OSIPP Discussion Paper : DP-2008-E-010

Entrepreneurial Competition and Its Impact on the Aggregate
EconomySeptember 3, 2008
Katsuya Takii
Associate Professor, Osaka School of International Public Policy (OSIPP)【キーワード】D21, D61, D83, L25, L26【要約】This paper models entrepreneurship as the entrepreneur's information
processing activity in order to predict changes in demand and reallocate
resources. The results show that allocative efficiency---and therefore aggregate
productivity---increases through intensified competition by entrepreneurs
grasping at opportunities. This fierce competition leads to price reductions that
result in the improvement of measured aggregate productivity. The pricereduction also forces relatively less able entrepreneurs to become workers. As
resources are then dealt with only by relatively talented entrepreneurs, thisselection effect also increases aggregate productivity. The paper also discusses
how the selection effect influences the distribution of firm size.
The financial support of the Japanese Ministry of Education, Culture, Sports,
Science and Technology (Grant-in-Aid for Scientific Research) is gratefully
acknowledged. Osaka School of International Public Policy, Osaka University,

1-31, Machikaneyama,
takii@osipp.osaka-u.ac.jp

Toyonaka,

1

Osaka,

560-0043,

Japan.

E-mail:1

Introduction

Diﬀerent researchers characterize entrepreneurs diﬀerently.

It is well known that

Knight (1921) emphasizes risk bearing as the main function of entrepreneurs while
Schumpeter (1947) views entrepreneurs as persons who do new things.

However,these are not the only functions of entrepreneurs discussed in the literature.
Many studies, for example, point out the prediction of idiosyncratic changes as
an important element of entrepreneurship.

Kirzner (1973) discusses the essential

role of entrepreneurial discovery in market processes. Schultz (1975) defines entrepreneurial ability as the ability to interpret new information and reallocate resources
to take advantage of profitable opportunities. More recently, Casson (2005) reviews
the literature and argues that judgmental decision-making is a common feature ofentrepreneurship.
Although these studies provide many valuable insights, they do not discuss how
entrepreneurship influences aggregate statistics such as aggregate productivity andthe distribution of firm size.

These considerations demand a model to locate theprediction of idiosyncratic changes within standard macroeconomics.
This paper models entrepreneurship as the entrepreneur’s information processing
activity in order to predict changes in demand and reallocate resources. It then analyzes how entrepreneurship influences aggregate productivity. When there is uncertainty in demand, entrepreneurs reallocate their attention to processing informationin order to predict the direction and magnitude of changes in demand.

Althoughthese changes can be inferred from a number of signals, some signals are more informative than others. Able entrepreneurs are assumed to know which signals are
more informative for prediction. Guided by prior knowledge, they process specificsignals in order to improve the accuracy of their predictions. Better prediction allows
entrepreneurs to reallocate resources toward more profitable opportunities, and this2

improves the allocative eﬃciency of the economy.
The paper shows that the benefits of reallocation are captured by a reduction inan aggregate price index through fierce competition among firms.

By processing

local information, firms can increase their outputs when demand for their products ishigh, enabling them to set a higher price. Conversely, firms can reduce outputs when
they set a lower price due to low demand. The strong correlation between price andoutput increases the firm’s expected revenues. However, when total expenditure isconstant, an increase in firm revenue involves a reduction in expenditure on goods
provided by other firms. This demand-stealing eﬀect results in a reduction in theaggregate price index and, therefore, an increase in measured aggregate productivity.
If ability diﬀers across agents, competition leads to selection among entrepreneurs.
This provides an additional mechanism that influences aggregate productivity. As the
reduction in the aggregate price index lowers expected profits and increases real wage
payments, less able entrepreneurs choose to be workers. Because more resources arethen dealt with only by high-ability entrepreneurs in equilibrium, allocative eﬃciency
and aggregate productivity in the economy again increase.
Note that this selection mechanism also influences the distribution of firm size.
As the increase in the ability diﬀerence across entrepreneurs forces less able entrepreneurs to become workers, it reduces the number of firms and increases the number ofavailable workers. Hence, it increases not only size diﬀerences across firms, but also
the average firm size.
The role of entrepreneurial ability in the economy is magnified when demand ismore volatile and goods more substitutable.

An increase in idiosyncratic risk in-creases the importance of prediction, whereas an increase in the substitutability ofgoods makes the demand for each good more sensitive to changes in tastes. Hence,both increase the benefits of prediction. As the increase in risk and the substitutabil-3

ity of goods increases the benefits from more able entrepreneurs, they force less able
entrepreneurs to be workers through a further reduction in the aggregate price. It isthen shown that increases in risk and the substitutability of goods increase not only
the average size diﬀerences, but also the average size of firms.
This paper is organized as follows. Section 2 reviews the literature and clarifies
the concepts used in the paper. Section 3 sets up the model. Section 4 analyzes
the equilibrium and derives real GDP as a function of entrepreneurship. Section 5analyzes how a change in the distribution of entrepreneurial ability influences real
GDP and the size distribution of firms.

Section 6 concludes by summarizing themain results and discussing some possible extensions.

2

Literature Review

Labor economists empirically investigate the importance of the ability to predictchanges.

Welch (1970) points out that the eﬀect of education on the production

process can be decomposed into two components. Higher education enables a worker
to increase the amount of output produced from a given quantity of inputs (theworker eﬀect); it also allows the worker to interpret information about the profitability of resource allocation, and this enables the worker to make better use of resources(the allocative eﬀect).

Schultz (1975) regards the allocative eﬀect of education as

the source of entrepreneurial ability and Takii (2003) compares these eﬀects of human intelligence in the context of managers’ human capital: managerial ability is
considered as the ability to organize and increase productivity and entrepreneurial
ability is considered as the ability to predict and react changes. This study follows
this distinction but allows entrepreneurs to process information based on knowledge
about the relationship between observable information and shocks in taste. Hence,
entrepreneurial ability is measured by the importance of knowledge for prediction.
4

Another strand of the literature suggests that information processing activities canalso be an important component of entrepreneurship [e.g., Fiet (1996) and Casson(2005)]. There are several features of entrepreneurial information processing: 1) By
processing information, entrepreneurs observe risk-reducing signals [e.g., Fiet (1996)].
2) Information must be specific [e.g., Hayek (1945), Fiet (1996) and Casson (2005)].
3) Entrepreneurial discovery occurs under imperfect competition [e.g., Fiet (1996) andKirzner (1997)]. 4) An entrepreneur is better suited to assess specific information
because of appropriate prior knowledge, which can be the result of previous experience[e.g., Fiet (1996) and Shane (2000)] and/or education [e.g., Schultz (1975)].

This

paper incorporates all of these features of entrepreneurship into a general equilibrium
model.
Because this paper defines entrepreneurial ability as the entrepreneur’s knowledge
about the relationship between observable information and changes in demand, theanalysis can be linked naturally to firm decisions under uncertainty.

In the nicesurvey by Hey (1979), the previous literature on firm decisions under uncertainty
takes information structure as given and analyzes its impact on firm’s decision in apartial equilibrium model.

In contrast, this paper assumes that entrepreneurshipcan convert information structure and analyzes how entrepreneurship influences theaggregate economy in a general equilibrium model.
It is also important to distinguish between diﬀerences in prediction accuracy andflexibility.

Economic theories of the firm investigate how firms adapt to changes

by examining firm flexibility. Many theorists investigate the tradeoﬀ between staticeﬃciency and flexibility (e.g., Mills and Schumann (1985) and Thesmar and Thoenig(2000)). Unlike these studies, we pay greater attention to a firm’s prediction ability.
In other words, we investigate the ability of firms to adapt appropriately to changes
in demand, whereas the existing literature on flexibility investigates the ability of5

firms to quickly or costlessly adapt to changes in demand. As flexibility improves
profits only when firms know how decisions should be changed, our results provide
useful complementary information for making inferences about the importance of
these theories.
This is not the first paper to incorporate a similar definition of entrepreneurial
ability into a general equilibrium model [e.g., Hassler and Mora (2000), Takii (2004;
2008)].

Hassler and Mora (2000) discuss the relation to social mobility and Takii(2004; 2008) considers the impact of entrepreneurial ability on aggregate productivity
and on expansionary fiscal policy, respectively. However, no earlier work explicitly
models how entrepreneurs process information to change information structure, whichis the essence of entrepreneurship in this paper, and no study examines the impact
of entrepreneurship on the size distribution of firms.

3

Model

In this section, we construct a model to analyze the aggregate impact of entrepreneurship. There are two occupations in an economy: entrepreneurs and workers. Eachfirm needs one entrepreneur to organize the firm. In this model, for simplicity, there
is no distinction between entrepreneurs, managers, plants, and firms. This simplification is made to develop a tractable model that focuses on information processing
activity as the essence of entrepreneurship.

The goods market is assumed to bemonopolistically competitive and each entrepreneur faces her own inverse demand
function and produces output by employing workers. Each demand curve is subject
to taste shocks that entrepreneurs cannot observe when they make production decisions. They process information and infer the direction and magnitude of these taste
shocks. We construct a model to analyze how these information processing activities
influence the aggregate economy.
6

Inverse Demand Function: There are heterogeneous households in an economy.
The jth household is assumed to maximize the following utility subject to its budget
constraint:
∙Z m

¸ α1
1
zi xαij di , α = 1 − ∈ (0, 1) ,ρUj ≡ maxxij
Z m i
pi xij di,s.t. Yjn =iwhere xij is the amount of good i consumed by the jth household, zi is a productspecific taste shock for product i, pi is the price for product i, m is the number of
products, and Yjn is the expenditure of the jth household.

As the parameter ρ isthe measure of the substitutability of goods, α also captures the substitutability ofgoods. Using standard procedures, the inverse demand function for good i can bederived as follows.
pi = zi xα−1
P α (Y n )1−α ,i∙Z m α¸ α−11
αα−1 1−αP =pi zi di
,
i
Z
Z
n
n
=
Yj dj, xi = xij dj,
Y

(1)
(2)
(3)

where P is the price index and Y n is nominal GDP. As we assume that the household
sector is the only sector that buys products, the total sum of consumer expenditure
is nominal GDP in this analysis.

A Firm’s Problem: An entrepreneur establishes a firm, employs workers, and produces output. A single firm produces one variety of each good and an entrepreneur
faces an inverse demand curve, equation (1).

However, when she undertakes pro-duction, she does not know what the realized taste shock is. The timing sequence7

is described as follows.

(1) A taste shock occurs.

(2) An entrepreneur processes

this information and infers the realized taste shock. (3) A firm produces nonstorable
goods by choosing the amount of employed workers. (4) The firm discovers the true
taste shock. (5) The firm sets its price to equate demand with the goods produced.
We now describe the behavior of a representative firm and omit subscript i where theintended meaning is clear.
Because the movement of z is not observed ex ante when z changes, the entrepreneur must predict the direction and magnitude of the change so as to appropriately
respond.

When the entrepreneur makes production decisions, z is not observed,© ªv
but noisy signals are observed, SAv ≡ sA
j j=1 , from which the realization of z canbe inferred.

As discussed later, the accuracy of prediction depends on the qual-

ity of signals, A, and the amount of signals, v, the entrepreneur actually processes.
We consider that the variable v captures the intensity of entrepreneurship.

It isassumed that the entrepreneur’s inference is based on a conditional distribution func¡¢tion, Qh z|SAv , where h measures the accuracy of an entrepreneur’s prediction forchanges in z. We specify the information structure later.

Note that because z is specific to a particular product, the required informationis also specific to the product. As only one firm produces this product, this is purely
firm-specific information.

Many researchers point out the importance of specificinformation in the entrepreneurial discovery process. The current model incorporates
the spirit of this argument.
The production function is assumed to be linear: x = M (v) L in labor where L isthe amount of labor input and M (v) is the productivity of a firm’s production process.
When an entrepreneur establishes a firm, she must set up the production process inorder to organize the firm. This activity is assumed to increase the productivity ofa firm’s production process. When an entrepreneur processes more signals, v, she

8may not be able to spend enough time setting up the production process. Hence, we
assume that M 0 (v) < 0.
It is assumed that the financial market is complete.
can hedge any idiosyncratic risk.

Therefore, entrepreneurs

Although financial investors cannot scrutinize

the information entrepreneurs observe, they are assumed to know the entrepreneurs’
eﬀort level and their ability, which influence the accuracy of information, h. Hence,
there is no distortion caused by asymmetric information. Entrepreneurs maximize
their firm’s expected profits:¡¢
π SAv = max
L½Z¾¡ Av ¢
x − wL ,pdQ z|Shs.t. p = zxα−1 P α (Y n )1−α , x = M (v) L,

where w is the wage rate.
The solution to the profit maximization problem reveals that the optimal price,
¡
¢
w
, on
p z, SAv , has to be equal to a constant markup, α1 , times marginal cost, M(v)average:¡¢p z, SAv = Rzw.
zQh (z|SAv ) αM (v)(4)

Note that the actual price deviates from the constant markup rule due to entreprez
neurs misunderstanding changes in tastes, R zQh (z|SAv ) . When the true z is discovered,

the entrepreneur realizes that it diﬀers from the predicted value. If the realized z is¡¢Rgreater than the predicted one, zQh z|SAv , then the entrepreneur can set a higher¡¢
R
price than the markup rule. If z is lower than zQh z|SAv , then the entrepreneurmust set a lower price.

Entrepreneurs produce more when they can set high prices, and produce less when
they must set low prices. Hence, expected revenue is influenced by the accuracy of9

prediction:∙¸ α
αP 1−α nY ,E [px] = [z (h) M (v) ]
w
"Z ∙Z
#1−α
1
¸ 1−α
¡
¢
¡
¢
z (h) ≡
dQhs SAv,zdQh z|SAv
1
α 1−α

h

where E h [px] =

R

¡
¢ ¡
¢
¡
¢
¡
¢
¡
¢
p z, SAv x SAv dQhzs z, SAv , Qhs SAv and Qhzs z, SAv is themarginal and joint distribution function of z and S Av .

Furthermore, as the expected amount of labor and the expected profits are proportional to E h [px] in this model, the accuracy of prediction also influences themthrough its impact on expected revenue.
Z

¡
¢
¡
¢
α h
L SAv dQhs SAv =E [px] ,
w
Z
¢
¡
¢
¡
π SAv dQhs SAv = (1 − α) E h [px] ,

(5)
(6)

where L (·) is an optimal labor function.

Information Structure: The accuracy of information can be a function of entrepreneurship. We assume that entrepreneurs use their ability to process information andmake sound judgments. In order to construct an operational model, we must specify
2

the information structure. Assume that ln z is normally distributed with mean − σ2z

and variance σ 2z . This means that the expected value of z is normalized at 1. The
random deviation of z from 1 represents ex ante unknown changes in taste.
Assume that there are many signals that are correlated with ln z with error:AsA
j = ln z + εj2σzwhere εAj is normally distributed with mean 0 and variance A .

Signals diﬀer by

their precision. People may learn the relationship between a signal, sA
j , and shock,10

ln z, from education or previous experience. But without knowing this relationship,
signals are of no value to entrepreneurs.

Researchers are split on their opinions

about how prior knowledge is obtained. However, all agree that entrepreneurs areheterogeneous in terms of the knowledge used to detect important information. Talented entrepreneurs are assumed to know which signal is more accurate. Hence, theparameter A can be interpreted as the measure of entrepreneurial ability.
Using their prior knowledge, entrepreneurs observe this signal v times. Let us
RR2(ln z− ln zdQh (z|SAv )) dQhzs (z, SAv )measure the accuracy of prediction by h = 1−
, whichσ2zis suggested by Takii (2007) as an estimable measure of the accuracy of prediction.
2

This measure is equivalent to R that is obtained when we regress ln z on

Pv

A
j sj

v

. It

is easy to show that h can be a function of A and v:1
h = η (Av) ≡Av.
Av + 1This shows that the intensity of entrepreneurship, v, and entrepreneurial ability, A,
increase h and, therefore, reduce perceived risk. This is consistent with arguments
by Fiet (1996). Applying Lemma 2 in Takii (2007) to the current problem, z (h) is
shown to be a function of A and v.
z (h) = exp

1

The measure h can be rewritten as h =ασ 2z η (Av).
2 (1 − α)

R RR[ ln zdQh (z|SAv )− ln zdQhz (z)]2 dQhs (SAv )σ 2z. This analysis

focuses on a multiplicative shock, and the shock and the corresponding signal are log-normally
distributed. In this case, we can show that signals are informative in the sense of Blackwell (1953)¡¢
R
if and only if the conditional expectation ln zdQh z|SAv is risky in the sense of Rothschild andStiglitz (1970).

The measure h captures this property. Of course, the variation of ln z can also¡¢Rinfluence that of ln zdQh z|SAv . Hence, we divide it by σ 2z so that h is independent of σ 2z .

Hence, h is not influenced by exogenous risk.

11Entrepreneurship: Processing information reduces the time available to set up
the production process due to their limited attention capacity. Hence, it lowers the
productivity of their firm’s production process.

Knowing this tradeoﬀ, entrepre-

neurs optimally choose the amount of information they observe.

We assume that

M (v) = exp (−βv) for a simple analysis. An entrepreneur’s decision problem is thensummarized as follows.

Π (A) ≡ max (1 − α) E η(Av) [px] ,
v
¸∙
¸ α
∙
αx (α, σ 2z , A, v) αP 1−α nη(Av)[px] = exp
Y ,
E
1−α
w
¡
¢
σ 2z η (Av)x α, σ 2z , A, v =− βv.
2 (1 − α)Define v (α, σ 2z , A) as the optimal solution to this problem.

Ignoring integer con-straints, we assume that the first-order conditions can approximate optimal conditions. The optimal conditions can then be described as follows.
q 2
σz A
−1
¢
¡
2β (1 − α)2β(1−α)2, if A ≥,v α, σ z , A =
A
σ 2z
¡
¢
2β (1 − α)v α, σ 2z , A = 0, if A <.
σ 2z

This shows that if entrepreneurs do not have enough prior knowledge, they choose
not to process information. Instead, they focus on preparation for the production
process so that they can produce more. However, the impact of ability on the amountof processing information is not monotonic. It is shown that¢¡2β (1 − α)8β (1 − α)vA α, σ 2z , A ≥ 0, if
≤A≤
.
2
σz
σ 2z
¡
¢
8β (1 − α).
vA α, σ 2z , A < 0, if A >σ 2z

When entrepreneurs know where an informative signal is, they have greater incentive
to process the information. However, if signals are extremely accurate, entrepreneurs
12do not require too many signals. For example, if entrepreneurs can perfectly observe
the movement of ln z, just one observation is suﬃcient to detect the movement in
ln z.
Note that increases in the substitutability of goods and risk increase the amountof information that entrepreneurs process:¢¢¡¡
2β (1 − α)vα α, σ 2z , A > 0, vσ2z α, σ 2z , A > 0, if A ≥.
σ 2z
An increase in risk increases the value of accurate information, while an increase in
the substitutability of goods makes demand more sensitive to taste shocks. Hence,
this also demands more information.
It is interesting to note that the substitutability of goods increases flexibility inchanging the amount of output. When goods are more substitutable, equation (1)suggests that the demand curve is flatter. Hence, an entrepreneur can change theamount of output without reducing prices greatly. An entrepreneur can exploit thebenefits from better information when she can easily change her decisions. Hence,an entrepreneur spends more time processing information when α is large.
Although an increase in entrepreneurial ability has ambiguous eﬀects on theamount of information processed, it unambiguously increases profits.

To see this,let us define a function x∗ such that x∗ (α, σ 2z , A) ≡ x (α, σ 2z , A, v (α, σ 2z , A)). Thenx∗ (α, σ 2z , A) can be solved as¢¡βx∗ α, σ 2z , A =
A

"s

#2
σ 2z A2β (1 − α),
− 1 , if A ≥2β (1 − α)σ 2z= 0, if A <2β (1 − α)
.
σ 2z

13

We can easily check that when A ≥ 2β(1−α)
,
σ2
z

¢
¡
x∗A α, σ 2z , A > 0,¢¢¡¡x∗α α, σ 2z , A > 0, x∗σ2z α, σ 2z , A > 0,¢¢¡¡x∗Aα α, σ 2z , A > 0, x∗σ2z A α, σ 2z , A > 0.

(7)
(8)
(9)

Equation (7) shows that entrepreneurial ability increases profits. In addition, equation (8) and equation (9) show that not only are profits increasing in α and σ 2z , but
α and σ 2z are also complementary in A.

That is, increases in risk and the substi-

tutability of goods increase the marginal profitability of entrepreneurial activities.
Let us define a function η ∗ such that η ∗ (A) = η [Av (α, σ 2z , A)]. Because entrepreneurial ability increases firm prediction, expected revenue is an increasing function of
A:

∙

¸∙
¸ α
αx∗ (α, σ 2z , A) αP 1−α nE[px] = exp
Y .
(10)
1−α
w
¸
∙
αx∗ (α,σ2z ,A)= M (v (α, σ 2z , A)) = 1 if and only if v (α, σ 2z , A) = 0.
Note that exp
1−α
¸
∙
∗
αx∗ (α,σ2z ,A)E η (A) [px]=≥ 1 is the relative expected
Hence, the component exp
1−α
E 0 [px]η∗ (A)

revenue gain from processing information.

In this paper, this is considered as thegain from entrepreneurship. Entrepreneurship improves expected revenue because it
allows entrepreneurs to produce more when demand is high and less when demand islow. As shown, this revenue gain is an increasing in α, σ 2z and A.

The arbitrage condition: An agent can be an entrepreneur or a worker.

Let

us assume that entrepreneurial ability is distributed along a continuous distributionfunction G (A). Let N denote the number of workers that have an ability less thanor equal to A. Then G (A) = N. It is convenient to define an inverse distributionfunction Γ such that A = Γ (G (A)).

The use of the inverse distribution function

makes it easier to analyze the model. The value Γ (N) represents the ability level14

that the NNs percentile from bottom agent obtains, where N s is the total population.
Because entrepreneurs can completely hedge their risks in the financial market,
they do not bear risk. As x∗A (α, σ 2z , A) ≥ 0, Π0 (A) ≥ 0. Because G (A) is continuous
distribution function, Γ0 (·) > 0. Hence, there is a unique cutoﬀ ability Γ (N ∗ ) withwhich an agent expects to earn the same income from being an entrepreneur or aworker:∗∗w = Π (Γ (N ∗ )) = (1 − α) E η (Γ(N )) [px] .

(11)

The second equation is derived because the expected profits are proportional to the
expected revenue.

Resource constraints: Using the function Γ we can define the labor market clearing
condition as follows.
∗

N =Z Ns Z
N∗

¢
¡
¢
¡
∗
L SAv dQsη (Γ(N)) SAv dN.

(12)

The left-hand side is the supply of labor and the right-hand side is the demand
for labor.

The equation states that the bottom N ∗ agents in the ability distrib-

ution become workers and the top N s − N ∗ agents in the ability distribution become entrepreneurs. As entrepreneurs with the ability Γ (N) demand labor by
R ¡ Av ¢ η∗ (Γ(N)) ¡ Av ¢L S
dQs
S , the right-hand side of equation (12) is the total demand

for labor.

As expected labor is proportional to expected revenue in equation (5), equation(12) can be rewritten as
α
N =
w
∗

Z Ns

∗

E η (Γ(N)) [px] dN.

N∗

15

(13)

4

The Distribution of Firm and Real GDP

In this section, we investigate the implications for the distribution of firm size andreal GDP. First, we show the existence and uniqueness of equilibrium and discuss theimplications on the distribution of firm size. Later, we derive real GDP and discuss
the eﬀect of entrepreneurial ability on aggregate productivity.

Equilibrium: This is derived from equation (11) and (13) where
Z Ns
N∗

exp

1−α ∗
α [x∗ (α, σ 2z , Γ (N)) − x∗ (α, σ 2z , Γ (N ∗ ))]
dN =
N .
1−α
α

(14)

This equation determines an equilibrium level of N ∗ . The existence and uniqueness of N ∗ can be easily proved. The formal proof is established in the Appendix.
Theorem 1 There exists a unique N ∗ .
Note that N ∗ contains several items of information. First, N ∗ is the number of∗workers. Second, N s −N ∗ is the number of firms. Third, N sN−N ∗ can be interpreted as
the average firm size. Fourth, it also relates to the average diﬀerences in revenue. As∙¸∗αx∗ (α,σ2z ,A)E η (A) [px]exp=
, the expected revenue diﬀerences due to diﬀerences in en1−α
E 0 [px]α[x∗ (α,σ2z ,Γ(N))−x∗ (α,σ2z ,Γ(N ∗ ))]. Hence, the av1−α∗∗2∗α[x (α,σ 2
R Nsz ,Γ(N ))−x (α,σ z ,Γ(N ))]dN
N ∗ exp
1−αerage revenue diﬀerences across firms can be measured by.
N s −N ∗trepreneurial ability are captured by exp∗

N
Equation (14) suggests that 1−αis equivalent to the average revenue diﬀerences
α N s −N ∗across firms.
Note that the average revenue diﬀerences can also be interpreted as the average
size diﬀerences. Equations (5) and (6) imply that the expected revenue diﬀerences aretranslated into diﬀerences in expected labor cost and diﬀerences in expected profits.

16

Hence, it is natural to interpret

R Ns

N ∗ exp

[ (

)

(

)] dN

∗
2
∗
α x∗ α,σ 2z ,Γ(N ) −x α,σ z ,Γ(N )1−α
N s −N ∗and, therefore,

1−α N ∗, as a measure of average size diﬀerences.
α N s −N ∗This interpretation suggests that the average firm size and the average size differences move together with N ∗ in the equilibrium. Applying the implicit function
theorem, we can show how the changes in α and σ 2z influence N ∗ . The formal proof
is established in the Appendix.
Proposition 1 There is a function N such that¡¢N ∗ = N α, σ 2z ; Γ , Nα > 0, Nσ2z > 0.
This proposition implies that if goods are substitutable and the economy is risky,
the number of workers is large, the average firm size is large, and the average size
diﬀerences across firms are large.
This is because ability is complementary to risk and the substitutability of goods.
A risky environment demands more accurate information. An increase in risk thenincreases the importance of prediction, whereas an increase in the substitutability ofgoods makes demand more sensitive to changes in tastes. Hence, they both increase
the benefits of able entrepreneurs and less able entrepreneurs are forced to be workers.
This results in an increase in the average firm size and the average size diﬀerences
across firms.
It is interesting to note that an increase in risk may increase size. The standard
argument emphasizes the tradeoﬀ between static eﬃciency and flexibility under uncertain demand.

As eﬃciency is easily converted to size and a risky environment

demands flexibility, there is a tradeoﬀ between size and risk [e.g., Mills and Schumann (1985)]. In fact, it would be possible to extend our model to derive a negative
relationship between risk and size by introducing irreversible investment.

In thissense, we cannot take this result as an empirically testable prediction. Rather, this17

proposition suggests that there is a neglected mechanism where risk increases size.
The relationship between risk and size may then be more complex than conventional
theories suggest.

Real GDP: In order to understand the impact on productivity, we derive real GDP
in this model. Note that the definition of nominal GDP implies
Z Ns
∗
n
Y =
E η (Γ(N)) [px] dN.

(15)

N∗

Substituting equation (15) into equation (13), nominal GDP is shown to be proportional to the total labor cost:
Yn =

wN ∗
.
α

(16)

In order to derive real GDP, we require an aggregate price index. Substituting
equation (10) into (15), the aggregate price index is shown to decline as a result ofentrepreneurship in the economy.
P =
z

∗

=

w
,
αz ∗
∙Z 1

(17)

αx (α, σ 2z , Γ (N))
dN
exp
1−α
N∗

¸ 1−α
α

.

Note that P is decreasing in z ∗ , which is a positive function of the total sum of
the relative expected revenue gain due to entrepreneurship. When all entrepreneurs
process information, they can increase revenue on average. However, total expenditure is given by Y n . This means that an increase in the revenue of a particular firm
results in the reduction of expenditure on other firms, bringing about a reduction in
the aggregate price index.
Dividing nominal GDP by the aggregate price index, real GDP can be expressed
as a function of the number of workers and z ∗ :Y =Yn= z∗N ∗.
P
18

This equation shows that z ∗ also represents the improvement in aggregate productivity due to entrepreneurship.

Entrepreneurship increases aggregate productivity

because it improves the allocative eﬃciency of the economy. Because entrepreneurship allows managers to provide output when needed, it also improves social welfare.
R
As shown when Y = Uj dj, the improvement in social welfare is captured by theincrease in real GDP.

Aggregate productivity z ∗ can be decomposed into a level eﬀect and a distributional eﬀect:z ∗ = Dist × Level,µZ N s¶ 1−ααα [x∗ (α, σ 2z , Γ (N)) − x∗ (α, σ 2z , Γ (N ∗ ))]Dist =
dN
exp
,
1−α
N∗
¢
¡
Level = exp x∗ α, σ 2z , Γ (N ∗ ) .

If there is no heterogeneity across agents, everybody has the same ability, Γ (N ∗ ).
Hence, only the level eﬀect, exp x∗ (α, σ 2z , Γ (N ∗ )), influences the productivity. It implies that exp x∗ (α, σ 2z , Γ (N)) can be interpreted as the productivity gain induced by
¶ 1−α
µ
α
R Ns
α[x∗ (α,σ 2z ,Γ(N ))−x∗ (α,σ 2z ,Γ(N ∗ ))]dN,
ability Γ (N). Then, the distribution eﬀect, N ∗ exp
1−αcan be interpreted as the weighted sum of productivity diﬀerences across firms. Thissummarizes the productivity improvement due to heterogeneous ability.
More importantly, equation (14) implies that the magnitude of the distributionaleﬀect is captured by the amount of workers in the economy. As the large productivity
gain from entrepreneurship brings about a large revenue gain, the weighted sumof productivity diﬀerences across firms are converted into the total sum of revenue
diﬀerences across firms.

On the other hand, large revenue diﬀerences mean that

relatively talented entrepreneurs are stealing large amounts of demand from less able
entrepreneurs.

Hence, marginal entrepreneurs are forced to be workers.
19

That

is, the distributional eﬀect influences aggregate productivity through the selection
mechanism.
Substituting equation (14) into the distributional eﬀect, real GDP can be rewritten
as follows.

∙

1−α
Y =
α

¸ 1−α
α

¢¤
£
¡
1
exp x∗ α, σ 2z , Γ (N ∗ ) [N ∗ ] α .

This equation suggests that real GDP is larger when the productivity of a marginal
firm and the number of workers are large.
distributional eﬀects.

The number of workers captures all

It means that the impacts of heterogeneity on real GDP

disappear once we control the number of workers.

5

The Distribution of Entrepreneurial Ability

In this section, we analyze how the distribution of entrepreneurial ability influences
the firm size distribution and aggregate productivity. Without loss of generality, we
assume that the population is equal to 1 and that ability is uniformly distributed.
RA 1
Suppose that G (A) = Al γAda. Then, Γ (N) = (γN + 1) Al . The parameter γlrepresents ability diﬀerences, while the parameter Al captures the overall level ofability. Real GDP and the equilibrium condition are rewritten as follows.
Y=(1 − α) N ∗
=
α

∙

1−α
α

Z 1

¸ 1−α
α

£
¡
¢¤
1
exp x α, σ 2z , (γN ∗ + 1) Al [N ∗ ] α ,α [x (α, σ 2z , (γN + 1) Al ) − x (α, σ 2z , (γN ∗ + 1) Al )]
dN.
1−α
N∗
i
h
−
1
. This condition guarantees that the most able
We assume that γ > 2β(1−α)
σ2 Al
exp

z

entrepreneurs have an incentive to processes information.

The following proposition can be proved. The formal proof is established in the
Appendix.

20

Proposition 2 Increases in both γ and Al increase Y .

On the other hand, whileN ∗ is clearly increasing in γ, the eﬀect of Al on N ∗ is ambiguous.
¡
¢
dY
dY
Nγ α, σ 2z ; γ, Al > 0,
> 0.
> 0,
dγ
dAl
The result shows that although increases in both γ and Al improve the productivity of economy, only a change in γ brings about a clear selection eﬀect.

When

all agents improve their abilities, marginal entrepreneurs may be able to survive
competition and remain as entrepreneurs due to their improved ability.
NAl (α, σ 2z ; γ, Al ) is ambiguous.

Hence,

However, an increase in γ means that able entre-

preneurs improve their ability more than less able entrepreneurs, and the marginal
entrepreneurs are selected to be workers. Because of this selection eﬀect, a change
in γ increases the average firm size and the average size diﬀerences.

6

Conclusions and Extensions

This paper models entrepreneurship as entrepreneur’s information processing activity
in order to predict changes in demand, and analyzes how entrepreneurship influences
aggregate productivity and the distribution of firm size. We emphasize the productive
impacts of competition and selection as a result of entrepreneurship.
Several extensions are considered. First, this paper emphasizes the importanceof processing information for the adaptation to idiosyncratic shocks as a neglectedrole of entrepreneurship. Modeling firm entrepreneurship as information processing
activity can address broader issues relating to entrepreneurship if the adjustment in
input costs and persistent shocks are incorporated into the model. The broader issues
that can be addressed include adopting a new technology, starting a new business,and developing a new product. These extensions are important so as to appreciate
all aspects of entrepreneurship.
21

Second, in this paper we assume that the distribution of ability is given. This isuseful for simple analysis. However, as the literature on entrepreneurship suggests,
ability can be acquired through previous experience and/or education. These investments change the future distribution of ability, while the expected future distributionof ability influences the incentive of entrepreneurs to invest. It would be interesting
to examine what equilibrium is then obtained. This extension is also left for future
research.

7

Appendix

The Proof of Theorem 1, and Proposition 1 and 2: Without loss of generality,
we prove the theorem and propositions under the assumption of N s = 1 and Γ (N) =(γN + 1) Al . Define a function B so thatZ 1
¢ 1−α ∗
¡ ∗α [x (α, σ 2z , (γN + 1) Al ) − x (α, σ 2z , (γN ∗ + 1) Al )]
2
N −
dN.
exp
B N , α, σ z , γ, Al ≡
α
1−α
N∗

As B (0, α, σ 2z , γ, Al ) < 0, B (1, α, σ 2z , γ, Al ) > 0 and BN (N ∗ , α, σ 2z , γ, Al ) |B(N ∗ ,α,σ2z ,γ,Al )=0 =1+ x∗3 (α, σ 2z , (γN ∗ + 1) Al ) γAl N ∗α> 0, the existence and uniqueness are obvious.

Note that¡¢
Bi (N ∗ , α, σ 2z , γ, Al ) |B(N ∗ ,α,σ2z ,γ,Al )=0
Ni α, σ 2z , γ, Al = −, i = α, σ 2z , γ, Al .
BN (N ∗ , α, σ 2z , γ, Al ) |B(N ∗ ,α,σ2z ,γ,Al )=0
Now, it is easy to derive that¡¢L∗
Bα N ∗ , α, σ 2z , γ, Al |B(N ∗ ,α,σ2z ,γ,Al )=0 = − − Fα ,α¡ ∗¢Bi N , α, σ 2z , γ, Al |B(N ∗ ,α,σ2z ,γ,Al )=0 = −Fi , i = σ 2z , γ, Al ,22

where

Z 1

⎡

α⎣

∗

x

(α, σ 2z , (γN + 1) Al )−x∗ (α, σ 2z , (γN ∗ + 1) Al )

⎤
⎦

α
dN,
Di exp1 − α N∗1−α∂x∗ (α, σ 2z , (γN + 1) Al ) − x∗ (α, σ 2z , (γN ∗ + 1) Al ), i = α, σ 2z , γ, Al .
Di =
∂i
h
i
Suppose that N ∗ ≤ γ1 2β(1−α)−
1. Then x∗ (α, σ 2z , (γN ∗ + 1) Al ) = 0. Because we2
σ z Alhiassume that γ > 2β(1−α)− 1 , equation (8) implies that Di is positive for some N
σ2 Al
Fi =

z

where i = α, σ 2z , γ, Al .

Hence Ni (α, σ 2z , γ, Al ) is positive for i = α, σ 2z , γ, Al . Supposehithat N ∗ > γ1 2β(1−α)−
1. Equation (9) implies that Dα and Dσ2z are still positive.
σ2 Alz

Hence, Nα (α, σ 2z , γ, Al ) and Nσ2z (α, σ 2z , γ, Al ) are positive.
DAl .
∗

∂x

(α, σ 2z , (γN + 1) Al )
∂γ∂N

β
=

We need to check Dγ andon¡¢ q σ2z (γN+1)Al
−
(1
−
γN)
1 − γN22β(1−α)(γN + 1)3 A2l¢¤£¡− (1 − γN)
β 1 − γN
βγN
2
≥
=
> 0.
3 2
(γN + 1) Al
2 (γN + 1)3 A2l

Hence, Dγ and Nγ (α, σ 2z , γ, Al ) are also positive. On the other hand, as∂x∗ (α,σ 2z ,(γN+1)Al )∂Al ∂Ncan be positive or negative, NAl (α, σ 2z , γ, Al ) is ambiguous.
dY
Because Nγ (α, σ 2z , γ, Al ) > 0, dY> 0. We need to check the sign of dA
.
dγ
l
∙
¸ 1−α
¡
¢¤
1
dY
1 1−α α £=exp x∗ α, σ 2z , (γN ∗ + 1) Al (N ∗ ) α −1 XAl ,
dAl
α
α
¸
∙
∗
¢
¡
∂x (α, σ 2z , (γN ∗ + 1) Al ) ∗∂x∗ (α, σ 2z , (γN ∗ + 1) Al ) ∗2
NN + 1+α
N
,
σ
,
α
.
γ,
A
XAl = α
A
l
z
l
∂Al
∂N ∗

dY
> 0. Q.E.D.
As it is shown that XAl > 0, dA
l

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