (k) = Bk β

800

0.156∗

0.301

1.95

800

-0.031∗

0.376

-0.39

∆ log z e (steady)800-0.036∗

0.380

-0.45

∆ log T¡¢∆ρuR(L(s)) f (k) = Bk β

800

0.139∗

0.088

1.74

800

0.053∗

0.322

0.66

800

0.024∗

0.315

0.30

587

0.028∗

0.152

0.35

592

0.023∗

0.153

0.29

800

-0.003

0.070

-0.04

800

-0.003

0.070

-0.04

∆ρuR(L(s)) (steady)¢¡∆h f (k) = Bk β∆h (steady)¡¢∆σ 2u f (k) = Bk β
∆σ 2u (steady)Table 1: Summary statistics (1988—1995)“steady” means that the steady state is assumed and “f (k) = Bk β means thatf (k) is assumed to be Cobb-Douglas for the purposes of estimation. ∗ indicatessigniÞcance at the 5% level.
The average of the measure of Þrms’ adaptability increased modestly (0.29% ∼0.35%). The modest increase in adaptability is conÞrmed by the simple correlation
between an unexpected shock and Þrms’ reactions to the shock (0.3% ∼ 0.66%).
The increase in the adaptability measure during the 1990s is interesting. Althoughadjustment is expected to be more diﬃcult during a recession, the data show that, on
average, Japanese Þrms improved their adaptability. This suggests that the measureof adaptability is not greatly aﬀected by recession.

As already discussed, the useof the correlation measure broadly corresponds to excluding the eﬀect of adjustment
costs. The data lend some support to this argument.
Small and opposing changes in Þrm-speciÞc productivity and adaptability and25

no movement in risk imply that w would not be greatly aﬀected by changes in thesevariables. This provides empirical justiÞcation for our assumption that w is constant.
Moreover, opposing changes in Þrm-speciÞc productivity and adaptability are also
consistent with our theory: an increase in adaptability lowers z e in the presence ofpolitical risk. Because the government’s share of output increased during the 1990s
in Japan [e.g., Hayashi and Prescott (2002)], our theory suggests that Þrms might
have responded more to political shocks during the 1990s.
Note here that there are fewer observations on ∆h than on ∆ρuR(L(s)) . To estimate∆h, we require a positive correlation in both periods.

Twenty-six percent of ourobservations do not satisfy this condition. To check the robustness of our results, we
also investigate regression analysis with the simple correlation below.

Regression results: Table 2 reports our regression results. All regressions showthat a change in Þrms’ adaptability increases TFP. This is consistent with the prediction of our theory: adaptability increases TFP once z e has been controlled for.
This result is robust. It passes several robustness checks.3
However, our theory is not consistent with risk having a negative eﬀect on TFP
growth: our theory predicts a positive eﬀect. This suggests that other factors should
be incorporated into our model. For example, if risk cannot be hedged in a Þnancial market, then the behavior of risk-averse entrepreneurs may explain the perverseÞnding.

Irreversible investment constitutes another possible explanation.

These

possibilities suggest further research.
3

We included employment or the number of establishments to control for scale eﬀects. We also

added the growth rate of employment to check whether the high correlation picks up the eﬀect ofgrowing Þrms. Including these variables does not change our results. We also used weighted leastsquares estimation using the square root of the number of cities as weights. This did not change
our results either.

26∆ log T F P = ψ 0 + ψ z ∆ log z e + ψ T ∆ log T + ψ σ ht ∆σ 2u + ψ h σ 2ut−1 ∆h + ε
f (k) = Bk β

f (k) = Bk β

steadysteady0.752∗∗∗

0.882∗∗∗

0.787∗∗∗

0.902∗∗∗(0.084)(0.104)(0.083)(0.099)0.522∗∗∗

0.547∗∗∗

0.510∗∗∗

0.546∗∗∗(0.020)(0.023)(0.019)(0.022)∆ log T

∆ log z eρuR(L(s))t ∆σ 2u

σ 2ut−1 ∆ρuR(L(s))-0.593∗-0.787∗∗∗(0.298)(0.272)0.781 ∗∗

1.139∗∗∗(0.294)(0.289)ht ∆σ 2uσ 2ut−1 ∆h-1.199∗-1.877∗∗∗(0.607)(0.479)2.415∗∗∗

4.136∗∗∗(0.587)(0.836)

0.068∗∗∗

0.051∗∗∗

0.065∗∗∗

0.051∗∗∗(0.014)(0.016)(0.014)(0.016)

Adj R-squared

0.526

0.550

0.539

0.571

Obs.

800

587

800

592

constant

Table 2: The eﬀect of Þrms’ adaptability on TFP“steady” means that the steady state is assumed and “f (k) = Bk β means thatf (k) is assumed to be Cobb-Douglas for the purposes of estimation.
signiÞcance at the 5% level.

∗∗

indicates signiÞcance at the 1% level.

∗

indicates∗∗∗

signiÞcance at the 0.5% level. Standard errors are in parentheses.

27

indicatesDespite this inconsistency, the signiÞcantly positive eﬀect of adaptability is interesting. Suppose the coeﬃcient for adaptability is 4.1 for example. Since average risk
is 0.035, a coeﬃcient of 4.1 implies that if a Þrm fully recognizes previously unpredicted changes in its environment, TFP increases by 14%, having controlled for thenegative externality. This means that if political risk is negligible, full recognition ofpreviously unpredicted changes raises TFP by 14%. An obvious problem is that wecannot distinguish political shocks from others. However, another extreme example
is informative. Suppose that political shocks are the only source of risk. When aÞrm fully recognizes previously unpredicted political shocks, equation (13) predicts
that TFP falls by 14%.
Of course, this is a rough estimate. Although we implicitly assume that aggregation eliminates measurement errors, this assumption might be questionable. However, measurement error typically causes eﬀects to be underestimated. Hence, ourestimates probably understate the real eﬀect.

Similarly, if h is heterogeneous in

an industry and a prefecture, the estimated relation between aggregate productivity
and the average value of h would understate the real relation. Again, our estimateswould understate the real relation.

Hence, the estimate obtained in this paper isconservative. A more accurate estimate requires more disaggregated data.

7

Conclusions and Extensions

This paper has presented a tractable macroeconomic model that enables a theoreticaland empirical investigation of a particular aspect of entrepreneurship–reacting appropriately to unexpected changes in the environment. We have shown that greater
adaptability raises total factor productivity (TFP) in a competitive economy. However, greater adaptability may lower TFP if opportunities are distorted.
Several extensions are being considered. First, although it has been assumed for28

simplicity that all shocks are idiosyncratic and not persistent, this assumption leads
us to underestimate the importance of entrepreneurship. An election or a revolution
represents an aggregate political shock. The introduction of new technology probably
generates persistent shocks. Incorporating these shocks into the model would be aninteresting extension.
Second, the source of shocks must be empirically identiÞed.

Although we Þnd

empirical evidence of a positive direct eﬀect of entrepreneurship on TFP, the overall
eﬀect is ambiguous. Identifying the source of shocks would enable estimation of theoverall eﬀect of entrepreneurship on TFP.
Third, investigating factors that enhance entrepreneurship would be interesting.
Although average adaptability is estimated by industry and prefecture in this paper,
we have said nothing about why adaptability diﬀers between industries and prefectures. Adaptability may be aﬀected by various factors, including inherited ability,
education, social networks, connections, the density of a region, regional communication systems, and organizations. Since we have a well-deÞned measure of adaptability,
it may be possible to identify empirically the factors that inßuence entrepreneurship.
Finally, it would be interesting to examine the extent to which a distorted reward
structure for entrepreneurs might reduce TFP. Macroeconomists have recently foundthat diﬀerences in unexplained exogenous productivity in the aggregate production
function explain a high proportion of income diﬀerences between countries [e.g., Halland Jones (1999) and Prescott (1998)].

Hayashi and Prescott (2002) argue thatpart of the ’lost decade’ in Japan can be explained by the slowdown in TFP growth.
Because the model in this paper relates entrepreneurs’ rent-seeking activities to TFP,
it may provide a suitable empirical framework within which to examine the extent towhich rent-seeking activities explain these problems.

29

8

Appendix

Mathematical Appendix:
Proof of Theorem 2: From the two resource constraints, (7) and (8), k can be
expressed as a function of the aggregate capital stock per unit of eﬀective labor in an
economy:

a

ka
,
k=
1−m

(24)

1−m
α
=
[1 − θ (k)] ,
m
1−α

(25)

where k a = TKN . From equations (3), (6) and (8), we derive that0

∈ (0, 1). DeÞne a function G (m, k):where θ (k) ≡ ff(k)k(k)·µ a ¶¸
k
1−m
α
a
1−θ
−
G (m, k ) ≡
1−α
1−m
m

(26)

We must show that for any k a ∈ (0, ∞), there exists an m ∈ (0, 1) that satisÞes
G (m, k a ) = 0.

(27)

It can be shown that limm→0 G (m, k a ) < 0 since when m converges to 0, the second
term of equation (26) goes to ∞ and the Þrst term is Þnite.

When m converges

to 1, the second term of equation (26) goes to 0 and the Þrst term is positive, since
limk→∞ θ (k) < 1 by assumption. This proves that there exists an m∗ ∈ (0, 1) thatsatisÞes equation (27). Moreover, the solution m∗ is unique, since it can be shownthat
G1 (m, k a ) |G(m,ka )=0
·
¸
αm 0
1−m
1
−
θ (k) k=m (1 − m)
m
1−α
1
α
=
[1 − θ (k) − mθ0 (k) k]m (1 − m) 1 − α½¾
1αf 00 (k) k[1 − θ (k)] [1 − mθ (k)] − m 0
θ (k) > 0.
=m (1 − m) 1 − α
f (k)
30

(28)

The derivation of the second equation uses G (m, k a ) = 0 and the derivation of thethird equation uses the deÞnition of θ (k). Given m, equation (24) uniquely solves fork, and, given a unique k, equation (3) uniquely solves for wr . Equations (1), (7) and(8) imply that
1−m
=
m

Z ·Z

h

zdQ (z|s)1¸ 1−αdQhs (s)·αf (k)αw + rk

1
¸ 1−α

,

(29)

Given unique values of wr and k, this equation uniquely solves for r and, therefore,
also uniquely solves for w. Finally, given unique values of w, r and k, equation (1)uniquely solves for L (s). Q.E.D.

Proof of Proposition 3: Aggregate output per unit of eﬀective labor in an economy,y a , can be expressed as follows:ay =where y =R Y (zs)

T L(s)mNyTR

L (s) dQhs (s)
,
TN

(30)

dQhzs (z, s). Since it can be shown that y = w+rk
, substituting thisαequation and equation (8) into equation (30) yields
ya =w + rk(1 − m) .
α(31)

Rearranging equation (29) yields"Z ·Z#(1−α) µ1¶α µ¶(1−α)¸ 1−αkw + rkmhh=zdQ (z|s)dQs (s)f.
α(1 − m)1−mApplying this equation to equation (31) yields the desired result. Q.E.D.

Proof of Proposition 4: The deÞnition of φ (k a ) implies that¶(1−α)µm
α−10a
φ (k ) = αf (k)f 0 (k) D,
1−m
¸
·
·
¸
f (k) (1 − α) (1 − m)dm
dm
where D ≡ 1 + k a + 0
−1
.
dk
f (k)
αm
dk a
31

(32)

Equation (27) implies D = 1. Hence φ0 (k a ) > 0.
To examine the second derivative, note that the Þrst derivative can be rewrittenas
φ0 (k a ) = α

θ (k) φ (k a )
.
ka

(33)

Hence the second derivative can be written as
φ0 (k a )
θ (k) k adk
where M = θ0 (k) a k a − θ (k) [1 − αθ (k)]dk
φ00 (k a ) = M

Hence, the sign of φ00 (k a ) is the same as that of M . To determine the sign of M ,two lemmas are required.
Lemma 13 The sign of m0 (k a ) is determined by the sign of θ0 (k):m0 (k a ) =θ 0 (k) m.
1 − θ (k) − θ0 (k) km

Proof. Equation (27) implies thatm0 (k a ) ==αθ0 (k)
1−α
¡
¢
,
0
1 1−m
α
−
θ
(k)
k
m
m
1−α

θ0 (k) m.
1 − θ (k) − θ0 (k) km£¡ ka ¢¤αThe derivation of the second equation uses the fact that 1−α
1 − θ 1−m
= 1−m
.
m

Equation (28) shows that 1 − θ (k) − θ 0 (k) km is positive.
Lemma 14 A rise in k a increases k:dk
1 − θ (k)>0=adk
(1 − m) (1 − θ (k) − θ0 (k) km)
Proof. Equation (24) implies thatdk
1 + km0 (k a )
.
=
dk a
1−m
The desired result follows from Lemma 13.
32

Applying Lemma 14 to the deÞnition of M , the following equation is derived:θ0 (k) [1 − θ (k)] k − θ (k) [1 − αθ (k)] (1 − θ (k) − θ0 (k) km).
1 − θ (k) − θ0 (k) km

M=0, the numerator becomes:
Since θ (k) = ff(k)k(k)
θ0 (k) [1 − θ (k)] k − θ (k) [1 − αθ (k)] (1 − θ (k) − θ0 (k) km)= θ (k)2 [1 − θ (k)] [α + m − 1 − αmθ (k)]+θ (k)

f 00 (k) k[1 − θ (k) + (1 − αθ (k)) mθ (k)]f 0 (k)Equation (27) implies that α + m − 1 − αmθ (k) = 0. Hence M < 0, and therefore,
φ00 (k a ) < 0.
Equation (27) also implies that limk→0 θ (k) < 1 guarantees that limka →0 m (k a ) ∈(0, 1). Hence, limka →0 φ (k a ) = 0 since limk→0 f (k) = 0, and limk→0 φ0 (k) = ∞ sinceequation (32) implies that limk→0 f (k) = 0 and limk→0 f 0 (k) = ∞ guarantee this.
Equation (27) also implies that limk→∞ θ (k) < 1 guarantees that limka →∞ m (k a ) ∈(0, 1).

Hence, equation (32) proves limk→∞ φ0 (k) = 0 since limk→∞ f 0 (k) = 0.

Q.E.D.

Proof of Theorem 6: Applying the standard Bayesian updating technique, it canbe shown that
Z

Using these results,R

udQhu (u|s) = hs,
V ar (u|s) = (1 − h) σ 2u .

zdF (z|s) can be expressed as follows:Z·¸
σ 2u hzdQ (z|s) = z exp hs −
,
2
h

e

(34)

³
´
2
where z e = exp µ + σ2u . Since the variance of s is σ 2u + σ 2ε , this can be written as
33

σ 2u
.
h

Using this result, it is easy to show that
"Z ·Z

h

zdQ (z|s)1¸ 1−αdQhs (s)#(1−α)= z e exp

ασ 2u h.
2 (1 − α)Hence, the desired result follows. Q.E.D.

Proof of Theorem 7: To prove the theorem, the following lemma is needed.
Lemma 15 Equation (12) is equivalent toZ ·Zhzi dQ (zi |si )

α
¸ 1−α

dQhs (s) =

Z ·Z

h

zi dQ (z|s)

1
¸ 1−α

dQhs (s)Proof. Since L (s) satisÞes equation (1),Z
(1 − zi ) [f (k) L (s)]α T dQhzs (z, s) m
α
¸ α
·Z
¸ 1−α
·
Z
αf (k) 1−αhhzi dQ (zi |si )dQzs (z, s)Tm=(1 − zi )w + rkHence, equation (12) implies
α
·Z
¸ 1−α
Z
h
0 = (1 − zi )zi dQ (zi |si )dQhzs (z, s) .
The desired result follows.
Since log z = µ + u, equation (34) implies that
Z µZ
and

Z µZ

h

zi dQ (zi |si )hzi dQ (zi |si )

1
¶ 1−α

α
¶ 1−α

1

dQhs (si ) = (z e ) 1−α exp

αdQhs (si ) = (z e ) 1−α exp

ασ 2u h,(1 − α)2α (2α − 1) σ 2u h,
2 (1 − α)2³´
2where z e = exp µ + σ2u . Using lemma 15, z e can be solved for as a function of hand σ 2u :z e = exp

−ασ 2u h.
(1 − α)34

The desired result follows from theorem 6. Q.E.D.

Proof of Proposition 8: The derivation of equation (17) is straightforward. Equation (18) is derived from the deÞnition of the Þrm’s production function. We utilize
the fact that random variables do not aﬀect k. Equation (20) is derived from thetwo Þrst-order conditions (1) and (2).

We now explain the derivation of equation(19). Equation (33) implies that
φ0 (k a ) k a= αθ (k) .
φ (k a )In addition, the Þrst-order condition (2) implies that1αθ (k) = R z[f (k)L(s)]α T
rkT L(s)Q.E.D.

dQz (z).

Proof of Theorem 10: Applying equation (34) to equation (21), it can be shown© eª 1
that L∗ = zw α [f (k a ) − f 0 (k a ) k a ] 1−α f (k a )µσ2 hL (s) = L exp hs − u
2
∗

1
¶ 1−α

.

h
i
2
1
hs − σu2 h . Hence,
Hence, the Þrm’s reaction to the shock R (L (s)) is given by (1−α)
the deÞnition of the correlation coeﬃcient implies that
ρuR(L(s)) =Q.E.D.

Data Appendix:
35

√
h.

Data description: Every year the Japanese Ministry of the Economy, Trade and Industry releases the Census of Manufacturing by city and industry. However, becauseof minor changes in the classiÞcation of industries and the integration and divisionof cities, the data that are released must be modiÞed for use in panel data analysis.
I-N Information Systems, Ltd undertakes this modiÞcation and thereby enables panel
data analysis of the behavior of the average establishment by city and industry.
Although the census covers all establishments in which four or more persons work
as employers or employees, if there are fewer than three establishments for an industry
in a city in any given year, data are not reported by the census, to maintain the privacyof establishments. To improve estimation of the correlation, we exclude entities on
which there are missing variables in any period.
The industries covered by the census are Food, Drink/tobacco/feed, Textiles,
Apparels, Lumber/wood products, Furniture/Þxtures, Pulp/paper, Printing, Chemicals, Petroleum/coal products, Plastic products, Rubber, Leather/leather products,
Pottery/glass products, Iron/steel, Non-ferrous metals, Metal products, General machinery, Electrical machinery, Transportation machinery, Precision tools, Weapons
and Other industries.

Y and wT L : Gross value added and labor expenses are divided by the number of
establishments. These values are then deßated by the GDP price deßator.
K : Fixed tangible assets are divided by the number of establishments. The replacement cost of the capital stock is then estimated from the following equation:
Kit+1 = Kit + (Fit+1 − Fit ) /pIt+1 , if Fit+1 > Fit ,= Kit + Fit+1 − Fit , if Fit+1 ≤ Fit ,where Ki1 = Fi1 /pI1 and Fit is average Þxed tangible assets per establishment and36

pIt is a price deßator for investment goods, which is taken from Keizai Tokei Nenkan(2002) (Annual Economic Statistics 2002) by Toyo Keizai Shinpo Sya. As we do nothave data on investment, this simpliÞed estimation method is used as an approximation, which is the approach taken by Nishimura, Nakajima and Kiyota (2003).
The Census of Manufacturing only reports Þxed tangible assets for establishments
in which the number of employers or employees is at least 10 [size group 2]. Hence,the capital stock of establishments with between four and nine employees or employers[size group 1] is estimated as follows. First, average labor expenses per establishmentof size groups 1 and 2 are estimated by city, industry and year. Average Þxed assets
per establishment are then regressed on average labor expenses per establishments
in size group 2 for each industry. A Þxed-eﬀects regression is used for this purpose.
The parameters of this regression are used to estimate the capital stock in size group1.
To estimate average labor expenses per establishment in size groups 1 and 2, thefollowing estimation method is used.

The Census of Manufacturing reports labor

expenses and the number of establishments for size groups 1 and 2 by industry andyear. Using these data, average labor expenses per establishment are estimated bysize group, industry and year. Assuming that the ratio of average labor expenses per
establishment in size group 1 to those in size group 2 is the same for each industry
and year, average labor expenses per establishment are estimated by city, industry
and year.
wT : wT is estimated by prefecture, industry and year by the weighted average of
deßated labor expenses over the number of employees. The number of establishments
is the weight. Because the number of employees is not reported, it is estimated. The
Census of Manufacturing reports the number of employees, the sum of employees andemployers, and the number of establishments by industry and year. Assuming that37

the ratio of employees per establishment to the sum of employers and employees per
establishment is the same for each industry and year, the number of employees is
estimated by city, industry and year.
r : The return to the capital stock is estimated by using
rt = pIt (it + δ) ,where it is the yield on 10-year government bonds and δ is the average depreciation
rate by industry over average Þxed tangible assets by industry.

The yield data

are from the homepage of Bank of Japan. Average depreciation and average Þxedtangible assets are taken from the Census of Manufacturing. As in Nishimura et al(2003), changes in the price deßator for investment goods are ignored, since this index
increased so much during the bubble in Japan that the user cost of capital became
negative. Because rt is only used to estimate the average capital share over time,
this simpliÞcation is unlikely to aﬀect our results. To check for robustness, we also
used the return to the capital stock, as in Hayashi and Prescott (2002). As this did
not change the results, we do not report them in this paper.

∆ log TFP : We estimate the average capital share of each Þrm from the sampleYover time by city and industry.
average of rK

Then the weighted average of thecapital share is estimated by prefecture, industry and period, with the number of
establishments in 1988 and 1996 as weights.

Unless otherwise stated, the same

weight is used to estimate the prefecture average. The average of the capital share
0

a

a

(k )k
over the period is chosen to estimate φ φ(ka) .

Value added, capital stock and thesum of employees and employers are aggregated by industry and prefecture in 1988
and 1996. Then ∆ log T F P is estimated as deÞned.
∆ log wT : wT in 1988 and 1996 is chosen for this estimation.
Y, Y∆ log ze : We estimate α, α (1 − β) and αβ from the sample averages of rK+wT
L wT L38

Y
and rK
over time by city and industry, respectively. Using the estimated α, α (1 − β)iRhR
Y (z,s)Y (z,s)hQand αβ,(z,s)
anddQhzs (z, s) are esα(1−α)
zs(wT L(s)) wT[wT L(s)]α(1−β) K(s)αβ wT (1−α)

timated by their sample averages over time by city, industry and period.

Theweighted averages of these values are calculated by prefecture, industry and period.
K, in 1988 and 1996 is also
The weighted average of the capital-labor expense ratio, wTLestimated by prefecture and industry. Using these values, ∆ log z e is then estimatedby prefecture and industry.
h : To implement corollaries 11 and 12, the correlations are estimated from the sample
averages over time by city, industry and period. Then, weighted averages of the correlations are estimated by prefecture, industry and period. The squared correlations
are then calculated when they are positive.
σ 2u : To estimate the variance, the standard deviation is estimated by city, industry
and period.

Then the weighted average of the standard deviation is estimated by

prefecture, industry and period.

The square of the average standard deviation isthen estimated.

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