OSIPP Discussion Paper : DP-2004-E-006

Entrepreneurial Efficiency: An Empirical Framework and Evidence
November 10, 2004
Katsuya Takii
Associate Professor, Osaka School of International Public Policy (OSIPP)【キーワード】D21, D61, O40【要約】This paper examines a particular aspect of entrepreneurship, namely
firms' activities in adapting to idiosyncratic environmental changes byappropriately reallocating resources. It presents an empirical framework thatexamines the social value of firms' abilities to predict and adapt to the movementof idiosyncratic shocks. Using the method, the quantitative effect of firms'
prediction ability on Total Factor Productivity (TFP) is investigated using datafrom Japan's Census of Manufacturing.
This paper contains material of a previous working paper, Takii (2003c). I would
like to thank Carmen Li, Boyan Jovanovic and seminar participants at the
University of Essex, the Kansai Macro Workshop, Osaka Prefecture University,
OSIPP and Sogang University for helpful comments. I also thank Kenjiro Hirata
for his excellent assistance. Financial support from the Japanese Ministry of
Education, Culture, Sports, Science and Technology (the Grant-in-Aid for
Scientific Research) is gratefully acknowledged. Osaka School of International

Public Policy, Osaka University, 1-31, Machikaneyama, Toyonaka, Osaka,
560-0043, Japan. E-mail: takii@osipp.osaka-u.ac.jp1

1

Introduction

It has long been recognized that adaptation to idiosyncratic shocks is vital in several
economic activities. In his well-known paper, Hayek (1945) regards “rapid adaptation
to changes in particular circumstances of time and place” as one of the most important economic problems.

Recently, much micro-evidence reconfirms the economic

significance of both idiosyncratic shocks and adaptations to these shocks. For example, Davis and Haltiwanger (1999) review the literature and insist that unobserved
idiosyncratic factors play a dominant role in explaining the redistribution of workers,
and Hubbard (2003) finds that advanced on-board computers significantly increase
capacity utilization in the trucking industry by improving dispatchers’ abilities tomake resource allocation decisions.
It can be conjectured that improvements in the ability to adapt to idiosyncratic
shocks will have significant influences on the aggregate economy.

One of the dif-ficulties in examining this in macroeconomic terms is that macroeconomics has a
tradition of working with an exogenously given aggregate production function. Because the relation between outputs and inputs is given by the aggregate production
function, there is no need for an economic agent who finds a productive use for inputs. As aggregate production function is a cornerstone of the neoclassical growthmodel, incorporating entrepreneurship in the aggregate production function will givea tractable tool with which to examine entrepreneurship in macroeconomics, whereentrepreneurship is defined as the activity of allocating resources in order to adapt
to idiosyncratic shocks.
This paper aims to accomplish this task and quantify the role of entrepreneurshipin macroeconomics. For this purpose, a firm’s entrepreneurial ability is modeled by
its ability to predict idiosyncratic changes in productivity. This paper derives an aggregate production function as a result of entrepreneurship and provides a tractable2

empirical framework to examine how firms’ abilities to predict and adapt to idiosyncratic changes in the environment influence an aggregate economy.
The concept of prediction ability in this paper aims to capture the soundnessof firms’ judgments about the economic impacts of idiosyncratic changes.

Firms

face several idiosyncractic changes every day: local area traﬃc increases, a new firmtakes away their many skillful engineers, a politician connected to a company loses anelection. As productivity is estimated by the ratio of measured output to input, thesefactors can potentially influence measured productivity. Hence, predicting change
in the productivity of one’s own firm means predicting how various changes in theenvironment directly influence production or sales.
When changes in productivity occur, the marginal products of inputs deviate from
input prices, and this generates opportunities for entrepreneurs to exploit. If entrepreneurs predict change and react to it appropriately, the deviation of the marginal
products of the inputs from input prices will be small. Hence, the improvement infirms’ prediction ability raises allocative eﬃciency and therefore increases productivity in the economy. It is shown that the increased prediction ability of firms raises
total factor productivity (TFP) of the aggregate production function in a competitive
economy.
A novel part of this paper is that it provides an empirical framework with whichto quantitatively examine the eﬀect of entrepreneurship on TFP. It is shown thatprediction ability can be measured by the squared correlation between a firm-specificshock and labor input.

In other words, appropriate adaptation to idiosyncraticchanges can be seen as evidence of better predictions about idiosyncratic changes inthe environment1 . Note that as most idiosyncratic shocks are unobservable, speci1

Hence, the derived measure can be alternatively interpreted. It measures the appropriatenessof a firm’s adaption to idiosyncratic changes. The previous version of this paper interpreted this as
the measure of adaptability.

3

fying all information that might influence entrepreneurs’ expectations is not feasible.
This paper shows that even if we do not know what entrepreneurs observe, we canstill infer the economic value of local information from their behavior .
We apply this method to investigate Japanese establishments during 1985—1999.
It is interesting to investigate how well establishments in Japan adapted to changes inthis period because it roughly corresponds to the Japanese boom in the 1980s and thelong recession of the 1990s. Our tentative estimate suggests that a rise in prediction
ability had a small but significant positive impact on TFP growth in Japan duringthat period.
A similar view of entrepreneurs is emphasized by Kirzner (1973).

Following

Hayek (1945), Kirzner argues that entrepreneurial discovery about previously unknown events is the engine of market equilibrating processes and insists that an
equilibrium analysis cannot capture the importance of entrepreneurial discovery. We
suggest that using an equilibrium model is beneficial. Particularly, we quantify theeﬀect of entrepreneurial discovery on TFP2 .
From a diﬀerent perspective, Schultz (1975) also defines entrepreneurial ability
as the successful interpretation of new information and allocation of resources toprofitable opportunities.

His idea is incorporated into an equilibrium model by

Holmes and Schmitz (1990), Hassler and Mora (2000) and Takii(2003a). However,no paper quantifies the social values of entrepreneurial ability, which is this paper’s
purpose.
This paper is organized as follows. The next section describes the model. Section
2

It may be argued that the method in this paper underestimates the role of entrepreneurial

discovery because, as Kirzner (1997) pointed out, the major part of uncertainty may not be measured
by Knightian risk. However, we believe that it is an important exercise to find a lower measure of
economic value that entrepreneurship can produce from the data (Kirzner, 1997). It might clarify
the benefits and limits of equilibrium analysis.

43 provides an empirical framework with which to examine the eﬀect of firms’ prediction abilities on TFP. Section 4 implements the methods empirically and reports
results from Japanese data. Section 5 concludes by summarizing the main results
and discussing possible extensions.

2

The ModelThis section extends a simple general equilibrium model presented by Takii (1994)
as being suitable for empirical study.

It lays the foundations for examining thequantitative eﬀects of entrepreneurship in the succeeding sections.
An agent can be an entrepreneur or a worker. Every firm needs an entrepreneur
to organize it. Firms are continuously distributed on [0, mN], where m ∈ (0, 1) is the
proportion of entrepreneurs in the total population, N. This implies that agents are
assumed to be identical.

Although the lack of heterogeneity among entrepreneurs

forced us to ignore a size distribution of firms as emphasized by Lucas (1978), it
allows us to focus on a diﬀerent economic problem: the eﬀect of entrepreneurshipon productivity in an economy. First the representative entrepreneur’s problem is
described, then resource constraints are presented.

The entrepreneur’s problem: An entrepreneur establishes a firm, 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 firm-specific productivity shock for the ith firm, and Yi , Ki and Li are
the amounts of the ith firm’s output, capital stock and labor input, respectively. The
parameters α and A measure the span of control and the management productivity
5respectively, and T measures the eﬀectiveness of labor, which is assumed to be influenced by common factors. It is assumed that F exhibits constant returns to scale in
K and L. By defining f (k) = F (k, 1), where k = TKL , we can express F (K, T L) as
a function of capital per unit of eﬀective labor in production: F (K, T L) = f (k) T L.
We assume that f 0 (.) > 0, f 00 (.) < 0, limk→0 f (k) = 0, limk→0 f 0 (k) = ∞ andlimk→∞ f 0 (k) = 0.
It is assumed that management productivity is a function of the eﬀectiveness of theentrepreneur. Because agents are homogeneous, the eﬀectiveness of the entrepreneuris the same as that of workers, which is given by T . Assuming that A = T 1−α , theproduction function can be written as
Yi = zi [f (ki ) Li ]α T.
There are three advantages in 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 firm’s profits are equivalent
to the returns to managerial input in a competitive environment (Mas-Colell et al.,1995). Secondly, this production function, which has constant returns to scale, results from maximizing total output in a hierarchical organization, as Rosen (1982) has
shown. Given Rosen’s model, managerial input, T , is required to supervise diﬀerent
tasks.

Thirdly, when T grows at a constant rate, this assumption guarantees theexistence of a balanced growth path, which is roughly consistent with the movementof macro data.
An entrepreneur has an important task other than a supervisory one. Because
the movement of zi is unpredictable ex ante, when zi changes, the entrepreneur must
predict the direction and magnitude of this change in order to respond appropriately.
When the entrepreneur makes production decisions, she/he observes a noisy signal, si ,from which the realization of zi can be inferred. It is assumed that the entrepreneur’s
6inference is based on a conditional distribution function, Qh (z|s), where h measuresthe entrepreneur’s ability to predict changes in z.

The conditional distributionfunction is the same for all entrepreneurs. It implies that all entrepreneurs share thesame knowledge about the relation between the productivity shock and the observablesignal. A more detailed information structure is subsequently specified.
Note that z is assumed to be an idiosyncratic shock.

Hence, the information

required to infer z must be local information. However, as both z and s are idiosyncratic, prices in this model do not depend on them. Hence, prices are predictablewithout knowing what others observe. That is, entrepreneurs do not need to know all
the local economic information because the price system summarizes the information
they need. This is the role of the price mechanism emphasized by Hayek (1945).
It is assumed that the financial market is complete.

Therefore, entrepreneurs

can hedge against idiosyncratic risks. Entrepreneurs maximize their firms’ expected
profits:
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
first-order conditions are:
w = α
r = α

Z

Z

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

zi dQh (zi |si ) [f (k (si )) L (si )]α−1 f 0 (k (si )) ,(2)

for any s, where k (s) and L (s) are the optimal levels of k and L. Note that these two
first-order conditions imply that entrepreneurs equate the wage rate (rental price) tothe expected marginal product of labor (capital), not to the actual marginal productof labor (capital). Unexpected idiosyncratic shocks cause marginal products to deviate from marginal costs, and these deviations provide opportunities for entrepreneurs7

to exploit. If entrepreneurs recognize these changes clearly, then they can seize theseopportunities. This is the aspect of entrepreneurship we emphasize in this model.
Expected profits are derived by substituting the two first-order conditions into
π (si ).
Z

∙

α
¸ 1−α
αf
(k)
T
π (s) dQhs (s) = (1 − α) z (h)w + rk
#1−α
"Z ∙Z
1
¸ 1−α
dQhs (si )z (h) =
zi dQh (zi |si )
1
1−α

(3)

Equation (3) shows that expected profit positively depends on z (h), the component
examined below. Note that k does not depend on si , shown from equations (1)and(2).

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 that theunpredictable component u summarizes an unexpected change in productivity. The
entrepreneur cannot observe u before making production decisions, but can observe
the signal s:s=u+ε
where ε is normally distributed with mean 0 and variance σ 2ε (h). We apply Takii’s(2003a) measure 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 predict
unexpected change, u, is defined as follows.
Definition 1 The measure of an entrepreneur’s ability to predict unexpected change8u is defined 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/he

reduces on average the conditional variance having observed s. To compare abilityRin diﬀerent environments, V ar (u|s) dQhs (s) is divided by σ 2u , which is the uncondi-

tional variance of u. The measure h ranges from 0 to 1. If the entrepreneur perfectlypredicts the change, h = 1, whereas if the entrepreneur does not predict change atall, it is h = 0.
It shows that¸
ασ 2u hz (h) = z exp,
2 (1 − α)
∙

e

n
o
σ2u
where z = exp µ + 2 . This equation shows that z (h) can be decomposed intoethe predicted productivity, µ, the risk from unpredicted changes, σ 2u , and prediction

ability, h. It shows that z (h) is an increasing function of h. This means that a risein h increases expected profits, a proposition empirically supported by Takii (2003b).
Rather, this paper examines the social value of prediction ability.

The arbitrage condition and resource constraints: Because entrepreneurs canhedge their risks in the financial market, they do not bear risk. As agents, they are
identical and can be entrepreneurs or workers, and expected profits must be equal tothe opportunity costs of being an entrepreneur, which is the wage rate in the labormarket.

Zπ (s) dQhs (s) = wT.

To close the model, the labor and capital markets must clear:Za
K = mNkT L (s) dQhs (s) ,
9

(4)

(5)

(1 − m) N = mN

Z

L (s) dQhs (s) ,(6)where K a is the aggregate capital stock. Equation (5) 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 firms, and kT L (st ) dQ (st ) is the averagefirm’s demand for capital. Equation (6) is the labor market clearing condition. The
left-hand side is the supply of labor, and the right-hand side is labor demand.

0

Aggregate Production Function: Let us define θ (k) = ff(k)k.
(k)

Let Y a andy a denote aggregate output and aggregate output per unit of eﬀective labor in anaeconomy, y a ≡ TYN , respectively. The following proposition is a direct application of
the results in Takii (2004) to the model with productivity growth. Hence, I omit theproof.
Proposition 2 Suppose that limk→∞ θ (k) < 1 and limk→0 θ (k) < 1.

Then, foraany ka ≡ TKN ∈ (0, ∞) , there exists an aggregate production function, φ (ka ) , which
satisfies equations (1), (2) (3), (4), (5) and (6):y a = z (h) φ (ka )¸∙ασ 2u he,where z (h) = z exp
2 (1 − α)(7)The derived aggregate production function is increasing and concave in ka ∈ (0, ∞),and satisfies the Inada conditions: φ0 (ka ) > 0, φ00 (ka ) < 0, limka →0 φ (k a ) = 0,limka →∞ φ0 (ka ) = 0, and limka →0 φ0 (k a ) = ∞.
When an unexpected change in productivity occurs, if entrepreneurs accurately
predict change, the deviations of the marginal productivities of inputs from input
prices would be small. Hence, an improvement in a firm’s prediction ability raises
allocative eﬃciency, and therefore increases economic productivity.
10

Proposition 2

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 firm’s prediction
ability raises GDP per unit of eﬀective labor in an economy.

3

A Framework for an Empirical Study

This section proposes an empirical framework to quantify the eﬀect of firms’ prediction abilities on TFP and presents several propositions linking unobserved parameters
to observable data. The next section demonstrates how to implement the propositions of this section using data from Japan’s Census of Manufacturing.
0

a

a

(k )k
The growth rate of TFP, gT F P , is usually defined as gT F P ≡ g Y − φ φ(ka) g K ,N

N

where g Y and g K are the growth rates of GDP per capita and capital stock perNN

capita, respectively. Theorem 2 implies that¶µ£¤φ0 (k a ) ka
α
2
2
g
hdσ
+
+
σ
dh
.
gT F P ≈ gze + 1 −Tuu
φ (k a )2 (1 − α)(8)Hence, we are primarily interested in the following regression equation:∆ log T F P = ψ 0 + ψ µ ∆µ + ψ 1σ ∆σ 2u + ψ T ∆ log T + ψ 2σ ht ∆σ 2u + ψ h σ 2ut−1 ∆h + ε, (9)where ψ 0 , ψ µ , ψ T , ψ 1σ , ψ 2σ and φh are constant parameters.

The growth rate of

T , ∆ log T , represents aggregate productivity growth. A change in µ, ∆µ, can beinterpreted as an increase in firm-specific productivity. After controlling these two
eﬀects, our theory predicts that a change in both risk and prediction ability has apositive eﬀect on the growth rate of TFP. The estimation of this equation requires
estimates of the variables, ∆ log T F P , ∆µ, ∆ log T , σ 2u and h, described below.

Estimation of ∆ log T F P and ∆ log T : First, we derive the equations that relate∆ log T F P and ∆ log T to observable variables, and then provide an interpretation11

of each equation. The following proposition explains the estimation of ∆ log T F P
and ∆ log T . Proof is given in the Appendix.
Proposition 3 If w is constant, the growth rate of TFP, ∆ log T F P , and aggregate
productivity growth, ∆ log T , can be estimated as follows:Y
φ0 (ka ) ka
K
−
∆ log
a
N
φ (k )
N∆ log T = ∆ log wT,∆ log T F P = ∆ log

0

a

(10)
(11)

a

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

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

(12)

zs

Y (z, s) ≡ z [f (k) L (s)] T, K (s) ≡ kT L (s) .

Equation (10) is the usual definition of TFP growth, except the method of estimating the elasticity of output with respect to capital is unusual. Equation (12)shows this can be estimated by the average capital share. Note that the definitionof the average capital share corresponds to the usual definition of capital share when
there is no random component.
Equation (11) 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 (11)reflects this intuition.
The wage rate per unit of eﬀective labor, wt , is an endogenous variable. It wouldchange when there is systematic change in µ, σ 2u and h.

A justification for thisassumption of constant wt is discussed in the next section.

Estimation of ∆µ and σ 2u : Next, we derive the equations for the estimation of ∆µand σ 2u . For this purpose, two diﬀerent assumptions are considered separately. The
12

first assumption is that kta is constant over time. This assumption is valid when theeconomy is in a steady state. The second assumption is that f (k) is Cobb—Douglas.
Therefore, two diﬀerent assumptions bring two diﬀerent estimates. These estimates
are used to check the robustness of the empirical results below.
Let E [x] and V ar (x) denote the expectation and the variance of x. First, we assume kta is constant. Given this assumption, we can derive the following proposition.
Proof is given in the Appendix.
Proposition 4 Suppose that wt and kta are constant. Then a change in firm specific productivity, ∆µ, and the measure of risk, σ 2u , can be estimated by a change in
expectation and the variance of log Y − α log wT L − (1 − α) log wT :∆µ = ∆E [log Z1 (z, s)] ,(13)

σ 2u = V ar [log Z1 (z, s)] ,

(14)

where
α = Rh1
Y (z,s)rK(s)+wT L(s)

i

,

(15)

dQhzs (z, s)

log Z1 (z, s) = log Y (z, s) − α log wT L (s) − (1 − α) log wT.
If constant kta is restrictive, an alternative method is to specify the production
function. Assume that f (k) = Bkβ . Then we can propose an alternative proposition. As the proof is similar to the proof of proposition 4, it is not repeated.
Proposition 5 Suppose that w is constant and f (k) = Bkβ . Then a change in firm
specific productivity, ∆µ, and the measure of risk, σ 2u , can be estimated by a change
in expectation and variance of log Y −α (1 − β) log wT L−αβ log K −(1 − α) log wT :∆µ = ∆E [log Z2 (z, s)] ,(16)

σ 2u = V ar [log Z2 (z, s)] ,

(17)

13

where11iiα (1 − β) = R h, αβ = R h,
Y (z,s)Y (z,s)h (z, s)h (z, s)
dQ
dQ
zs
zs
wT L(s)rK(s)log Z2 (z, s) = log Y (z, s) − α (1 − β) log wT L (s) − αβ log K (s) − (1 − α) log wT.
Note that log Z1 (z, s) and log Z2 (z, s) can be interpreted as a firm-specific shock.
log Z1 (z, s) is the component of value added that cannot be explained by the labor
expenses or wage rates, and log Z2 (z, s) is the component of value added that cannot
be explained by the labor expense, capital input or wage rates. Hence, both measure
the levels of productivity excluding the contribution represented by wage rates. As
the movement of aggregate shocks must be captured by that of wage rates, boththe estimated logZ1 (z, s) and log Z2 (z, s) exclude aggregate shocks in productivity.
Hence, the average firm-specific productivity is the component of E [log Z1 (z, s)] andE [log Z2 (z, s)]. Similarly, σ 2u can be interpreted as the measure of risk due to changes
in firm-specific productivity.

Estimation of h: Next, we explain the estimation of h. It is subsequently shownthat h can be estimated by the correlation between the unexpected shock and thereaction to the shock. If a firm recognizes the change and reacts to it, this correlationmust be high. To confirm this intuition, we define the reaction to the shock.
Definition 6 The firm’s reaction to the shock R (L (s)), is defined as the logarithmof the deviation of 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
0
L =
f (k) .
α [f (k) − f (k) k]
w

∗

14

(18)

Equation (18) is derived by substituting z e into the first-order conditions (1) andR(2) for zi dQh (zi |si ). Using the definition of the firm’s reaction to the shock, thefollowing theorem is proved in the Appendix.

Theorem 7 A firm’s ability to predict idiosyncratic shocks, h, can be estimated by
the correlation between u and R (L (s)):

where

£
¤2
h = ρuR(L(s)) , ρuR(L(s)) ≥ 0,¢¡R
u R (L (s)) − R (L (s)) dQhs (s) dQhus (u, s)ρuR(L(s)) = qR
.
¢2
R¡
R
u2 dQu (u) R (L (s)) − R (L (s)) dQhs (s) dQhs (s)R

Theorem 7 shows that h can be estimated by the squared correlation between theunexpected shock and the firm’s reaction to the shock. The proof is based on thefirst-order condition (1). The entrepreneur employs more than the predicted level oflabor input when it is believed that a positive productivity shock has been realized,
and employs less than the predicted level of labor when it is believed that a negative
one has occurred. When the entrepreneur’s belief is accurate, then the correlationmust be larger.
To implement this idea, we need to estimate L∗ . This involves the estimation ofan unknown function f (·). However, if ka is constant, it is shown that f (k) is also
constant. Because the correlation coeﬃcient is invariant to an aﬃne transformationof a variable ( ρXY = ρX(ηY +ι) , where η and ι are constant ), the correlation between
the unexpected shock and the reaction to the shock can be estimated without using
the function f (·). The following corollary can be proved from the definition: u and
R (L (s)).
Corollary 8 If wt and kta are constant, the correlation between the unexpected shockand the 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 .
15

Corollary 8 shows the main factor aﬀecting h is the correlation between value
added and labor expenses. This is a fairly crude measure of prediction ability. If we
could explicitly model information that entrepreneurs observe, a more accurate measure might be obtained. However, observable data are less likely to reflect the ideasof Hayek (1945) and Kirzner (1973), who emphasized the importance of unobservable
local information.
The correlation measure reflects the value of local information. To understandthis, it is helpful to modify the equations in corollary 8.
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.

(19)
(20)

As argued before, equation (19) measures firm-specific productivity, as an aggregate productivity shock must also increase wT . The two equations show that h canbe measured by the correlation between a firm-specific shock and labor input.

Itmeans that prediction ability can be estimated by how appropriate the adaptation to
the idiosyncratic changes are. Hence, despite potential problems, it is likely that thecorrelation measure contains useful information about the ability of firms 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 that determines firms’ predictions, 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 are lower not only in the covariance of value added and labor
expenses, but also in their variances.
Similarly to the estimation of ∆µ and σ 2u , if constant ka is a restrictive assumption
16

Alternatively, we can assume f (k) = Bkβ .

Then the following corollary is easily

proved.
Corollary 9 Suppose that w is constant and f (k) = Bkβ .

Then, the correla-tion between the unexpected shock and the reaction to the shock can be estimatedby the correlation between log Y − α (1 − β) log wT L − αβ log K − (1 − α) log wT andlog wT L − α (1 − β) log wT L − αβ log K − (1 − α) log wT .

4

Evidence from Japanese Data

This section implements empirical methodology using data from Japan.

First, I

describe the data and how to estimate each variable from this data.

Secondly,summary statistics are reported. Finally, the regression results are reported.

Data description: Proxies for Y , K, wT L, wT and r were constructed mainly fromthe Census of Manufacturing in Japan for 1985—99, provided by I-N Information
Systems, Ltd. Every year the Japanese Ministry of Economy, Trade and Industry
releases the Census of Manufacturing by city and industry. It covers all establishments in which four or more persons work as employers or employees.

However,

because of minor changes in the classification of industries and the integration anddivision of cities, the data released must be modified for use in panel data analysis.
I-N Information Systems, Ltd undertakes this modification and thereby enables panel
data analysis of the behavior of the average establishment by city and industry. Moredetails of the data and the construction of variables are given in the Appendix.
We split the data into two periods: 1985—91 and 1992—99. These periods roughly
correspond to before and after the burst of Japan’s economic bubble. We estimate
E [log Z1 (z, s)], E [log Z2 (z, s)], σ 2u and h using the constructed Y , wT L, wT , r and17

K, by city, industry and period. This estimation is based on sampled averages overtime.

It gives us estimates for the average establishments by city, industry andperiod.

Then, we estimate the representative values for the aggregate production

function from the weighted average of E [log Z1 (z, s)], E [log Z2 (z, s)], σ 2u and h byprefecture, industry and period, with the number of establishments in 1988 and 1996
as weights.

Finally, we take the diﬀerence of these representative values in two

periods. Hence, ∆µ, ∆σ 2u and ∆h are estimated by prefecture and industry.
In order to estimate TFP growth, we estimate the aggregate output, aggregate
capital and aggregate number of workers by prefecture and industry in 1988 and 1996.
Then ∆ log T F P is estimated by prefecture and industry. That is, we estimate theaggregate production function’s TFP growth between 1988 and 1996 by prefecture
and industry. We also estimate the weighted average of wT by prefecture and industry in 1988 and 1996. Then we estimate ∆ log wT from the diﬀerence between 1988
and 1996 by prefecture and industry. Our regression analyses use these estimatedvariables, ∆ log T F P , ∆ log wT , ∆µ, σ 2u , h, ∆σ 2u and ∆h, which span a cross-sectionof prefecture and industry. Precise definitions of variables are presented in the Appendix.
We need discussions for our empirical strategies. Firstly, our aggregate production
function is based on prefecture and industry. This is necessary for an empirical study.
Although we assume that parameters, α, β, µ, σ 2u , h are the same in theory for all
firms, it is less likely. The chemical industry demands more capital than the textile
industry.

Hence, the capital share in the chemical industry would be larger.

will certainly influence α and β.

It

The measure of h is supposed to capture theinformation advantages of a firm. A big city may have this advantage. Hence, it
is more reasonable to aggregate variables by region and industry. There is also analternative reason. For the purpose of estimation, we must assume that w is constant.

18

There could be objections that a constant w is not consistent with our theory, because
shifts in z e , h and σ 2u change w. However, as workers can move between regions andindustries, w is approximately the same in all regions and industries in a competitive
labor market. To the extent that z e , h and σ 2u do not change on average, a constantw is justified. In fact, the data support this assumption, as is subsequently shown.
Hence, to satisfy the assumption of constant w, we need to work with regional data.
Secondly, for the purpose of estimating unobserved variables, we split the data
into two periods. One of the assumptions of this estimation is the constant k a . It
requires the economy to be in a steady state. However, the steady-state assumption
is not consistent with the regression equation (3), because if the economy is in a´³
φ0 (ka )ka
steady state, gT F P ≈ 1 − φ(ka ) gT . In order to maintain consistency in our

analysis, we implicitly assume that the economy was in a steady state during 1985—
91, and 1992—99.

A large shock is assumed to have occurred around 1991, whichcaused the economy to move away from a steady state. The values of E [log Z1 (z, s)],E [log Z2 (z, s)], h and σ 2u during the transition period are approximated by the steadystate values of E [log Z1 (z, s)], E [log Z2 (z, s)], h and σ 2u .

Because the Japanese

economic bubble burst in 1991, it is not unreasonable to assume that the steady statechanged around this year.

Furthermore, the robustness of empirical results basedon these assumptions is checked by empirical results based on the assumptions off (k) = Bkβ .

Summary statistics: Table 1 reports the summary statistics of our estimates for∆ log T F P , ∆µ, ∆ log T , ∆σ 2u and ∆h. The annual TFP growth rate is about 2%
and aggregate productivity growth is 1.7%. This means that aggregate productivity
growth accounts for most growth in TFP, which is broadly consistent with the steadystate assumption.

19

On average, firm-specific productivity declined slightly (-0.0018 ∼ -0.0058), andthe level of idiosyncratic risk remained constant.

The average of this measure offirms’ prediction ability increased modestly (0.0029 ∼ 0.0035). This modest increasein prediction ability is confirmed by a simple correlation between an unexpected shock
and firms’ reactions to the shock (0.003 ∼ 0.0066). The increase in the prediction
ability measured during the 1990s is interesting. Although adjustment is expected
to be more diﬃcult during a recession, the data show that, on average, Japanesefirms become better able to adapt to idiosyncractic changes.

This suggests thatthe measure of prediction ability is not greatly aﬀected by recession.

As alreadydiscussed, the use of the correlation measure corresponds broadly to excluding theeﬀect of adjustment costs. The data lend some support to this argument.
Note that the movement of firm-specific productivity growth, risk and prediction
ability is much smaller than the movement of TFP growth. Moreover, changes infirm-specific productivity and prediction ability are opposite, and there is no movement in risk.

These observations imply that w would not be greatly aﬀected bychanges in these variables. This provides empirical justification for our assumptionthat w is constant.
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 our observations
do not satisfy this condition.

This apparent irrationality may indicate that someassumptions might be unrealistic. In particular, we assume that all firms know theunconditional mean of the shock, and we define an unexpected change as the deviationfrom the unconditional mean. If a firm’s subjective belief about the unconditionalmean of the shock diﬀers from the objective one, a negative correlation might bepossible. To check the robustness of the results, we used the simple correlation as
an alternative measure of prediction ability. Using this simple correlation, we can20

Variable

Obs. Mean (1988-1996) Std. Dev. Mean (annual)∆ log T F P¡¢∆µ f (k) = Bkβ8000.165∗

0.298

0.0206800-0.046∗0.287-0.0058∆µ (steady)800-0.014∗0.294-0.0018∆ log T¡¢∆ρuR(L(s)) f (k) = Bk β

800

0.139∗

0.088

0.0174800

0.053∗

0.322

0.0066800

0.024∗

0.315

0.003

587

0.028∗

0.152

0.0035592

0.023∗

0.153

0.0029800-0.0030.070-0.0004800-0.0030.070-0.0004∆ρuR(L(s)) (steady)¡¢∆h f (k) = Bkβ∆h (steady)¡¢∆σ 2u f (k) = Bkβ∆σ 2u (steady)Table 1: Summary statistics 1 (1988—1995)“Steady” means that the steady state is assumed, and “f (k) = Bkβ ” means thatf (k) is assumed to be Cobb—Douglas for the purposes of estimation. ∗ indicatessignificance at the 5% level.
Variable¡¢ρuR(L(s))t f (k) = Bk βρuR(L(s))t (steady)¡¢ht f (k) = Bk β

ht (steady)¡¢σ 2ut−1 f (k) = Bkβ
σ 2ut−1 (steady)Obs. Mean Std. Dev.

period

800

0.258

0.242

1992-1999

800

0.243

0.241

1992-1999

587

0.132

0.124

1992-1999

592

0.128

0.125

1992-1999

800

0.034

0.063

1985-1991

800

0.035

0.063

1985-1991

Table 2: Summary statistics 2“Steady” means that the steady state is assumed and “f (k) = Bkβ ” means thatf (k) is assumed to be Cobb—Douglas for the purposes of estimation.
21

use all the observations.
Table 2 reports the summary of statistics for σ 2ut−1 and ht .

Subscript t − 1means the variables are estimated data during 1985—1991, and subscript t means thevariables are estimated data during 1992—99.

The level of prediction ability, h, isabout 0.13 during 1992—1999. It seems like a small number, which is confirmed fromthe simple correlations, ρuR(L(s))t , which is around 0.25.

The simple correlations

ρuR(L(s))t are larger than h because h is the square of the simple correlations. This
small number might imply that the adaptation to shocks are not always appropriate.
On the other hand, it may be simply a result of measurement errors. The issue ofthe measurement errors is discussed below.

Regression results and their interpretation: Table 3 reports our regression
results.

All regressions show that a change in firms’ prediction ability increases

TFP. This is consistent with the predictions of our theory: an increase in prediction
ability increases TFP34 .
3This result is robust. For robustness checks, we included employment or the number of estab-lishments to control scale eﬀects. We also added the growth rate of employment to check whether
high correlation picks up the eﬀect of growing firms. Including these variables did not change results. We also used weighted least squares estimations using the square root of the number of cities
as weights. Some regression results are insignificant, but the coeﬃcients were always positive.
4

Takii (2004) shows that if political shocks are important, improvements in prediction ability

cause negative externality: if somebody is good at seizing opportunities, it reduces the opportunitiesof others. In this case, Takii (2004) shows that µ is endogenously chosen byµ=−ασ2u h σ 2u−.
(1 − α)2Although individual entrepreneurs react to political shocks given µ, as these reactions do not produce
a new value in the economy, prediction ability lowers µ. If this indirect eﬀect is important, although
we find positive coeﬃcient of ψ h , then the overall eﬀect of an increase in h could be negative. This22∆ log T F P = ψ 0 + ψ µ ∆µ + ψ 1σ ∆σ 2u + ψ T ∆ log T + ψ 2σ ht ∆σ 2u + ψ h σ 2ut−1 ∆h + ε
f (k) = Bkβ

f (k) = Bkβsteadysteady0.802∗∗∗

0.981∗∗∗

0.769∗∗∗

0.954∗∗∗(0.096)(0.118)(0.096)(0.114)

0.558∗∗∗

0.603∗∗∗

0.534∗∗∗

0.597∗∗∗(0.029)(0.035)(0.029)(0.034)

0.018

0.303

0.110

0.954∗∗∗(0.139)(0.302)(0.141)(0.281)∆ log T∆µ∆σ 2uρuR(L(s))t ∆σ 2u

σ 2ut−1 ∆ρuR(L(s))-0.399-0.681(0.388)(0.364)0.866∗

1.236∗∗∗(0.345)(0.330)ht ∆σ 2uσ 2ut−1 ∆h-1.799-4.17∗∗∗(1.440)(1.029)1.620∗

6.680∗∗∗(0.775)(0.977)0.079∗∗∗

0.062∗∗∗

0.066∗∗∗

0.045∗(0.016)(0.019)(0.016)(0.018)Adj R-squared

0.389

0.413

0.397

0.426

Obs.

800

587

800

592

constant

Table 3: The eﬀect of firms’ adaptability on TFP“Steady” means that the steady state is assumed and “f (k) = Bkβ ” means thatf (k) is assumed to be Cobb—Douglas for the purposes of estimation.
significance at the 5% level.

∗∗

indicates significance at the 1% level.

∗

indicates∗∗∗

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

23

indicatesIn order to examine its importance, we took the largest value of our estimate,ψ h = 6.7, for our exercise. As average risk is 0.035, a coeﬃcient of 6.7 implies that
if h changes from 0 to 1, TFP increases by 23%. At first glance, it does not look
like a small number.

However, note that ∆h increases annually by 0.0035.

Thismeans that an increase in h raised annual TFP growth rate by 0.08% in Japan. As
annual TFP growth is about 2%, the contribution of an increase in prediction ability
on TFP growth is less than 5%. This is a small number.
Although we implicitly assume that estimating a representative h for the aggregate production function eliminates measurement errors, this assumption might bequestionable.

A measurement error typically causes eﬀects to be underestimated.

This might be a reason for this small number. In order to examine this concern, we
estimated the coeﬃcient using a diﬀerent approach. Note the theory predicts
ψh =

α
.
2 (1 − α)We estimated average α from our sample. Our estimated average α = 0.54 when we
assume f (k) = Bkβ and 0.51 when we assume the steady state. Using this number,ψ h = 0.64 when we assume f (k) = Bkβ and 0.56 when we assume the steady state.
Both numbers are much smaller than 6.7. Hence, diﬀerent approaches do not change
results and the impact is small.
The estimated α = 0.54 is smaller than the commonly used number in the literature: Atkeson and Kehoe (2000) use 0.85 and Chang (2000) uses 0.8. Probably, it
is because Japanese establishments are much smaller than in the US. However, it isinteresting to estimate our ψ h using a common estimate. Take α = 0.85. It means
possibility is examined by the following regression equation.
∆µ = φ + φ1σ ∆σ 2u + φT ∆ log T + φ2σ ht ∆σ 2u + φh σ 2ut−1 ∆h + ε.
We find that φh is typically positive and not significant. Hence, this concern is dismissed.

24that ψ h = 2.8. Again this is smaller than 6.7. Hence, our tentative conclusion isthat the eﬀect of entrepreneurship on productivity growth is small.
This result is tentative because our estimates probably understate real eﬀects.
Firstly, a large measurement error is likely to lower estimated ∆h and σ 2u . As the main
reasons for the small contribution of h comes from small ∆h and σ 2u , the measurement
error is likely to lower the contribution of h. Secondly, if h is heterogeneous in anindustry and a prefecture, the estimated relation between aggregate productivity andthe average value of h would understate the real relation. Although heterogeneous
h is an attractive assumption, it requires estimating h from a smaller sample andamplifies measurement errors. Hence, to solve these problems, more disaggregateddata are required.

5

Conclusions and Extensions

This paper has presented an empirical framework investigating the social value ofthe ability to predict and adapt to idiosyncratic productivity shocks, which aims tocapture the soundness of firms’ judgments about economic impacts of idiosyncracticchanges. It provides a novel method of investigating the economic value of unobserved
information in macroeconomics.
Several extensions can be considered.

Firstly, more accurate data are called

for. Because of possible measurement errors, much disaggregate data is required for
precise estimations. This data requirement is one of the diﬃculties in implementing
our method. Recently, much evidence has been presented by plant-level data. Useof this data will improve understanding about entrepreneurship.
Secondly, incorporating an adjustment cost of investment and firms’ entry andexit might be important. As the main purpose of this paper is to position entrepreneurship in the aggregate production function, these factors are not considered in the25model. However, the consideration of these factors allowed the examination of the
dynamics of industry ( Lucas and Prescott (1971), Jovanovic (1982) and Hopenhayn
and Rogerson (1993)). Once we introduce the adjustment costs of investment andfirms’ entry and exit, a diﬀerence between persistent shock and independent shock
becomes important. A persistent shock together with the adjustment costs of investment raises the importance of prediction. This point is partially examined in Takii(2000), who finds a positive impact of prediction ability on the average Tobin’s Q.
However, its macro impacts have not yet been investigated.
Finally, investigating factors that enhance entrepreneurship is interesting.

Al-

though average prediction ability is estimated by industry and prefecture in this paper, we have said nothing about why prediction ability diﬀers between industries andprefectures. Prediction ability may be aﬀected by various factors, including inherited
ability, education, social networks, connections, region density, regional communication systems, and organizations.

As we have a well-defined measure of prediction

ability, it may be possible to empirically identify factors influencing entrepreneurship.

6

Appendix

Proof of Proposition 3: The derivation of equations (10) and (11) is straightforward. We explain the derivation of equation (12). Takii (2004) shows thataφ (k ) = fµ

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

26

(21)

The definition of φ (ka ) implies thatµ¶(1−α)mα−10af 0 (k) D,
φ (k ) = αf (k)
1−m
¸
∙
¸
∙
f (k) (1 − α) (1 − m)
dm
dm
−1
.
where D ≡ 1 + k a + 0
dk
f (k)
αm
dka

(22)

Equation (21) implies D = 1. Hence, it is shown that
φ0 (ka ) ka
= αθ (k) .
φ (ka )In addition, the first-order condition (2) implies that1αθ (k) = R z[f (k)L(s)]α T
rkT L(s)Q.E.D.

dQz (z).

Proof of Proposition 4: Equations (13) and (14) are immediately derived fromthe definition of the firm’s production functions. We utilized the fact that randomvariables do not aﬀect k.

Equation (15) is easily derived from the two first-orderconditions (1) and (2). Q.E.D.

Proof of Theorem 7: 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:¸∙Zσ 2u hhe,zdQ (z|s) = z exp hs −
2
27

(23)

´
³
2
where z e = exp µ + σ2u . Applying equation (23) to equation (18), it can be shown© eª 1
that L∗ = zw α [f (ka ) − f 0 (ka ) ka ] 1−α f (ka )µ¶ 1σ 2u h 1−α.
L (s) = L exp hs −2∗

1
Hence, the firm’s reaction to the shock R (L (s)) is given by (1−α)

the definition of the correlation coeﬃcient implies that
ρuR(L(s)) =

h
i
σ2u h
hs − 2 . Hence,√h.

Q.E.D.

Data Appendix:
The Detail of Data description: Although the census covers all establishments inwhich four or more persons worked as employers or employees, by city, industry and
year, 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 privacy of establishments.
To improve estimations of the correlation, we exclude entities for which there are
missing variables in any period.
The industries covered by the census are food, drink/tobacco/animal feed, textiles, apparel, lumber/wood products, furniture/fixtures, pulp/paper, printing, chemicals, petroleum/coal products, plastic products, rubber, leather/leather products,
pottery/glass products, iron/steel, nonferrous metals, metal products, general machinery, electrical machinery, transportation machinery, precision tools, weapons andother industries.

Data Construction: As shown below, the estimation of variables requires several
steps. This might cause the reader to be suspicious. Although reported results are
based on the following estimation method, we conducted several regressions to gain28

preliminary results using diﬀerent methods of estimations. The results are more orless the same as our reported results.

Y and wT L : Gross value added and labor expenses are divided by the number of
establishments. These values are then deflated by the GDP price deflator.
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:
Kciy+1 = Kciy + (Fciy+1 − Fciy ) /pIy+1 , if Fciy+1 > Fciy ,= Kciy + Fciy+1 − Fciy , if Fciy+1 ≤ Fciy ,where Kci85 = Fci85 /pI85 and Fciy are average fixed tangible assets per establishmentof cth city and ith industry in year y, and pIy is a price deflator for investment goods inyear y, which is taken from Keizai Tokei Nenkan (2002) (Annual Economic Statistics
2002) by Toyo Keizai Shinpo Sya.

As we do not have data on investment, thissimplified estimation method was used as an approximation, which is the approachtaken by Nishimura, Nakajima and Kiyota (2003). The subscript y spans the period
between 1985 and 1999.
The Census of Manufacturing only reports fixed tangible assets for establishments
in which the sum of employers and employees is at least 10 [size group 2]. Hence,the capital stock of establishments in which the sum of employers and employees is
between four and nine workers [size group 1] is estimated as follows. First, averagelabor expenses per establishment in size groups 1 and 2 are estimated by city, industry
and year. Average fixed assets per establishment are then regressed on average labor
expenses per establishments and city dummies in size group 2 for each industry. The
parameters of this regression are used to estimate the capital stock in size group 1.
The Census of Manufacturing reports labor expenses for size groups 1 and 2 by29

industry and year, but not by city. Hence, we need to estimate this by city, industry
and year.

For this purpose, the following estimation method is used.

First, theaverage labor expenses per establishment are estimated by size group, industry andyear. Average labor expenses per establishment for each size group are estimated bycity, industry and year from the following equation:wT Lsciy =wT LsiywT Lciy ,wT L1iy + wT L2iy

where wT Lsciy is the average labor expense per establishment in the cth city, ith industry and size group s in year y, wT Lciy is the average labor expense per establishment
in the cth city and ith industry in year y, and wT Lsiy is the average labor expense
per establishment in the ith industry and size group s in year y.
wT : wT is estimated by prefecture, industry and year, using the weighted average ofdeflated labor expenses over the number of employees. The number of establishments
is the weight. Because the number of employees is not reported, this is estimated.
The Census of Manufacturing reports the number of employees, the sum of employees
and employers, and the number of establishments by industry and year. Assumingthat the ratio of employees per establishment to the sum of employers and employees
per establishment is the same in each industry and year, the number of employees
was estimated by city, industry and year:
Lciy ≡

µ

L
E+L

¶

(E + L)ciy ,iywhere Lciy is the number of employees per establishment in the cth city and ith
industry in year y, (E + L)ciy is the sum of employers and employees per establishment¡ L ¢is the ratio of employees perin the cth city and ith industry in year y, and E+Liy
establishment to the sum of employers and employees per establishment in the ith
industry in year y.

30

r : The return to the capital stock is estimated by using
riy = pIy (iy + δ i ) ,where iy is the yield on 10-year government bonds in year y, and δ i is the average
depreciation rate over average fixed tangible assets of the ith industry.

The yield

data are from the website of The Bank of Japan. Average depreciation and average
fixed tangible assets are taken from the Census of Manufacturing. As in Nishimura
et al. (2003), changes in the price deflator for investment goods are ignored, as thisindex increased so much during the bubble in Japan that the user cost of capital
became negative. Because r is only used to estimate the average capital share overtime, this simplification is unlikely to aﬀect our results.

To check for robustness,

we also used the return to the capital stock in Hayashi and Prescott (2002), which istaken from Hayashi’s website. As this did not change the results, we do not reportthem in this paper.

∆ log TFP : We estimated the average capital share of each firm from the sampleYaverage of rK
over time by city and industry.

Then the weighted average of thecapital share was estimated by prefecture, industry and period, with the numberof establishments in 1988 and 1996 as weights. Unless otherwise stated, the same
weights are used to estimate the prefecture average. The average of the capital share
0

a

a

(k )k
over the period was chosen to estimate φ φ(kValue added, capital stock and thea) .

sum of employees and employers were aggregated by industry and prefecture in 1988
and 1996. Then ∆ log T F P was estimated as defined.
¸ µ 0 a a¶ ∙¸∙φ (k ) k
Kpi96
Ypi96
Ypi88
Kpi88
∆ log T F Ppi = log
−
log
,
− log
− log
Npi96
Npi88
φ (ka )
Npi96
Npi88
pi
⎤
⎡
µ 0 a a¶Xφ (k ) k
1
nci96
1
n
1 ⎢X
⎥
P
P ci88
=
+
⎦,
⎣
P
P
Y
Yciy
ciy
a
φ (k )
2 c∈C {85≤y≤99} riy Kciy
{85≤y≤99} riy Kciyc∈Cp nci96c∈Cp nci88
pi
c∈C
p

p

15

15

31

where Ypiy , Npiy , Kpiy are aggregate output, the sum of employees and employers, thecapital stock of the pth prefecture and ith industry in year y, and nciy is the numberof establishments in the cth city and ith industry in year y.

Moreover, riy is thereturn to the capital stock of the ith industry in year y, and Cp is the set of cities in
prefecture p.
∆ log wT : wT in 1988 and 1996 is chosen for this estimation.
∆ log (wT)pi = log (wT)pi96 − log (wT)pi88
where (wT)piy is the weighted average of the wage rate in the pth prefecture and ith
industry in year y.
Y∆µ : We estimate α, α (1 − β) and αβ from the sample averages of rK+wT, Y and
L wT LY
over time by city and industry, respectively.
rK

Using the estimated α, α (1 − β) andαβ, E [log Z1 (z, s)] and E [log Z2 (z, s)] are estimated by their sample averages overtime by city, industry and period. The weighted averages of these values are calculated by prefecture, industry and period. Using these values, ∆µ is then estimatedby prefecture and industry:
(P
)
(P
)
X
X
nci96
nci88
{92≤y≤99} [log Z]ciy{85≤y≤91} [log Z]ciyPP∆µpi =
−
8
n
7
ci96
c∈Cp
c∈Cp nci88
c∈C
c∈C
p

p

where [log Z]ciy = log Yciy − αci log [wT L]ciy − (1 − αci ) log [wT ]piy or [log Z]ciy =log Yciy − [α (1 − β)]ci log [wT L]ciy − [αβ]ci log Kciy − (1 − αci ) log [wT ]piy andαci = P1
Yciy
{85≤y≤99} riy Kciy +(wT L)ciy

15, [α (1 − β)]ci = P

1
Yciy{85≤y≤99} (wT L)ciy15, [αβ]ci = P

1
Yciy{85≤y≤99} riy Kciy

15

[wT L]ciy means the average wage payments of the average establishments in the cth
city and ith industry in year y, and [wT ]piy is the weighted average of the wage ratein the pth prefecture and ith industry in year y. Cp is the set of cities in prefecture
p.
32

.

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.
⎧
⎡v
⎤⎫2
P
h
i2
uP
⎪
⎪
[log
Z]
y∈Θ
ciy
⎪
⎪
t
⎨ X ⎢ut y∈Θt [log Z]ciy −
⎥⎬
¡ 2¢
mt
n
cix
⎢
⎥
P
,
σ u pit =⎣⎪mt
ncix ⎦⎪
⎪
⎪
c∈C
p
c∈C
⎭
⎩ p

where mt is the number of years during period t, ncix is the number of establishments in the cth city and ith industry in year x.
t = 2 then x = 96.

If t = 1 then x = 88, and if

Θt is the set of years during period t, where t ∈ {1, 2} andΘ1 = {85, 86, 87, 88, 89, 90, 91} and Θ2 = {92, 93, 94, 95, 96, 97, 98, 99}. See the construction of ∆µ for the definition of [log Z]ciy , and Cp .
h : To implement corollaries 8 and 9, the correlations were 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.

£¤ρuR(L(s)) pit
hpit

⎧
⎫
⎪
⎪
P
P
h
i
h
i
⎪
⎪
P
⎪
⎪
y∈Θt log Zciyy∈Θt log Wciy
⎪
⎪
⎨
⎬
log
Z
log
W
−
−
X
ciy
ciy
y∈Θt
mt
mt
ncix
P
s
,
=
∙
¸2
∙
¸2
P
P
⎪ P
P
⎪
c∈Cp ncix ⎪y∈Θt log Zciyy∈Θt log Wciyc∈Cp ⎪⎪⎪y∈Θt log Zciy −y∈Θt log Wciy −
⎪
⎪
mt
mt
⎩
⎭
mt

£
£
¤2
¤
= ρuR(L(s)) pit , if ρuR(L(s)) pit ≥ 0.

mt

where log Wciy = log [wT L]ciy − αci log [wT L]ciy − (1 − αci ) log [wT ]piy or log Wciy =log [wT L]ciy − [α (1 − β)]ci log [wT L]ciy − [αβ]ci log Kciy − (1 − αci ) log [wT ]piy .

Seethe construction of ∆µ for the definition of [log Z]ciy ,αci , [α (1 − β)]ci , [αβ]ci and Cp ,and the construction of σ 2u for the definition of ncix , Θt and mt .

33

References[1] Atkeson, Andrew and Patrick, Kehoe J. (2000), “The transition to a new economy following the Second Industrial Revolution,” Working Paper 606, Federal
Reserve Bank of Minneapolis.
[2] Chang, Yongsung (2000), “Wages, Business Cycles, and Comparative Advantage,” Journal of Monetary Economics, 46, 143—171.
[3] Davis, Steven J. and Haltiwanger, John (1999), “Gross Job Flows,” In Handbookof Labor Economics, Volume 3, Ashenfelter, Orley and Card, David (eds), 2711—
2805.
[4] Hassler, John and Mora, José V., Rodríguez (2000), “Intelligence, Social Mobility, and Growth,” American Economic Review, 90, 888—908.
[5] Hayashi, Fumio and Prescott, Edward C. (2002), “The 1990s in Japan: A Lost
Decade,” Review of Economic Dynamics, 5, 206—235.
[6] Hayek, Friedrich (1945), “The Use of Knowledge in Society,” American Economic
Review, 519—530.
[7] Hubbard, Thomas, (2003), “ Information, Decisions, and Productivity: OnBoard Computers and Capacity Utilization in Trucking,” American Economic
Review, 1328—1352.
[8] Hopenhayn and Rogerson (1993), “Job Turnover and Policy Evaluation: A General Equilibrium Analysis,” Journal of Political Economy, 101, 915—938.
[9] Holmes, T. J., and Schmitz, J. A. (1990), “A Theory of Entrepreneurship and its
Application to the Study of Business Transfers,” Journal of Political Economy
98, 265—294.
34

[10] Jovanovic, Boyan (1982), “Selection and the Evolution of Industry,” Econometrica, 50, 649—670.
[11] Kirzner Israel M. (1973), Competition and Entrepreneurship, University of
Chicago Press.
[12] Kirzner, Israel M. (1997), “ Entrepreneurial Discovery and the Competitive Market Process: An Austrian Approach,” Journal of Economic Literature, XXXV,
60-85.
[13] Lucas, Robert E. (1978),“On the Size Distribution of Business Firms,” Bell Journal of Economics, 9, 508—523.
[14] Lucas, Robert E., Jr. and Prescott, Edward C.(1971), “Investment and Uncertainty,” Econometrica, 39, 659—682.
[15] Mas-Colell, Andreu, Whinston, Michael and Green, Jerry R. (1995), Microeconomic Theory, New York, Oxford University Press.
[16] Nishimura, Kiyohiko, Nakajima, Takanobu, and Kiyota, Kozo (2003), “Ushinawareta 1990 nendai, Nihon Sangyo ni Naniga Okottanoka?–Kigyo no Sannyu
Taisyutsu to Zenyouso Seisansei,” RIETI Discussion Paper Series 03-J-002.
[17] Rosen, Sherwin (1982), “ Authority, Control and the Distribution of Earnings,”
Bell Journal of Economics, 311—323.
[18] Schultz, Theodore W. (1975), “The Value of the Ability to Deal with Disequilibria,” Journal of Economic Literature, 13, 827—846.
[19] Takii, Katsuya (2000) “Prediction ability and Investment under Uncertainty,”
Discussion Paper Series 518, University of Essex.

35[20] Takii, Katsuya (2003a), “Prediction Ability,” Review of Economic Dynamics 6,
80—98.
[21] Takii, Katsuya (2003b), “The Value of Adaptability,” OSIPP Discussion Paper:
DP-2003-E-003.
[22] Takii, Katsuya (2003c) “Entrepreneurial Eﬃciency,” OSIPP Discussion Paper:
DP-2003-E-010[23] Takii, Katsuya (2004), “Entrepreneurial Eﬃciency: Theory,” OSIPP Discussion
Paper: DP-2004-E-005.

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