OSIPP Discussion Paper : DP-2004-E-005

Entrepreneurial Efficiency: Theory
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 models firms' entrepreneurial ability according to their ability
to predict changes in productivity (i.e., their prediction ability), and derives anaggregate production function as a result of entrepreneurship. An increase infirms' prediction ability improves allocative efficiency in a competitive economy,
but can reduce it when opportunities are distorted. It is shown that prediction
ability can aggravate distortions in the presence of political risk.

Becauseefficiency affects the total factor productivity (TFP) of an economy, the model can
explain how entrepreneurship influences TFP.
This paper contains material of a previous working paper, Takii (2003b). I would
like to thank Boyan Jovanovic and seminar participants at the University ofEssex, the Kansai Macro Workshop, Osaka Prefecture University, OSIPP andSogang University for helpful comments. 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 is well known that entrepreneurship plays an essential role in several economic
activities. When new technology is discovered, when demographic change occurs or
when 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 to develop a formal model of entrepreneurship in macroeconomics.

Theabsence of entrepreneurship from macroeconomics can be explained in part by theextensive use of an exogenously given aggregate production function. As the relation
between output and inputs is given by the aggregate production function, there is noneed for an economic agent who finds a productive use for inputs.
Of course, we agree that the aggregate production function contains many usefulproperties in macroeconomics. In particular, it can account for the main features of
economic growth in many OECD countries. However, recent evidence from plantlevel data suggests that the weaknesses of the traditional aggregate production function may not be negligible. This evidence shows 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)].
The usefulness of the aggregate production function will be greatly improved if we
can incorporate the ability to allocate resources in the aggregate production function.
This paper models a firm’s entrepreneurial ability by the ability to predict changes
in the productivity of that firm (i.e., a firm’s prediction ability), and analyze how entrepreneurship influences an aggregate economy, where entrepreneurship is defined as
the activity of allocating resources to size opportunities. In this model, the aggregate
production function is derived from firms’ reactions to these changes. Therefore, we
can analyze how firms’ prediction ability aﬀects the macroeconomy, while retaining
the advantages of the aggregate production function.
2

Although economists estimate productivity from data, in reality, it is never observed. It has to be predicted by managers.

Individual managers can observe asequence of changes in the environment and predict what might influence their production or sales. They might observe that traﬃc in their local area increases, thatnew firms attract many skillful engineers to their firms or that a politician who isconnected with their company loses an election. Since productivity is estimated by
the ratio of measured output to input, all of them can potentially influence measured
productivity. Hence, predicting a change in the productivity of their own firm means
predicting how various changes in the environment directly influence production orsales. That is, prediction ability in this paper aims to capture the soundness of firms’
judgment about economic impacts of the changes.
When a change in productivity occurs, the marginal products of inputs deviate
from input prices, and this generates opportunities for entrepreneurs to exploit. If
entrepreneurs appropriately predict the change and react to it, the deviation of themarginal products of the inputs from input prices will be small. Hence, the improvement in firms’ prediction ability raises allocative eﬃciency, and therefore increases
productivity in the economy.

It is shown that the increased prediction ability offirms raises the total factor productivity (TFP) of the aggregate production function
in a competitive economy.
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 gen-3

erates negative externalities: when an entrepreneur is good at taking opportunities,
there are fewer opportunities for others. It is shown that increased firms’ prediction
ability can lower TFP.
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) defines entrepreneurial ability as the ability to interpret new information and allocate resources to profitable opportunities. Althoughboth researchers insist that equilibrium analysis is not suitable for understanding theimportance of entrepreneurship, we suggest that there are benefits 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.
This strategy has been adopted by Holmes and Schmitz (1990) and Takii (2003a).
Holmes and Schmitz (1990) incorporate a Schultzian entrepreneur into an equilibrium
analysis, and emphasize the importance of the division of labor between entrepreneurs
and managers. Takii (2003a) examines the allocation of prediction ability in a market
economy. This paper diﬀers from theirs in two respects: (1) it identifies a social roleof entrepreneurship as influencing allocative eﬃciency; and (2) it incorporates thisrole into the aggregate production function. Since the aggregate production function
is a cornerstone of the neoclassical growth model, the model potentially can be applied
to broad topics.
In order to demonstrate the usefulness of the model, we extend the model to analyze the eﬀect of rent-seeking activities on an aggregate economy. The result contributes to a controversy in the literature on pressure groups. Becker (1983) showsthat competition between pressure groups can lead to eﬃcient allocation.

On theother hand, Tullock (1967) argues that rent-seeking activities waste resources. The
detrimental eﬀect of entrepreneurs’ rent-seeking activities has recently been reempha-

4

sized in the context of economic growth [e.g., by Baumol (1990), Murphy et al. (1991)
and Holmes and Schmitz (2001)]. In contrast to the previous literature, this paper
shows that, even if rent-seeking activities are costless, competition between pressuregroups 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 underestimatedby the existing literature.
The negative eﬀect of prediction ability on TFP also has an alternative implication. Since prediction ability is determined by the accuracy of information, theresult means that accurate information can reduce welfare. Blackwell (1953) showsthat accurate information always improves a decision maker’s ex ante utility in an
individual decision problem. On the other hand, Hirshleifer (1971) points out that
accurate information can lower welfare in a pure exchange economy because it makes
risk sharing more diﬃcult. Eckwert and Zilcha (2001) introduce a production sector and investigate a trade-oﬀ between the welfare gain due to a better allocation of
resources and the welfare loss due to worse risk sharing. This paper provides an alternative reason why accurate information lowers welfare: more accurate information
can aggravate distortion in a distorted economy.
This paper is organized as follows. The next section describes the model. Section
3 defines the measure of prediction ability and analyzes the macroeconomic eﬀect offirms’ prediction ability. Section 4 extends the model to analyze the eﬀect of politicaluncertainty.

Section 5 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 entrepre5

neurship in the next section.
An agent can be an entrepreneur or a worker. Each firm needs one entrepreneur
to organize the firm. In this model, for simplicity, there is no distinction between
entrepreneurs, managers and firms. This simplification is made to develop a tractablemodel that focuses on firms’ entrepreneurship.
Firms are continuously distributed on [0, mN], where m ∈ (0, 1) is the proportionof entrepreneurs in the total population, N. This implies that agents are assumed
to be identical. Although the lack of heterogeneity among entrepreneurs forces us toignore the distributional aspect of the original model in Lucas (1978), it allows us tofocus on a diﬀerent economic problem: the eﬀect of entrepreneurship on productivity
in an economy. In fact, owing to this simplification we derive a rather simple condition that the derived aggregate production function exhibits concavity and Inada
conditions, which are essential assumptions for the neoclassical growth model.
The representative entrepreneur’s problem is described first, and then the market
equilibrium is defined.

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 [F (Ki , 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.
It is assumed that F exhibits constant returns to scale in K and L.

By definingf (k) = F (k, 1), where k = K, we can express F (K, L) as a function of capital per
Lworker in production: F (K, L) = f (k) L.

We assume that f 0 (.) > 0, f 00 (.) < 0,limk→0 f (k) = 0, limk→0 f 0 (k) = ∞ and limk→∞ f 0 (k) = 0.
6

The parameter αmeasures the span of control.

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

decreasing returns to scale in Ki and Li .
Idiosyncratic changes occur everywhere in reality. However, some influence productivity; others not.

The random shock zi summarizes changes in productivity

due to all idiosyncratic changes.

Because the movement of zi is unpredictable exante, when zi changes the entrepreneur must predict the direction and magnitudeof 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 therealization of zi can be inferred. It is assumed that the entrepreneur’s inference isbased on a conditional distribution function, Qh (z|s), where h measures the entrepreneur’s ability to predict changes in z. The conditional distribution function is thesame 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 specified 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 financial market is complete. Therefore, entrepreneurs canhedge against any idiosyncratic risk. Entrepreneurs maximize their firm’s expected7profits:
π (si ) = max
k,L

½Z

α

¾

h

zi [f (ki ) Li ] dQ (zi |si ) − wLi − rki Li ,where w is the wage rate 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.

Because theproduction function is strictly concave in k and L and satisfies the Inada conditions,
there exists a unique interior solution, and the first-order conditions are necessary
and suﬃcient for the maximization problem.
Note that the two first-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).

Idiosyncratic shocks cause marginal

products to deviate from marginal costs, and these deviations provide opportunities
for entrepreneurs to exploit. If entrepreneurs predict the direction and magnitude of
the changes clearly, they can take the opportunities. This is the aspect of entrepreneurship that we emphasize in this model, and that is also stressed by Kirzner (1973)
and Schultz (1975).
The following is derived from the two first-order conditions:wf (k (si )) − f 0 (k (si )) k (si )=.
rf 0 (k (si ))(3)This equation implies that the capital stock per worker in production k (si ) does notdepend on the realization of the signal si .

Hence, we denote this by k.

As theright-hand side of equation (3) is strictly increasing in k, k is uniquely determined by
w
.
r

8

Expected profits are derived by substituting the two first-order conditions into
π (si ):Z

π (s) dQhs (s) =Z(1 − α)(w + rk) L (s) dQhs (s) .
α(4)Equation (4) shows that expected profits are proportional to the costs of production.

The arbitrage condition: Because entrepreneurs can completely hedge their risks
in the financial market, they do not bear risk. As agents are identical and can beentrepreneurs or workers, expected profits 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) = w.

(5)

Hence, equations (4) and (5) implyw=

Z(1 − α)(w + rk) L (s) dQhs (s) .
α(6)Resource constraints: To close the model, the labor and capital markets mustclear:
K a = mNkT
(1 − m) N = mN

Z

Z

L (s) dQhs (s) ,L (s) dQhs (s) ,

(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 firms and k L (st ) dQ (st ) is the averagefirm’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 defined as follows:9Definition 1 A market equilibrium is {L (·) , k, w, r, m} that satisfies the following
conditions.
1. The firm’s profit maximization conditions: equations (1) and ( 3).
2. The arbitrage condition: equation (6).
3. The resource constraints: equations (7) and (8).

0.
Let us define 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 ka ≡ KN ∈ (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 becomes

infinite, the labor share does not converge to 0. That is, the wage rate must increaseat a faster rate than employment falls in the limit. This guarantees that m has asolution in (0, 1).

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

Let y a denote aggregate output per capita.

The following propositionshows that entrepreneurs’ ability to predict changes in the productivity of the firms
can influence the TFP of the aggregate production function. The proof is given in
the Appendix.
10

Then for any ka ∈ (0, ∞), there

Proposition 3 Suppose that limk→∞ θ (k) < 1.
exists an aggregate production function:y a = z (h) φ (ka )where z (h) ≡

"Z ∙Z

a

φ (k ) ≡ f

Ã

(9)
¸ 1

zdQh (z|s)

ka
1 − m (ka )and m (ka ) ∈ (0, 1) is a solution of
"

!α

Ã

α
ka
1−θ
1−α
1−m

1−α

#(1−α)

dQhs (s),m (k a )(1−α) [1 − m (ka )]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
satisfies 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 ka ∈ (0, ∞), andsatisfies the Inada conditions:
φ0 (ka ) > 0, φ00 (ka ) < 0.
φ0 (ka ) = ∞lim φ (ka ) = 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 satisfies all the important assumptionsof the aggregate production function in the neoclassical growth model.
11

3

The Macroeconomic Eﬀects of Entrepreneurship

In this section, we specify the information structure that entrepreneurs can access,and we examine the macroeconomic eﬀects of entrepreneurship. TFP, z (h), is shown
to be an increasing function of prediction ability.
Since a firm’s prediction ability is determined by the accuracy of information,
it can be expressed by informativeness, in the sense of Blackwell (1953). Without
loss of generality, the distribution functions are assumed to be continuous and theconditional distribution Qh (zi |si ) can be derived fromdQh (zi |si )h= qz/s(zi |si ),dzih(si |zi )qz (zi )
qs/z
h
qz/s
(zi |si ) = R h,
qs/z (si |zi )qz (zi )dzi

hwhere qs/z
(si |zi ) is a conditional density function given zi and qz (zi ) is a prior density

function zi . Assume that h is ordered by the informativeness, in the sense of Blackwell(1953). That is, h1 ≥ h2 if and only if there exists a nonnegative function φ (s2i |s1i ),which satisfies the following three relations:

Z

³

Z

φ s2i |s1i ds2i = 1, a.e. ∀s1i ,Z

φ s2i |s1i ds1i ∈ (0, ∞) , a.e. ∀s2i .

´

1

³

´

³

´

³

´

2

³

´

h
h
φ s2i |s1i qs/z
s1i |zi ds1i = qs/z
s2i |zi , a.e. ∀zi , ∀s2i

Blackwell (1953) proves that if h is ordered by this criterion, information structure
h1 brings higher ex ante utility than information structure h2 for any utility function. Using Blackwell’s (1953) definition of informativeness, it is shown that z (h) is
increasing in h.
Theorem 5 If h is ordered by informativeness in the sense of Blackwell (1953), thenz (h) is increasing in h.
12

1

h
Proof. As E h [zi |si ] 1−α is convex in E h [zi |si ] and E h [zi |si ] is convex in qz/s
(zi |si ),1hE h [zi |si ] 1−α is convex in qz/s
(zi |si ). From Theorem 2 in DeGroot (1970, p. 436),

the result is immediate.
The result shows that if firms have more accurate information, the productivityof the economy increases.

When changes in productivity occur, if entrepreneurs

can accurately predict the direction and magnitude of the changes, the deviationsof actual marginal productivities of inputs from input prices will be small. Hence,
an improvement in firms’ prediction ability raises allocative eﬃciency, and therefore
increases the productivity of the economy.

The Operational Measure of h: In order to make the model more operational, we
make a further assumption on information structure. Assume that log z comprises a
predictable component µ 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 a change in productivity. The signal s
is correlated with u with an error:
s=u+ε
where ε is normally distributed with mean 0 and variance σ 2ε (h). We can apply
Takii’s (2003a) measure of prediction ability in this context. Let Qhu (u|s) denote theconditional distribution of u given s. The measure of prediction ability , u, is defined
as follows.
Definition 6 The measure of prediction ability about u is defined by:
h=1−

R

V ar (u|s) dQhs (s)
,
σ 2u
13

where V ar (u|s) =

R³

R

´2

u − udQhu (u|s)

dQhu (u|s).

This measure implies that the entrepreneur accurately predicts u when she reduces on average the conditional variance having observed s.
in diﬀerent environments,

R

To compare ability

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 u, h = 1, whereas if the entrepreneur does not predict u at all, h = 0.
Using the definition 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 predicts u, the variance of thenoise term is smaller. When h = 1, the variance is 0, and when h = 0, the varianceis infinite.
An advantage of this measure is that it can be estimated. The proof is found in
Takii (2004).
Proposition 7 In the steady state, the measure of prediction ability, h, can be estimated by the correlation between log Y − α log L − log w and log L, where α =R

1Y (z,s)[ rK(s)+wL(s)]dQhzs (z,s).

The correlation measure reflects the value of unobserved local information, the im-portance of which is emphasized by Hayek (1945). Note that log Y − α log L − log w
measures firm-specific productivity, since an aggregate productivity shock must also
increase w.

Hence, the proposition shows that h can be measured by the correla-tion between a firm-specific shock and labor input.

It means that an appropriate

adaptation to a firm-specific shock can be interpreted as evidence of high prediction
ability1 . Since a firm-specific shock is idiosyncratic, the information needed to adapt
1

In fact, Takii (2003b) argues that h can be interpreted as the measure of adaptability.

14

the shock must be local information. Takii (2004) estimates the measure of prediction ability and conducts an empirical study, which provides evidence that supports
the prediction of this paper.

We refer to Takii (2004) for more discussion of theempirical implementation of the model.

Macroeconomic eﬀects of entrepreneurship: Using this measure, z (h) can be
decomposed into productivity, risk and prediction ability. The proof is given in the
Appendix.
Theorem 8 For any ka ∈ (0, ∞), a rise in h increases GDP per worker in an
economy, y a :"#ασ 2u hy a = z e φ (ka ) exp.
2 (1 − α)(11)The theorem provides an operational model of entrepreneurship in macroeconomics. It explicitly shows how prediction ability influences the TFP of the aggregate production function. Since the aggregate production function is a cornerstoneof the neoclassical growth model, the theorem potentially can be applied to broad
topics. One application is demonstrated in the next section.

4

Political Risk and Entrepreneurship

In the previous section, it was argued that entrepreneurship can improve allocativeeﬃciency and the TFP of an economy, because when entrepreneurs predict changes intheir firms’ productivity correctly, they can equate the marginal products of inputs totheir prices. However, in this section, we use an example to show that if opportunities
are distorted, prediction ability can lower the TFP of an economy. For this purpose,
idiosyncratic distortions in Restuccia and Rogerson (2003) are introduced in ourmodel.
15

Assume that government subsidizes firms and that all subsidies are financed byincome tax:
zi = (1 − τ ) (1 + Gi )where τ is the constant average and marginal income tax rate and Gi is the subsidy
rate for the ith firm.

The tax is proportional to output. Since output must be

equal to the sum of profits, wage payments and rental costs of capital, the tax is also
proportional to all of them in market equilibrium. Hence, τ is a short-cut expression ofa flat-rate income tax on all income of a representative household. It is also assumed
that the amount of subsidy is also proportional to output and that the technologicalproductivity of each firm is unity.

A similar distortion appears in Restuccia andRogerson (2003). Their notion of ‘output tax’ is the same as our 1 − zi . Restuccia
and Rogerson (2003) argue that ‘output tax’ allows us to analyze a generic family of
idiosyncratic distortions, which causes the reallocation of resources across firms.
The subsidy rate has two components: a predictable component g (Ri ) and an
unpredictable component ui .
log (1 + Gi ) = g (Ri ) + ui
where Ri is the rent-seeking activity of the ith firm 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 and that the cost of rent-seeking activities is 0 when Ri ≤ R∗ andinfinite when Ri > R∗ .

Then, because entrepreneurs are identical, all choose thesame level of 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.
16

There is an additional constraint that does not appear in the previous section, agovernment’s budget constraint:τZ(1 + Gi ) [f (k) L (si )]α dQhus (ui , si ) mN =Z

Gi [f (k) L (si )]α dQhus (ui , si ) mN
(12)

where L(s), k and m are the market equilibrium solutions to the previous problem.
The right-hand side represents tax revenue and the left hand side represents total subsidies. In order to satisfy equation (12), µ is endogenously determined. Rearranging
the equation, it is rewritten as

0=
Note thatRZ(1 − zi ) [f (k) L (si )]α dQhzs (zi , si ) mN.

[f (k) L (si )]α dQhzs (zi , si ) mN and

R

(13)

zi [f (k) L (si )]α dQhzs (zi , si ) mN are

the values of aggregate output before and after the transfer of income, respectively.
Hence, equation (13) requires that the income transfer does not change aggregate
output. However, since L (s) is chosen when entrepreneurs expect income transfer,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 firms’ prediction ability lowers TFP when there ispolitical risk. The proof is given in the Appendix.
Theorem 9 Suppose that µ satisfies equation (13 ). Then, for any ka ∈ (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

(14)

Political risk reduces TFP because political risk generates a negative externality:
when an entrepreneur is good at taking opportunities, there are fewer opportunities17

for others. In fact, it is shown that when equation (13) is satisfied, 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, prediction ability lowers µ tosatisfy equation (13). This generates a negative externality.
Two comments are warranted.

First, equation (14) shows that, if there is nopolitical risk, competition between pressure groups leads to eﬃcient allocation, as
suggested by Becker (1983). More importantly, equation (14) 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.

Hence,

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

5

Conclusions and Extensions

This paper models firms’ entrepreneurial ability by the ability to predict changes in
the productivity of firms, which is aimed at capturing the soundness of firms’ judgment about economic impacts of changes in the environment, and analyzes how entrepreneurship influence the aggregate economy. We have shown that greater prediction
ability raises total factor productivity (TFP) in a competitive economy. However,greater prediction ability may lower TFP if opportunities are distorted.
An advantage of the model is that we can empirically examine its predictions.
Takii (2004) conducts an empirical study with this model. Using data from the Census of Manufacturing in Japan, 1985—1999, Takii (2004) estimates firms’ prediction
ability and examines the eﬀect of prediction ability on TFP by prefecture and indus18

try. Takii (2004) suggests that a rise in prediction ability had a small but significant
positive impact on TFP growth in Japan during that period.
Several extensions are being considered. First, although it has been assumed forsimplicity 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.
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)].

Because the model in this paper relates

entrepreneurs’ rent-seeking activities to TFP, it may provide a suitable empiricalframework within which to examine the extent to which rent-seeking activities explainthese problems.

6

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:
k=
a

ka
,
1−m

(15)

where k a = KN . From equations (3), (6) and (8), we derive1−mα=[1 − θ (k)] ,
m
1−α

19

(16)

0

where θ (k) ≡ ff(k)k∈ (0, 1). Define a function G (m, k):
(k)
"

Ã

ka
α
G (m, ka ) ≡
1−θ
1−α
1−m

!#

−

1−m
m

(17)

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

(18)

It can be shown that limm→0 G (m, ka ) < 0 since when m converges to 0, the second
term of equation (17) goes to ∞ and the first term is finite.

When m converges

to 1, the second term of equation (17) goes to 0 and the first term is positive, since
limk→∞ θ (k) < 1 by assumption. This proves that there exists an m∗ ∈ (0, 1) thatsatisfies equation (18). Moreover, the solution m∗ is unique, since it can be shown
that

=
=
=
=

G1 (m, k a ) |G(m,ka )=0
∙
¸
1−m
αm 0
1
−
θ (k) km (1 − m)m
1−αα1[1 − θ (k) − mθ0 (k) k]m (1 − m) 1 − α("#)
α
f 00 (k) k
1
1 − θ (k) − mθ (k) 1 − θ (k) + 0
m (1 − m) 1 − α
f (k)
(
)
α
1
f 00 (k) k[1 − θ (k)] [1 − mθ (k)] − m 0
θ (k) > 0.
m (1 − m) 1 − α
f (k)

(19)
(20)
(21)

The derivation of the second equation uses G (m, ka ) = 0 and the derivation of thethird equation uses the definition of θ (k). Given m, equation (15) uniquely solves fork, and, given a unique k, equation (3) uniquely solves for wr . Equations (1), (7) and(8) imply that
1−m Z
=
m

∙Z

¸ 1

zdQh (z|s)

1−α"αf (k)αhdQs (s)w + rk

# 1

1−α

.

(22)

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.
20

Proof of Proposition 3: Aggregate output per unit of eﬀective labor in an economy,y a , can be expressed as follows:R

mNy L (s) dQhs (s)
,
y =
N
a

where y =R Y (zs)
L(s)

(23)

dQhzs (z, s). Since it can be shown that y = w+rk
, substituting thisαequation and equation (8) into equation (23) yields
ya =w + rk(1 − m) .
α(24)Rearranging equation (22) yields"Z ∙Z#(1−α) Ã!α µ¸ 1¶(1−α)
1−αkmw + rkhh=
zdQ (z|s)dQs (s)f.
α(1 − m)1−mApplying this equation to equation (24) yields the desired result. Q.E.D.

Proof of Proposition 4: The definition of φ (k a ) implies that
µ

¶

(1−α)
m
f 0 (k) D,
φ (k ) = αf (k)
1−m "
"
#
#
dm
dm
f (k) (1 − α) (1 − m).
where D ≡ 1 + k a + 0
−1
dk
f (k)
αm
dka
0

α−1

a

(25)

Equation (18) implies D = 1. Hence φ0 (ka ) > 0.
To examine the second derivative, note that the first derivative can be rewrittenas
φ0 (k a ) = α

θ (k) φ (ka )
.
ka

(26)

Hence the second derivative can be written as
φ0 (ka )φ (k ) = M
θ (k) kadk
where M = θ0 (k) a ka − θ (k) [1 − αθ (k)]
dk
00

a

Hence, the sign of φ00 (ka ) is the same as that of M. To determine the sign of M,
two lemmas are required.
21

Lemma 10 The sign of m0 (ka ) is determined by the sign of θ0 (k):m0 (ka ) =θ0 (k) m.
1 − θ (k) − θ0 (k) km

Proof. Equation (18) implies that
G2 (m, ka ) |G(m,ka )=0
G1 (m, ka ) |G(m,ka )=0α1
θ0 (k)
− 1−α
1−m
,
1
α
[1 − θ (k) − mθ0 (k) k]m(1−m) 1−αm0 (ka ) = −=θ0 (k) m.
=
1 − θ (k) − θ0 (k) km

The derivation of the second equation uses equation (19). Equation (21) shows that1 − θ (k) − θ0 (k) km is positive.
Lemma 11 A rise in ka increases k:
1 − θ (k)
dk
=
>0
dka
(1 − m) (1 − θ (k) − θ0 (k) km)
Proof. Equation (15) implies that
1 + km0 (ka )
dk
.
=
dka
1−m
The desired result follows from Lemma 10.
Applying Lemma 11 to the definition of M, the following equation is derived:M=θ0 (k) [1 − θ (k)] k − θ (k) [1 − αθ (k)] (1 − θ (k) − θ0 (k) km).
1 − θ (k) − θ0 (k) km

Now equations (19) and (20) imply"#

f 00 (k) k
θ0 (k) k = θ (k) 1 − θ (k) + 0,
f (k)and equation (19) and (21) imply
1 − θ (k) − θ0 (k) km = [1 − θ (k)] [1 − mθ (k)] − m
22f 00 (k) kθ (k) .
f 0 (k)Hence the numerator becomes:
θ0 (k) [1 − θ (k)] k − θ (k) [1 − αθ (k)] (1 − θ (k) − θ0 (k) km)= θ (k) [1 − θ (k)] [1 − θ (k) − (1 − αθ (k)) (1 − mθ (k))]
f 00 (k) k[1 − θ (k) + (1 − αθ (k)) mθ (k)]+θ (k) 0f (k)= θ (k)2 [1 − θ (k)] [α + m − 1 − αmθ (k)]f 00 (k) k[1 − θ (k) + (1 − αθ (k)) mθ (k)]+θ (k) 0f (k)Equation (18) implies that α + m − 1 − αmθ (k) = 0. Hence M < 0, and therefore,
φ00 (ka ) < 0.
Equation (18) also implies that limk→0 θ (k) < 1 guarantees that limka →0 m (ka ) ∈(0, 1). Hence, limka →0 φ (ka ) = 0 since limk→0 f (k) = 0, and limk→0 φ0 (k) = ∞ sinceequation (25) implies that limk→0 f (k) = 0 and limk→0 f 0 (k) = ∞ guarantee this.
Equation (18) also implies that limk→∞ θ (k) < 1 guarantees that limka →∞ m (k a ) ∈(0, 1).

Hence, equation (25) proves limk→∞ φ0 (k) = 0 since limk→∞ f 0 (k) = 0.

Q.E.D.

Proof of Theorem 8: Applying the standard Bayesian updating technique, it canbe shown that
Z

Using these results,R³

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

zdF (z|s) can be expressed as follows:

2

´

Z

"

#

σ 2u hzdQ (z|s) = z exp hs −
,
2
h

e

(27)

where z e = exp µ + σ2u . Since the variance of s is σ 2u + σ 2ε , this can be written as
σ2u.
hUsing this result, it is easy to show that
"Z ∙Z

h

¸ 1

zdQ (z|s)1−α#(1−α)

dQhs (s)23= z e exp

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

Proof of Theorem 9: To prove the theorem, the following lemma is needed.
Lemma 12 Equation (13) is equivalent to
Z ∙Z

¸ α

h

1−α

zi dQ (zi |si )dQhs (s) =

Z ∙Z

¸ 1

h

zi dQ (z|s)

1−αdQhs (s)Proof. Since L (s) satisfies equation (1),ZZ=

(1 − zi ) [f (k) L (s)]α dQhzs (z, s) m
(1 − zi )

∙Z

¸ α

1−α

zi dQh (zi |si )"αf (k)
dQhzs (z, s)
w + rk

# α

1−α

m

Hence, equation (13) implies
0=Z(1 − zi )∙Z¸ α

zi dQh (zi |si )1−αdQhzs (z, s) .

The desired result follows.
Since log z = µ + u, equation (27) implies that
Z µZ

and

Z µZ

³

¶ 1

h

1−α

zi dQ (zi |si )¶ α

zi dQh (zi |si )
2

´

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)2where z e = exp µ + σ2u . Using lemma 12, z e can be solved for as a function of hand σ 2u :z e = exp

−ασ 2u h.
(1 − α)The desired result follows from theorem 8. Q.E.D.

24

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