GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.174

Free Entry, Market Diffusion, and Social Inefficiency
with Endogenously Growing Demand

Hiroshi Kitamura, Akira Miyaoka, and Misato Sato

February 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Free Entry, Market Diffusion, and Social Inefficiency
with Endogenously Growing Demand∗
Hiroshi Kitamura†

Akira Miyaoka‡

Misato Sato §February 10, 2011

Abstract
This paper analyzes market diffusion in the presence of oligopolistic interaction
among firms. Market demand is positively related to past market size because of consumer learning, networks, and bandwagon effects. Firms enter the market freely in eachperiod with fixed costs and compete in quantities. We demonstrate that free entry leads
to a socially inefficient number of firms over time, and that the nature of the inefficiencychanges as the market grows: the number of firms is initially insufficient but eventually
excessive. This is in contrast with previous findings in the theoretical literature.

JEL Classification Codes: D11, L11, L14.
Keywords: Free Entry; Market Diffusion; Intertemporal Externalities; Entry Regulation.
∗
We especially thank Katsuya Takii, Shingo Ishiguro, Roberto Samaniego, and Junichiro Ishida for helpful
discussions and comments. We also thank David Flath, Keizo Mizuno, Keiichi Morimoto, Mitsuru Sunada,
Ryoji Yoshioka, the conference participants at the Japanese Association for Applied Economics, and the seminar
participants at Osaka University and Osaka Prefecture University for constructive comments. We finally thankHaruna Tsuchiya for outstanding research assistance. The first author gratefully acknowledges financial support
from JSPS Grant-in-Aid for Scientific Research (A) No. 22243022 organized by Masaki Nakabayashi. The
second author gratefully acknowledges financial support from the Global COE program “Human Behavior andSocioeconomic Dynamics” of Osaka University. The usual disclaimer applies.
†
Corresponding Author: Faculty of Economics, Sapporo Gakuin University, 11 Bunkyodai, Ebetsu,
Hokkaido, 069-8555, Japan. E-mail: kitamura@sgu.ac.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043,
Japan. E-mail: jge013ma@mail2.econ.osaka-u.ac.jp§
Graduate School of Economics, George Washington University, 2115 G street, NW Monroe Hall 340 Washington DC 20052, USA. E-mail: smisato@gwmail.gwu.edu

1 Introduction
The number of suppliers changes over the lifetime of a product market. Gort and Klepper(1982) investigate 46 new products in the US from their initial introductions up to 1981.1
They characterize the evolution of markets as having five stages. In Stage 1, the number offirms in the market is small. In Stage 2, the number drastically grows. In Stage 3, it reaches
a maximum. In Stage 4, there is a shakeout of firms, and the number rapidly decreases. In
Stage 5, the number stabilizes. The time pattern in the growth of the number of firms from
Stage 1 to Stage 3 is characterized as “S-shaped diffusion.”2
In the related theoretical literature, market diffusion from Stage 1 to Stage 3 is regarded
as resulting from intertemporal externalities, such as learning by doing (Jovanovic and Lach
(1989)), firms’ learning of the market demand (Rob (1991)), or intertemporal consumption
externalities (Vettas (2000) and Kitamura (2010)). In this literature, the number of producers
under free entry is socially insufficient over time.
The aim of this paper is to investigate market diffusion with oligopolistic interaction
among firms theoretically. The previous literature assumes that firms are atomistic pricetakers whose production levels are exogenously determined. Although this assumption is innocuous in explaining S-shaped diffusion, oligopolistic interaction may be a nonnegligible
element when we discuss entry regulation policy for new industries.
To explore the importance of oligopolistic interaction, we construct a model of market
diffusion following Kitamura (2010). In his model, the market grows because of intertemporal consumption externalities, and the market demand depends positively on the previous
period’s market size. His approach allows us to analyze the diffusion model with oligopolistic
interaction by comparing free-entry diffusion, in which the number of firms is determined by
the zero-profit condition, and socially optimal diffusion, in which the number of firms maxi1

There are several papers on the growth of markets. See, for example, Klepper and Graddy (1990), Jovanovic
and Macdonald (1994), and Klepper (1997).
2
S-shaped diffusion is not an isolated phenomenon. Empirical evidence shows that the interfirm and intrafirm
diffusion of new technology tends to be S-shaped: see the seminal work of Griliches (1957), Mansfield (1968),
and Stoneman (2002) for a survey of technological diffusion. In addition, the S-shaped interhousehold diffusion
is treated as a stylized fact in the marketing literature. For example, this phenomenon is observed in color
televisions (Karshenas and Stoneman (1992)), fax machines (Economides and Himmelberg (1995)), clothes
dryers (Krishnan, Bass, and Jain (1999)), and mobile phones (Gamboa and Otero (2009)).

1

mizes social welfare.3 The novel dimension here is that not only the number of firms but also
the output per firm is endogenously determined.4
We demonstrate that the existence of oligopolistic interaction among firms does not play
an essential role in explaining S-shaped diffusion, however, more importantly, it is crucial inconsidering entry regulation policy in new industries. First, we find that S-shaped diffusion
arises in free-entry equilibrium in our model with oligopolistic interaction. This result canbe explained using the logic in the previous literature. Therefore, this finding implies that theassumption that firms are atomistic price takers does not play a crucial role in explaining thefundamental mechanism of S-shaped diffusion.
Second, we have an inefficiency result that differs from the previous literature, in which
the number of firms under free entry is found to be inefficiently small over time. In ourmodel, in contrast, the nature of the inefficiency (i.e., too few firms or too many) depends onthe degree of market maturity: when the market is in the growing phase, the number of firms
under free entry is insufficient. However, as the market enters the mature phase, the numberof firms becomes excessive.
This result provides a new policy implication. According to the previous literature, entry
into markets should be encouraged over time. Based on our result, however, entry regulations should be changed depending on the phase of market growth: entry should be initiallyencouraged but eventually restricted.
To understand this result, we begin by considering the case where the output level isexogenously determined, as in the previous literature. When the output level of each firmis exogenous, new entry only leads to the demand shift effect; that is, new entry today increases tomorrow’s demand because of intertemporal consumption externalities. This effectcan be regarded as the future benefit of increasing the current number of firms. The previous literature concludes that the number of firms under free entry is socially insufficient overtime because firms under free entry do not internalize the future benefit from intertemporal3

One of the important elements to develop in a model of market diffusion is demand structure. In the modelsof Rob (1991) and Vettas (2000), the demand curve is a horizontal straight line, and we cannot examine the roleof oligopolistic interaction. In contrast, the demand curve is downward sloping in Kitamura (2010). This allows
us to analyze the model of diffusion with oligopolistic interaction.
4
Bergemann and Välimäki (1997) analyze market diffusion with strategic behavior. In their model, firm
output is endogenously determined, but the number of firms is fixed exogenously.

2externalities when they enter the market.
In contrast, when oligopolistic interaction exists, as in our model, new entry today also
leads to a “business stealing effect”; that is, new entry causes existing firms to reduce their
output levels today. This effect can be regarded as the current loss of increasing the current
number of firms. Therefore, in our model, the socially optimal number of firms depends onthe magnitudes of the future benefit from demand shift and of the current loss from businessstealing. Since S-shaped diffusion arises when the demand shift effect is initially stronger but
eventually weaker, the number of firms under free entry is initially insufficient but eventually
excessive.
Our findings suggest that although it is useful to express the fundamental mechanism of
S-shaped diffusion via a framework in which firms are atomistic price takers, we need totake into account the oligopolistic interaction among firms when applying the model to entry
regulation policy. Our analysis may especially apply to entry regulation policy when thegovernment in a developing country encourages or regulates new entry in infant industries,
which are already mature in developed countries and where an intertemporal consumption
externality is most likely to be observed. When we discuss the entry regulation policy insuch situations, the framework that ignores oligopolistic interaction may yield misleading
predictions.
This paper is related to the literature concerned with the Excess Entry Theorem.5 Mankiw
and Whinston (1986) show that the number of firms under free entry is socially excessive
because oligopolistic interaction leads to a business stealing effect. Our model can be interpreted as examining the social inefficiency of free entry from a dynamic perspective.
In addition, this paper is related to the literature concerned with intertemporal consumption externalities. In a number of markets, demand may be positively related to past market
size. There are several reasons for this phenomenon.6 First, past market size may play animportant role in the market in the presence of network externalities.7 Second, past market
5

There are a number of theoretical works on the Excess Entry Theorem. See, for example, Spence (1976),
Dixit and Stiglitz (1977), and Suzumura and Kiyono (1987). For recent works, see Ghosh and Morita (2007),
who analyze a vertical oligopoly model and show that free entry leads to a socially insufficient number of firms.
6
One of the established literatures focuses on rational addiction, where a consumer’s utility is positively
related to the volume of his/her own past consumption (Becker and Murphy (1988)).
7
See Katz and Shapiro (1994) for a survey. Goolsbee and Klenow (2002) empirically examine the impor-3

size may be a signal of popularity, and consumers often desire a popular product becauseof the bandwagon effect, which is purely psychological.8 The existence of media reports
and advertisements informing consumers of past sales may support this hypothesis.9 Finally,an increase in past market size may provide more information about product quality to consumers.10
The rest of this paper is organized as follows. Section 2 sets up the model. Section
3 introduces a free-entry equilibrium and a socially optimal equilibrium. Section 4 analyzes
the social inefficiency of free-entry diffusion. Section 5 gives concluding remarks. The proofsof all results are provided in the Appendix.

2 ModelThis section develops the model, which follows Kitamura (2010). The new dimension hereis the oligopolistic interaction among firms: the output per firm is endogenously determined. This modeling strategy is designed to clarify the importance of oligopolistic interaction among firms.
We characterize the consumers’ behavior in 2.1 and the firms’ behavior under free entry
in 2.2. Then, we introduce the timing of the game in 2.3. We assume that time is discreteand that the horizon is infinite. In this paper, it is also assumed that the market is a perishable
goods market or a service market in which the service fee is charged in every period.

tance of learning and network externalities in the diffusion of home computers. They find that people are more
likely to buy their first home computer in areas where a large fraction of households already own computers or
when a large share of their friends and family already own computers.
8
See Leibenstein (1950), a seminal work on the bandwagon effect. Becker (1991) studies restaurant pricing
where consumer demand is positively related to market size. Biddle (1991) develops an empirical model of the
bandwagon effect and shows that the current demand is positively related to past demand levels.
9
See, for example, Monteiro and Gonzalez (2003).
10
See Caminal and Vives (1996), who theoretically analyze the importance of past market share as a signal of
product quality. Berndt, Pindyck, and Azoulay (2003) empirically examine the role of past sales in the demandfor pharmaceuticals. They find empirical evidence that the past sales of a drug have a positive effect on bothits value to consumers and its rate of diffusion at the brand level. See also Grinblatt, Keoharju, and Ikäheimo(2008), who find that the purchases of neighbors influence a consumer’s purchases of automobiles.

42.1 Consumers
There are a number of mass unit consumers for all periods. Each consumer has a different
preference for a product. Let θ be the type of consumer, which is stationary for all periods andis uniformly distributed on the interval [0, 1]. We also assume that the number of consumers isa/b, where a > 0, b > 0. The market size, defined as the number of consumers who purchasethe product, at period t is denoted by Qt . The consumers’ willingness to pay depends on
the previous period’s market size because of the intertemporal consumption externality. We
assume the following reservation price for consumers of type θ at periods t = 1, 2, ..., ut (θ):
Assumption 1.
ut (θ) = U(θ, Qt−1 ) = aθ + σ(Qt−1 ),(1)where σ(Qt−1 ) > 0 represents the intertemporal consumption externality and has the following properties: σ(0) = 0, σ0 (Qt−1 ) > 0, σ00 (Qt−1 ) < 0, σ000 (Qt−1 ) > 0, limQt−1 →0 σ0 (Qt−1 ) = ∞,and limQt−1 →∞ σ0 (Qt−1 ) = 0.
From Assumption 1, it is easy to see that the intertemporal consumption externality has
the following two properties: it is strictly increasing in the previous market size; however, itsdegree, or equivalently the benefit of the externality, is strictly decreasing. This assumption
guarantees that the market size converges to a finite number.
A consumer of type θ pays pt for the product and enjoys consumer surplus of ut (θ) − pt .
The consumer purchases the product if and only if the consumer surplus is nonnegative, i.e.,ut (θ) − pt ≥ 0. Then, the inverse demand function at period t, P(Qt−1 , Qt ), becomes:
a + σ(Qt−1 ) − bQt 0 ≤ Qt ≤ ba ,P(Qt−1 , Qt ) = 0
Qt > ab ,(2)

for all t = 1, 2, ..., and 0 ≤ Qt−1 ≤ a/b. It is easy to see that the inverse demand is a strictlyincreasing function of the previous period’s market size but a strictly decreasing function of
the current period’s market size.11
11

In the models of Rob (1991) and Vettas (2000), the demand is perfectly elastic: the demand curve is ahorizontal straight line. This demand structure does not allow us to analyze the role of oligopolistic interaction,
where firms compete in quantities. In contrast, the downward-sloping demand here allows us to introduce
oligopolistic interaction to the model of market diffusion.

5

2.2 Firms under Free Entry
Firms under free entry are identical. In contrast to Kitamura (2010), firms in this paper
compete in quantities, and the output per firm is endogenously determined. For every period,
there are incumbents and a large (infinite) number of potential entrants. When potential
entrants decide to enter the market, entry must incur a setup cost f > 0, which is the initialinvestment in purchases such as machines. We assume that machines are perfectly durable
and can be operated in an environment of constant returns to scale for all periods. Let c > 0
be the marginal production cost, i > 0 be a constant interest rate, and β ≡ 1/(1 + i) denote thediscount factor.

2.3 Timing
For each period, a period game consists of a two-stage game, as follows:
1. Stage 1: Entry Decision
In Stage 1, potential entrants decide whether to enter the market or not.12 If they enter
the market with setup cost f , they compete with incumbents in Stage 2 and earn profits.
If they do not, their profits for the period are zero. Let Nt be the number of incumbents
at period t and let nt be the number of new entrants in Stage 1 of period t. By definition,P
we have nt = Nt − Nt−1 for all t = 1, 2, .... Assuming that N0 = 0, we have Nt = tτ=1 nτ .
If the demand is small and the fixed cost is high or the discount factor is low, then entry
may not occur in the first period. The following assumption guarantees first-period
entry:
Assumption 2.
(a − c)2> (1 − β) f.
4b(3)Assumption 2 implies that the number of new entrants in the first period is at least 1,
i.e., N1 ≥ 1. If inequality (3) does not hold, then first-period entry is not profitable and12

In the models of Rob (1991) and Vettas (2000), the incumbent decides whether or not to exit the market
because of demand uncertainty. In contrast, exit never occurs here because there is no demand uncertainty, as
in Kitamura (2010).

6

does not occur. Since this condition holds for all following periods, the entry neveroccurs.
2. Stage 2: Production
In Stage 2, firms in the market compete in quantities to maximize operation profits forthe period. We assume that the equilibrium in this stage is symmetric. Let qit be theequilibrium output of firm i at period t. We also define pt as the equilibrium price inperiod t and πit = [pt − c]qit as the equilibrium operation profits of firm i in period t.

3 Equilibrium
This section provides the characterization of free-entry equilibrium and socially optimal equilibrium. We first characterize the post-entry equilibrium at period t given the number of incumbents Nt−1 and the number of new entrants nt in 3.1. Then, we characterize free-entry
equilibrium and show that free-entry diffusion becomes S-shaped in 3.2. Finally, we characterize socially optimal diffusion in 3.3.

3.1 Post-Entry Equilibrium
Given the number of firms in Stage 1, firms in Stage 2 compete in quantities and choose
their output levels to maximize their profits for the period, πit .13 The post-entry equilibrium isdetermined by the market clearing condition, the firms’ profit-maximizing behavior, and thesymmetry property. Now, we define the post-entry equilibrium as follows:
Definition 1. Given Nt−1 and nt , the post-entry equilibrium consists of sequences {Qt , pt , qit }that simultaneously satisfy the following conditions:1. Firm output is symmetric for all t = 1, 2, ...:
qit = qt for all i.
13(4)This implies that when firms choose their output levels, they do not take into account the entry of new firms
in future periods. In other words, firms play the stage game Nash equilibrium at each period. Although it would
be more realistic to assume that firms take future entry into account, the analysis would become considerably
more complicated.

7

2. The market clears for all t = 1, 2, ...:
Qt = Nt qt .

(5)

3. The market price is determined by the inverse demand for all t = 1, 2, ...:

pt = P Qt−1 , Qt .

(6)

4. Each firm’s output is the best responses to other firms’ outputs for all t = 1, 2, ...:h iqit = arg max P Nt−1 qt−1 , (Nt − 1)qt + qit − c qit for all i.

(7)qit ≥0According to the above definition, we identify the properties of the post-entry stage equilibrium. From equation (2) and equilibrium conditions (4)–(7), the output per firm becomes:
qt =1 a + σ(Nt−1 qt−1 ) − c.
Nt + 1b(8)Then, we obtain the post-entry equilibrium price and operation profits per firm, respectively:
pt = c +(a + σ(Nt−1 qt−1 ) − c),Nt + 1(9)and

1(a + σ(Nt−1 qt−1 ) − c)2.
(10)(Nt + 1)2b
It is easy to see that the post-entry equilibrium has the standard properties of the Cournot–πt =Nash equilibrium under linear demand and constant marginal cost. For the convenience of theanalysis in the following sections, we summarize these properties of post-entry equilibrium
as follows:
Lemma 1. The post-entry equilibrium has the following properties:
1. Aggregate output is strictly increasing in the number of firms and is bounded:a + σ(Nt−1 qt−1 ) − c∂Nt qt> 0 and lim Nt qt =
.
Nt →∞
∂Nt
b

(11)

2. Output per firm is strictly decreasing in the number of firms and converges to zero:∂qt< 0 and lim qt = 0.
Nt →∞
∂Nt
8

(12)

3. Equilibrium prices (operation profits per firm) are strictly decreasing in the number offirms and converge to the marginal cost (zero):∂pt∂πt< 0, lim pt = c and< 0, lim πt = 0.
Nt →∞
Nt →∞
∂Nt
∂Nt

(13)

One of the significant properties of the post-entry equilibrium is that the entry generates
an externality effect called a business stealing effect; that is, the new entry decreases theincumbents’ output levels. As proved in Mankiw and Whinston (1986), this effect makes
the free-entry equilibrium socially excessive in the static model. In the previous literature on
market diffusion, the output per firm is exogenously determined under free entry. Therefore,
the second property of Lemma 1 does not hold. This leads to different welfare implications
for market diffusion.

3.2 Free-Entry Diffusion
In this subsection, we first characterize the free-entry diffusion in 3.2.1 given the post-entry
equilibrium outcome derived in 3.1. Then, we examine the time pattern of free-entry diffusionand show that it becomes S-shaped by using the logic in the previous literature in 3.2.2.
3.2.1 Characterization of free-entry equilibrium
Let net be the free-entry equilibrium number of new entrants at period t, and let Nte be thefree-entry equilibrium number of firms at period t. For simplicity, we treat N as a continuous
variable. We define R(Nt−1 , nt ) as the discounted sum of future operation profits at periodt, which is composed of the direct operation profits at period t and the discounted future
operation profits,14 i.e.:R(Nt−1 , nt ) = πt + βR(Nt , nt+1 ),(14)

for all t = 1, 2, .... In Stage 1 of each period, potential entrants enter the market as long as
the present value of net profits is positive, i.e., R(Nt−1 , nt ) > f . Therefore, in each period, thenumber of new entrants satisfies the zero-profit condition, defined as follows:15
14

Because entrants and incumbents in Stage 2 are symmetric at each period, and the horizon is infinite, they
have the same present value of their future revenue streams.
15
One of the important factors for the existence of market diffusion is that the horizon is infinite. When thehorizon is infinite, the zero-profit condition holds with a positive number of entrants at each period. In contrast,9Definition 2. Given the post-entry equilibrium outcome, the free-entry equilibrium consistsof the sequence {nte }∞0 , which satisfies the zero-profit condition for all periods, i.e.:e, net ), with equality if net > 0.
f ≥ R(Nt−1(15)

According to the above definition, the properties of the free-entry equilibrium under thetransition process are identified. From the zero-profit condition (15), the operation profits inthe free-entry equilibrium become πet = (1 − β) f for each period. In addition, from equationsp(9) and (10), the price in the free-entry equilibrium becomes pet = c + (1 − β)b f for all
periods. These properties imply that both the equilibrium profits and the equilibrium priceare constant. By substituting equation (10) into the zero-profit condition (15), the market
diffusion under free-entry equilibrium is summarized as follows:
Proposition 1. Let N e be the number of firms in the steady state under the free-entry equilibrium. Suppose that N0e = 0. Then, for all t = 1, 2, ..., there exists a unique net > 0 thatsatisfies (15), while the free-entry equilibrium output per firm, qet , and number of firms satisfy
the following conditions:
1. The output per firm is constant over time:r(1 − β) feqt =, for all t = 1, 2, ....
b(16)

2. The number of firms is an increasing function of the number in the previous period:q(1−β) fea + σ Nt−1−cb
− 1, for all t = 1, 2, ...,
(17)
Nte =
p
(1 − β)b f
and it satisfies Nte ∈ [0, N e ] for all t = 1, 2, ..., and monotonicity, N0e = 0 and Nte → N e
as t → ∞.
Proof. See Appendix.
if the horizon is finite, then incumbents and entrants are not symmetric. To hold the zero-profit condition, theentrants need to achieve higher profits than the incumbents, and the number of incumbents should decrease.
Therefore, under a finite horizon, the zero-profit condition does not lead to a positive number of entrants at eachperiod.

10

Q.E.D.
The dynamic system of equation (17) is summarized in Figure 1. Figure 1 shows that√p
we have σ0 (N e (1 − β) f / b)/b < 1 in the steady state.16 This indicates that the number offirms under free entry does not reach the steady state as long as the degree of consumption
externality is strong enough.
The constant values of equilibrium output, profits per firm, and equilibrium price haveseveral implications. First, the time pattern of the free-entry equilibrium number of firms
coincides with that of the aggregate output. In addition, the constant equilibrium price implies
that the firms under free entry act as if they were price takers whose output levels were
exogenously determined. Therefore, the free-entry equilibrium, in this paper, basically has
the same properties as in the previous literature on market diffusion.
3.2.2 S-shaped market diffusion
We now examine the time pattern of free-entry diffusion and show that it becomes S-shaped(initially convex and eventually concave) when the externality effect is initially strong enough.
Note that the degree of consumption externality decreases as the market size increases. This
makes the number of firms under free entry monotonically converge to the steady state, andthe time pattern eventually becomes concave. Therefore, the time pattern of free-entry diffusion becomes S-shaped if and only if the number of new entrants initially increases.
Note that in free-entry diffusion, the numbers of firms in the first and second periods arepp
N1e = [a − c]/ (1 − β)b f − 1 and N2e = [a + σ(N1e qe1 ) − c]/ (1 − β)b f − 1, respectively. While
the former does not depend on the consumption externality, the latter does. This implies thatthe strong degree of consumption externality leads to the high market growth from Period 1
to Period 2 and the convex time pattern of market diffusion in the early periods:
Proposition 2. The time pattern of free-entry diffusion becomes S-shaped if and only if:p a − c − (1 − β)b f pa − c − (1 − β)b f < σ.
(18)bProof. See Appendix.
16

√
p
e
e
e
(1 − β) f / b)/b.
Note that by differentiating (17) with respect to Nt−1
, we have ∂Nte /∂Nt−1= σ0 (Nt−1

11Q.E.D.
From inequality (18) and the properties of σ(·), it is easy to see that the initial convexityof the free-entry diffusion is observed in the environment of small values of initial market
size. This follows from the low market demand, low discount factor, and high production andsetup costs. In addition, given these parameters, the strong consumption externality effectcontributes to the initial convexity of free-entry diffusion. From Figure 1, it is obvious that the√pefree-entry diffusion has a convex time pattern provided that σ0 (Nt−1(1 − β) f / b)/b > 1.
Therefore, we conclude that the S-shaped diffusion is observed for low initial market size anda strong consumption externality effect.
The mechanism of the initial convexity of free-entry diffusion, explained above, is thesame as that in Vettas (2000) and Kitamura (2010), in which firms are small atomistic pricetakers. Therefore, our result implies that the oligopolistic interaction does not play an essential role in explaining the fundamental mechanism of S-shaped diffusion.

3.3 Socially Optimal Diffusion
A social planner sets the number of firms to maximize the social welfare, given the postentry equilibrium outcome derived in 3.1. Let not be the number of new entrants and Nto be
the number of firms set by the planner at period t, and let qot be the output per firm undersocially optimal planning at period t. Now, we define the socially optimal equilibrium asfollows:
Definition 3. Given the post-entry equilibrium outcome, socially optimal planning satisfies
the following Bellman equation:
Z (N o +not )qot
t−1
o
o
o
o
+not ), (19)V(Nt−1 ) = max[a+σ(Nt−1
qot−1 )−bQ]dQ−(Nt−1+not )qot c−not f +βV(Nt−1
o
nt ≥0

0

subject to equation (8).
The interpretation of equation (19) is as follows. The present value of the sum of future
welfare is current welfare plus the discounted next-period value of the sum of future welfare.

12Note that the social planner is unable to control directly the output per firm in Stage 2 (postentry stage) but is able to do so indirectly by controlling the number of firms.17 We now
characterize the socially optimal equilibrium as follows:
Proposition 3. Suppose that N0o = 0. Then, the optimal diffusion path satisfies the following
second-order difference equation:of = R(Nt−1, not )∂qo
+ [pot − c]Nto t∂nth∂qo i o o
+ βσ0 (Nto qot ) qot + Nto t Nt+1
qt+1
∂nt
h ∂qot+1 ∂qot+1 io+ β[pot+1 − c]Nt+1
−
,
∂Nt
∂nt+1

(20)

where R(Nt−1 , nt ) = πt + β f .
Equation (20) shows that the marginal expansion cost is equal to the marginal social
benefit, which is composed of four elements. The first term on the right-hand side of equation(20) is the present value of future operation profits. The second term is welfare loss from thebusiness stealing effect, which is captured as ∂qot /∂nt : the new entry reduces the current
output per firm. This term is regarded as the current loss: increasing current entry reduces
the social welfare. The third term is the future benefit from the demand shift effect, in which
an increase in the number of firms directly raises demand in the subsequent period. The lastterm is the future benefit following the business creating effect, ∂qot+1 /∂Nt − ∂qot+1 /∂nt+1 : the
new entry indirectly raises the output per firm in the subsequent period through the demand
shift effect.
The last three terms represent the intertemporal trade-off. The current loss gives theplanner an incentive to restrict the number of firms. However, the planner has a competing
incentive to raise the number of firms because of future benefits. The optimal planning isdetermined by the magnitudes of these losses and benefits.18
17

Therefore, this equilibrium is interpreted as the second-best equilibrium rather than the first-best equilibrium, in which the planner would be able to control directly the output per firm. If that were the case, thefirst-best equilibrium number of firms would be 1 because of economies of scale.
18
In socially optimal diffusion, in contrast to free-entry diffusion, the number of firms in the first period

13Note that the new dimension here beyond the previous literature is the existence of theintertemporal trade-off in the optimal planning. This trade-off has not been addressed in
the previous literature concerned with market diffusion. In past works, the business stealing
effect and business creating effect are not considered, although there exists a benefit frominternalizing intertemporal externalities, which are engines of the market growth.19

4 Social Inefficiency and Time DependenceThis section analyzes the social inefficiency of free-entry diffusion by comparison with socially optimal diffusion. Note that the only difference between free-entry diffusion and socially optimal diffusion is whether the intertemporal trade-off exists or not. We first explore
the social inefficiency of free-entry diffusion when the market is in the growing phase in 4.1.
Then, we examine the case when the market is in the mature phase in 4.2.

4.1 Social Inefficiency in the Growing Phase
To begin the analysis, we rewrite equation (20). Let the current loss of increasing the numberoof firms at period t be µ(Nt−1, Nto ), and let the future benefit of increasing the number ofoofirms at period t be λ(Nt−1, Nto , Nt+1). Then, we rewrite equation (20) with the linear demand

function as follows:oooo(1 − β) f = π(Nt−1, Nto ) − µ(Nt−1, Nto ) + λ(Nt−1, Nto , Nt+1
),

where

1
(a + σ(Nt−1 qt−1 ) − c)2,(Nt + 1)2b
Ntπ(Nt−1 , Nt ),µ(Nt−1 , Nt ) =Nt + 1βσ0 (Nt qt ) Nt+1 qt+1 Nt+1 + 2
λ(Nt−1 , Nt , Nt+1 ) =µ(Nt−1 , Nt ).
b
Nt qt Nt+1 + 1π(Nt−1 , Nt ) =

(21)

(22)
(23)
(24)

depends on, and is positively related to, the degree of intertemporal consumption externality. Therefore, thestrong externality effect leads to an initially large number of new entrants in socially optimal diffusion. This
makes the S-shaped time pattern more difficult to obtain than in free-entry diffusion.
19
Since the output per firm is exogenously determined in this literature, ∂qot /∂nt = 0 and ∂qot /∂Nt−1 = 0 for
all t = 1, 2, .... It is easy to see that the business stealing effect and the business creating effect in equation (20)are absent, but the future benefit from the demand shift effect still exists.

14eNote that the free-entry diffusion satisfies (1 − β) f = π(Nt−1, Nte ) for all t = 1, 2, .... There-

fore, the difference in the number of firms depends on the magnitude of the current loss,ooo, Nto ), and the magnitude of the future benefit, λ(Nt−1, Nto , Nt+1), which depends on theµ(Nt−1demand shift effect and the business creating effect.
Furthermore, the future benefit at period t is determined by the degree of discounting, β,othe degree of consumption externality, σ0 (Nto qot ), the growth rate of the market size, [Nt+1
qot+1 −oNto qot ]/Nto qot , and the number of firms at period t + 1, Nt+1
. By comparing equations (23) and(24), it is seen that future benefits are produced by a higher discount factor, a stronger degreeof consumption externality, higher market growth, and a smaller number of firms. In thisenvironment, free entry leads to a socially insufficient number of firms:ooo
Proposition 4. Suppose that µ(Nt−1, Nto ) ≷ λ(Nt−1, Nto , Nt+1), i.e.:ooqot+1 Nt+1+2
βσ0 (Nto qot ) Nt+1
1≷.
ob
Nto qot Nt+1
+1

(25)

e
o
Then, for all Nt−1
R Nt−1, we have Nte ≷ Nto .

Proof. See Appendix.
Q.E.D.
Note that we have N0e = N0o = 0. Therefore, Proposition 4 implies that, at Period 1, thefree-entry equilibrium number of firms becomes socially insufficient, N1e < N1o , when the
future benefit is larger than the current loss, µ(N0o , N1o ) < λ(N0o , N1o , N2o ).
In addition, by interpreting Proposition 4 differently, we see that the free-entry equilibrium number of firms is more likely to be socially insufficient at early periods. At early
periods, (a) the degree of consumption externality is strong,20 (b) the growth rate of the market size is high,21 and (c) the number of firms is small. Therefore, we conclude that thefree-entry equilibrium number of firms tends to be socially insufficient when the market is in
the growing phase.
Note that the right-hand side of (25) is larger than 1 as long as βσ0 (Nto qot )/b ≥ 1.
Note that insufficient entrants appear even if βσ0 (Nto qot )/b < 1. This occurs if the growth rate of the market
is high enough and the number of firms is small enough.
20

21

15

4.2 Social Inefficiency in the Mature Phase
Next, we turn to the analysis of the mature phase. Let N o denote the steady-state number ofofirms in the socially optimal diffusion that satisfies Nto = Nt+1
= N o . Then, in the steady state,

we can rewrite equations (21)–(24) as follows:(1 − β) f = π(N o ) − µ(N o ) + λ(N o ),where

(a + σ(Nq) − c)2,(N + 1)2 b
Nµ(N) =π(N),N+1
βσ0 (Nq) N + 2µ(N).
λ(N) =b
N+1π(N) =

(26)

(27)
(28)
(29)

Note that (1−β) f = π(N e ) holds in free-entry diffusion. Thus, as in the analysis of the growing
phase, the difference in the number of firms depends on the magnitudes of the current loss,µ(N o ), and future benefit, λ(N o ). As the market becomes mature, the degree of consumption
externality and the growth rate of the market size become lower, and the number of firms
operating in the market increases. Compared with the growing phase, these changes in themarket environment lower the magnitude of the future benefit relative to the current loss,oooi.e., λ(Nt−1, Nto , Nt+1)/µ(Nt−1, Nto ) ≥ λ(N o )/µ(N o ).22 More importantly, the following lemmashows that the future benefit becomes smaller than the current loss in the steady state:
Lemma 2. In the steady state, the future benefit becomes smaller than the current loss, i.e.,µ(N o ) > λ(N o ). More precisely, we have:1>

βσ0 (N o qo ) N o + 2.
bNo + 1(31)

Proof. See Appendix.
22

By comparing equations (24) and (29), it is easy to see that:ooooλ(Nt−1, Nto , Nt+1
) βσ0 (Nto qot ) Nt+1
qot+1 Nt+1
+ 2 βσ0 (N o qo ) N o + 2 λ(N o )
=
≥
=
,
o
o
o o
o
µ(Nt−1 , Nt )b
Nt qt Nt+1 + 1b
N o + 1 µ(N o )obecause σ00 (·) < 0, ∂Nt qt /∂Nt > 0, and Nto ≤ Nt+1
≤ No.

16

(30)

Q.E.D.
Lemma 2 implies that, in the steady state, the social benefit of increasing the number offirms, π(N) − µ(N) + λ(N), becomes smaller than the private benefit of firms entering themarket, π(N). Therefore, it is optimal for the social planner to slow down the rate of new
firms entering as the market becomes mature. The following proposition shows that, in thesteady state, the socially optimal number of firms is smaller than the free-entry equilibrium
number of firms:
Proposition 5. In the mature phase (steady state), the free-entry equilibrium number of firms
is socially excessive, i.e., N e > N o .
Proof. See Appendix.
Q.E.D.
From Propositions 4 and 5, we conclude that the free-entry equilibrium number of firms
tends to be initially insufficient but eventually excessive. This is in contrast to the results ofprevious studies, where the number of firms in the free-entry diffusion is socially insufficient
over time. Table 1 and Figure 2 present a numerical example in which the number of firms
under free entry is initially insufficient but eventually excessive.23
The result in this section implies that oligopolistic interaction between firms can be an
important factor for the discussion of entry regulation policy in new industries. According to
the previous literature, since the number of firms in free-entry diffusion is socially insufficient
over time, encouraging entry is a desirable policy regardless of the phase of market growth. In
contrast, this paper provides a different policy implication. The entry regulation policy should
be changed depending on the phase of market growth: entry should be initially encouraged
but eventually discouraged.

5 Concluding Remarks
This paper models market diffusion in the presence of oligopolistic interaction. In contrast to
the previous literature on market diffusion, not only the number of new entrants but also the23

In this example, the steady state of the socially optimal diffusion path is locally saddle stable.

17

output per firm is endogenously determined by the oligopolistic interaction. In this setting,
we explore the social inefficiency of free-entry diffusion.
The major result reported here is that the social inefficiency of free-entry diffusion depends on the time period. The free-entry equilibrium number of firms tends to be initiallyinsufficient but eventually excessive. This result is in contrast to the previous literature not
only on market diffusion, where the free-entry equilibrium is found to be socially insufficient
over time, but also on the Excess Entry Theorem, where free entry is found to be socially
excessive in the static model.
These results provide important policy implications for entry regulations in new industries. The entry regulation policy needs to be changed depending on the time and the degreeof market growth: entry should be initially encouraged but eventually discouraged. It is possible to improve social welfare by giving subsidies to early entrants but taxing late entrants
without violating intertemporally balanced budget constraints.24
There are several issues that require future research. First, as the number of firms becomesstable, there is a possibility of market restructuring. A stable number of firms may generatecollusion between firms. Another instance of market restructuring is horizontal mergers. If
there exist cost synergies, market maturity may lead to horizontal mergers, which reduce
the number of firms and improve welfare (see Davidson and Mukherjee (2007)). This may
explain the shakeout of firms corresponding to Stage 3 to Stage 5 in Gort and Klepper (1982).
Second, there is concern about whether or not our results hold under other intertemporalexternalities. Although we use only intertemporal consumption externalities in our model, we
predict that our results would hold even in the presence of other intertemporal externalities.
Finally, there is concern about the generality of our results. Our results are obtained in termsof a parametric example; however, they may apply in a more general setting. We hope that
our study helps researchers to address these issues.

24

The optimal subsidy or tax is equal to the difference between the right-hand side of equation (15) and theright-hand side of equation (20), that is, equal to the last three terms of equation (20). If these terms are positive(negative), then subsides (tax) should be given (imposed).

18

Appendix
Proof of Proposition 1eee∈ [0, N e ). Then, R(Nt−1, net ) = f for all Nt−1
∈ [0, N e ).
We first guess that net > 0 for all Nt−1e
, Nte ) with respect to Nte , we have:
By solving (1 − β) f = π(Nt−1e)=
Nte = N(Nt−1ea + σ(Nt−1
qet−1 ) − c− 1,p(1 − β)b f(32)

for all t = 1, 2, .... Together with equation (8), we obtain equations (16) and (17). Frompe
the properties of σ(·), we have N1e = N(N0e ) = [a − c]/ (1 − β)b f − 1 > 0, N 0 (Nt−1) > 0,e0e0eee →0 N (Ne
N 00 (Nt−1) < 0, limNt−1
t−1 ) = ∞, and limNt−1 →∞ N (Nt−1 ) = 0. Therefore, N(Nt−1 )ecrosses the Nte = Nt−1
line only once, and there is a unique steady state, N e . We finally verify
ethat net > 0. From Figure 1, it is easy to see that we have net > 0 for all Nt−1
∈ [0, N e ).

Q.E.D.

Proof of Proposition 2
From Figure 1, the time pattern of free entry eventually becomes concave. Therefore, it
becomes S-shaped if and only if 2N1e < N2e . From equation (17), we obtain inequality (26).
Q.E.D.

Proof of Proposition 4
We prove the first case. Let 1 >o qo N o +2
βσ0 (Nto qot ) Nt+1eo
t+1 t+1
o +1 and Nt−1 ≥ Nt−1 . Suppose in negation thatb
Nto qot Nt+1

Nte ≤ Nto . Then, from the properties of π(Nt−1 , Nt ), we would have the following inequalities:oee, Nto ).
, Nto ) ≥ π(Nt−1
π(Nt−1, Nte ) ≥ π(Nt−1

(33)

o
e
, Nto ) ≤ (1 − β) f .
Because π(Nt−1, Nte ) = (1 − β) f in the free-entry equilibrium, we have π(Nt−1ooo). In the same
, Nto , Nt+1
This is a contradiction to equation (20) because µ(Nt−1, Nto ) > λ(Nt−1way, we can prove the second case.
Q.E.D.
19

Proof of Lemma 2
Suppose in negation that:

βσ0 (N o qo ) N o + 2.
bNo + 1
Then, we would have the following inequalities:
1≤

(34)

βσ0 (N o qo )N o qo N o + 2 σ(N o qo ) N o + 2
<
,
(35)
b
No + 1b
No + 1
where the last inequality follows from the properties of σ(·). By substituting qo = (a +N o qo ≤σ(N o qo ) − c)/(b(N o + 1)) into inequality (35), we have"#o oσ(N o qo ) N o + 2σ(N o qo ) N o + 2
o oo a + σ(N q ) − c
⇔N
<
(36)
N q <
b
No + 1b(N o + 1)b
No + 1
N o [a − c]⇔< σ(N o qo ).
(37)
2
By using inequality (37), we have:" o#2
N o [a−c]
N + 2 (a − c)2 (a − c)2(a + σ(N o qo ) − c)2 (a + 2 − c)2
o
>
= o
>
. (38)
π(N ) =b(N o + 1)2b(N o + 1)2
N +1
4b
4b
Note that inequality (34) implies that µ(N o ) ≤ λ(N o ). Then, together with inequality (38) andAssumption 2 (inequality (3)), we have:(a − c)2> (1 − β) f.
(39)
4bHowever, this contradicts the equilibrium condition (26). Therefore, inequality (31) holds.
π(N o ) − µ(N o ) + λ(N o ) >Q.E.D.

Proof of Proposition 5
As mentioned in Subsection 3.2, we have σ0 (N e qe )/b < 1 in the steady state of free-entry
diffusion. Hence, we assume that N e > N̂, where N̂ is such that σ0 (N̂ q̂)/b = 1. To begin the
proof of Proposition 5, we have the following lemma:
Lemma 3. Let Q(N) = Nq. Then, for N ∈ (N̂, ∞),1. Q0 (N) → 0 and Q(N) → m ∈ (0, ∞) as N → ∞.
2. π(N) is strictly decreasing in N and approaches zero as N becomes larger: π0 (N) < 0
and limN→∞ π(N) = 0.
20

Proof of Lemma 3
1. By differentiating Q(N) with respect to N, we have:0

Q (N) =

N
[1 − N+1]q(N)0N σ (Q(N))
1 − N+1
b

> 0,

(40)

for N ∈ (N̂, ∞). It is easy to see that Q0 (N) → 0 as N → ∞. Then, by using L’Hôpital’srule, we obtain:
σ0 (Q(N))NQ0 (N) o
N→∞
b
b
N
n a + σ(Q(N)) − c on σ0 (Q(N)) [1 − N+1]Q(N) o
= lim
+ lim
0
N→∞
N→∞
b
b
1 − N σ (Q(N))

lim Q(N) = limN→∞n a + σ(Q(N)) − c

+

N+1
b
N
N
nh a + σ(Q(N)) − c ih
σ (Q(N)) [1 − N+1 ] N+1 io= lim
1+
N σ0 (Q(N))
N→∞bb
1 − N+1b
0(41)

n a + σ(Q(N)) − c o.
N→∞b
Since σ(·) satisfies the Inada condition, and a > b, there exists a unique m ∈ (0, ∞)= limsuch that m = [a + σ(m) − c]/b.
2. By the properties of Q(N), it is easy to see that π(N) → 0 as N → ∞. Next, we showthat π(N) is strictly decreasing in N. By differentiating π(N), we have:
2π(N)[b − σ0 (Nq)]< 0,π0 (N) = −N[b − σ0 (Nq)] + b(42)for N ∈ (N̂, ∞). Thus, π(N) is strictly decreasing in N.
This completes the proof of Lemma 3.
Q.E.D.
Now, we turn to the proof of Proposition 5. Suppose in negation that N e ≤ N o . Then,because π(N) is strictly decreasing in N ∈ (N̂, ∞) by Lemma 3, we have the following inequality:
π(N e ) ≥ π(N o ).

(43)

Since N e satisfies π(N e ) = (1−β) f , we have π(N o ) ≤ (1−β) f . However, this is a contradiction
to (26) because µ(N o ) > λ(N o ) by Lemma 2. Therefore, we have N e > N o .
Q.E.D.
21

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24

N te

N e t = N e t −1

Ne
N 4e
N te =

N 3e()

a + σ N te−1 [(1 − β ) f ] b −1/ 2 − c−1[(1 − β )bf ]1/ 2
1/ 2

N 2e

N1e
0

Ne
Figure 1

Nte−1

160

140

Number of firms

120

100

80

OptimalFree entry

60

40

20

0
period 0

period 1

period 2

period 3

period 4

period 5

period 6

period 7

period 8

period 9 period 10

Time

Figure 2
The optimal and the free-entry equilibrium paths for
a = 8, b = 0.5, c = 5, f = 18, β = 0.97, σ (Qt −1 ) = 6(Qt −1 )0.5

Time

Optimal Path

Free-entry Pathperiod 000period 1

20.68957495

4.773502692

period 2

48.58768294

30.49189952

period 3

71.5955845

69.77412988

period 4

86.02806271

103.1004422

period 5

93.94248156

124.2976272

period 6

98.00723622

136.0106333

period 7

100.0294918

142.054915

period 8101.0202103

145.0721357

period 9101.5019707

146.5542658

period 10

101.7353969

147.2766772

period 11

101.8483019

147.6274666

Table 1