Discussion Papers In Economics
And BusinessFree Entry, Market Diffusion, and Social Inefficiency
with Endogenously Growing Demand
Hiroshi Kitamura, Akira Miyaoka and Misato Sato

Discussion Paper 11-04-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFree Entry, Market Diffusion, and Social Inefficiency
with Endogenously Growing Demand
Hiroshi Kitamura, Akira Miyaoka and Misato Sato

Discussion Paper 11-04-Rev.

November 2012

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFree Entry, Market Diffusion, and Social Inefficiency
with Endogenously Growing Demand∗
Hiroshi Kitamura†

Akira Miyaoka‡

Misato Sato §November 23, 2012

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 the nature of theinefficiency under free entry can change as the market grows, and more importantly, thatS-shaped diffusion can be a signal that the number of firms under free entry is initiallyinsufficient, but eventually excessive.

JEL Classification Codes: D11, L11, L14.
Keywords: Free Entry; Market Diffusion; Intertemporal Externalities; Oligopolistic Interaction; S-shaped Diffusion.
∗
We especially thank Katsuya Takii, Shingo Ishiguro, Roberto Samaniego, Junichiro Ishida, Noriaki Matsushima, Reiko Aoki, David Flath, Makoto Hanazono, Hiroaki Ino, Masayuki Kanazaki, Jong In Lee, Keizo
Mizuno, Toshihiro Matsumura, Keiichi Morimoto, Mitsuru Sunada, Ryoji Yoshioka, the conference participantsat the Japanese Economic Association, the Japanese Association for Applied Economics, and 2011 Institutions
and Economics International Conference, and the seminar participants at Osaka University, Osaka Prefecture
University, and Kwansei Gakuin University for helpful discussions and comments. We also thank Hideshi Itohand two anonymous referees for constructive comments and suggestions. We finally thank Haruna Tsuchiya foroutstanding research assistance. The first author gratefully acknowledges financial support from JSPS Grantin-Aid for Research Activity start-up No. 22830075, for Young Scientists (B) No. 24730220, and for Scientific
Research (A) No. 22243022, organized by Masaki Nakabayashi. The second author gratefully acknowledges
financial support from Grant-in-Aid for JSPS Fellows No. 24·1593 and the Global COE program “Human
Behavior and Socioeconomic Dynamics” of Osaka University. The usual disclaimer applies.
†
Corresponding Author: Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku,
Kyoto-City, Kyoto, 603-8555 Japan. E-mail: hiroshikitamura@cc.kyoto-su.ac.jp Phone: +81-75-705-1584‡
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. In particular, the time pattern in the growth of the numberof firms from Stage 1 to Stage 3 is characterized as “S-shaped diffusion.” Gort and Klepper(1982) and other papers observe this phenomenon in several markets, such as computers,
television products (Gort and Klepper, 1982), and generic drugs (Ching, 2010).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; Kitamura, 2010).3 The common features of these studies are thatfirms are atomistic price takers whose production levels are exogenously determined, and thatthe number of firms under free entry is socially insufficient over time.
This paper aims to investigate theoretically market diffusion with oligopolistic interaction
among firms. We develop a dynamic model of market diffusion following Kitamura (2010).
In his model, the market grows because of intertemporal consumption externalities, throughwhich 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 maximizes social welfare.4 The
1There are several papers on the growth of markets. See, e.g., Klepper and Graddy (1990), Jovanovic andMacdonald (1994), and Klepper (1997).
2
Empirical evidence shows that the inter- and intrafirm diffusion of new technology tends to be S-shaped. See
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 and found 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).
3
See also Vettas (1998), who develops a model of two-sided learning to explain S-shaped diffusion.
4
One of the important elements to develop 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 role1

novel dimension here is that the output per firm is endogenously determined.5
We demonstrate that the existence of oligopolistic interaction turns out to be crucial in thenature of inefficiency of free entry. In the previous literature, the number of firms under freeentry is insufficient over time. In our model, in contrast, the nature of the inefficiency (i.e.,
too few firms or too many) can change depending on the degree of market maturity, and, more
importantly, S-shaped diffusion can be a signal that the number of firms under free entry is
initially insufficient, but eventually excessive.
To understand this result, we begin by considering the case where the output level is exogenously determined, as in the previous literature. When the output level of each firm isexogenous, new entry only leads to the demand-shift effect; i.e., new entry today increases
tomorrow’s demand because of intertemporal consumption externalities. This effect can 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 over time
because firms under free entry do not internalize the future benefit from intertemporal externalities when they enter the market.
In contrast, when oligopolistic interaction exists, new entry today also leads to a “businessstealing effect”; i.e., new entry causes existing firms to reduce their output levels today. Thiseffect 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 on the magnitudes of the
future benefit from demand shift and of the current loss from business stealing. Because, asin Kitamura (2010), an S-shaped time pattern of free-entry diffusion arises for the initiallystronger but eventually weaker demand-shift effect, S-shaped diffusion can be a signal of theinitially insufficient, but eventually excessive number of firms under free entry.
These results provide 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 initially encouraged, but eventually restricted. In particular, this policy recommendation tends to applyof 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.
5
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

to the industries where S-shaped diffusion has already been observed in other countries.
This paper is related to the literature examining the social inefficiency of free entry.
Mankiw and Whinston (1986) and Suzumura and Kiyono (1987) show that the numberof firms under free entry is socially excessive because oligopolistic interaction leads to abusiness-stealing effect.6 This finding is often called the “excess-entry theorem.”7 Recently,
on the other hand, several papers show that free entry can be socially insufficient under different circumstances. Ghosh and Morita (2007a, b) show that if we consider the vertical
relationship between firms, then free entry can lead to a socially insufficient number of firms.
Mukherjee (2010) also obtains the result of insufficient entry by considering the external
economies of scale under which the production cost of each firm decreases as more firms
enter the market.8
These results show a trade-off between negative and positive externalities of new entry
in a static environment and which externality becomes dominant depends on the exogenous
parameters in the model. In contrast, this paper focuses on the intertemporal trade-off between the current loss and future benefit of new entry and shows that the relative size of theseexternalities depends not only on the exogenous parameters, but also on the phase of market
growth. In addition, our analysis within a dynamic framework also enables us to explore therelationship between social inefficiency of free entry and the time pattern of market diffusion.
Therefore, our approach allows us to shed light on the importance of a dynamic perspective
in analyzing the inefficiency of free entry.
This paper is also related to the literature concerned with intertemporal consumption externalities.9 Some empirical studies actually report that demand can be positively related topast market size. Goolsbee and Klenow (2002) empirically examine the importance of learn6

See also von Weizsacker (1980) and Perry (1984).
According to Suzumura and Kiyono (1987) and Suzumura (2012), there are two types of excess-entry
theorem, the “first-best” or “second-best.” In the former (latter), a social planner is assumed to be (un)able tocontrol each firm’s output or price level in the post-entry stage. In this paper, we assume a “second-best” social
planner. Therefore, the socially optimal diffusion in our paper corresponds to a “second-best” social welfare
optimum, and the terms “insufficient” or “excessive” are used from a “second-best” social welfare perspective.
8
Spence (1976) and Dixit and Stiglitz (1977) show that the number of firms under free entry can be socially insufficient if consumers prefer product diversity. In addition, Mukherjee (2011) shows that under the
Stackelberg leader–follower structure, free entry leads to a socially insufficient number of firms.
9Intertemporal consumption externalities here are also related to rational addiction, where a consumer’sutility is positively related to the volume of his/her own past consumption (Becker and Murphy, 1988).
73

ing 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.10 In addition, Berndt, Pindyck, and Azoulay (2003) present empiricalevidence that the past sales of a drug have a positive effect on both its value to consumers andits rate of diffusion at the brand level.11 In a (generic) drug market, the qualities of drugs tend
to be initially uncertain for doctors and patients. Therefore, the widespread use of the drug inprevious periods can be a signal that it is sufficiently safe or effective.12
The rest of this paper is organized as follows. Section 2 sets up the model. Section 3 introduces free-entry and socially optimal equilibria. Section 4 analyzes the social inefficiencyof free-entry diffusion. Section 5 gives concluding remarks. All proofs are provided in the
Appendix.

2 ModelThis section develops the model following Kitamura (2010). The new dimension here is theoligopolistic interaction among firms: the output per firm is endogenously determined. Thismodeling strategy clarifies 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.

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 and
10

For a survey of the literature on network externalities, see Katz and Shapiro (1994).
Caminal and Vives (1996) analyze the importance of past market share as a signal of product quality. Monteiro and Gonzalez (2003) analyze the role of past-sales advertising as a corporate strategy. Grinblatt, Keoharju,
and Ikäheimo (2008) find that the purchases of neighbors influence a consumer’s purchases of automobiles.
12
There is also the bandwagon effect (Leibenstein, 1950). 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.
114

is 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 strictlyincreasing in the previous period’s market size, but strictly decreasing in the current period’s
market size. Note that our demand structure differs from that of Rob (1991) and Vettas(2000). In their models, the demand is perfectly elastic: the demand curve is a horizontal
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 demandhere allows us to introduce oligopolistic interaction to the model of market diffusion.13
13

In addition to the perfectly elastic demand, Rob (1991) and Vettas (2000) assume demand uncertainty andfirms’ learning. If we introduced demand uncertainty into our model, an additional externality associated withlearning the demand would emerge and the analysis would be considerably more complicated.

5

2.2 Firms
In this paper, in contrast to Vettas (2000) and Kitamura (2010), firms compete in quantities,
and the output per firm is endogenously determined. For every period, there are incumbents
and a large (infinite) number of identical potential entrants. When potential entrants decideto enter the market, they must incur a setup cost f > 0, which is the initial investment
in purchases such as machines. We assume that machines are perfectly durable and canbe operated in an environment of constant returns to scale for all periods. To simplify the
analysis, we assume that the scrap value of machines is zero. Let c > 0 be the marginal
production cost and β ∈ [0, 1) denote the discount factor.

2.3 Timing
For each period, a period game consists of a two-stage game, as follows.
2.3.1 Stage 1: Entry Decision
In Stage 1, potential entrants decide whether to enter the market.14 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, we have nt = Nt − Nt−1
P
for all t = 1, 2, .... Assuming that N0 = 0, we have Nt = tτ=1 nτ . If the demand is small andthe 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 the first-period entrants is larger than one, i.e.,
N1 > 1. If inequality (3) does not hold, the first-period entry is not profitable and does notoccur. Because this condition holds for all following periods, the entry never occurs.
14

Because the setup cost is not recoverable, and the equilibrium operating profits become nonnegative in theenvironment of linear demand and constant marginal cost, exit never occurs in this model.

62.3.2 Stage 2: Production
In Stage 2, firms in the market compete in quantities to maximize operating profits for theperiod. We assume that the equilibrium in this stage is symmetric. Let qit be the equilibrium
output of firm i at period t. We also define pt as the equilibrium price in period t and πit =[pt − c]qit as the equilibrium operating profits of firm i in period t.

3 Equilibrium
This section provides the characterization of free-entry equilibrium and socially optimal diffusion. 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 socially optimal diffusion in 3.2 and 3.3, respectively.

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 .15 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.

(4)

2. The market clears for all t = 1, 2, ...:
Qt = Nt qt .
15

(5)

If we assumed that firms decide their output levels to maximize the discounted sum of future operating
profits, then the analysis of socially optimal diffusion would become considerably more complicated while theresults in the free-entry equilibrium are unchanged. See also footnotes 18 and 20.

7

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 response 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 operating 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 analysis in thefollowing 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:∂Nt qta + σ(Nt−1 qt−1 ) − c> 0 and lim Nt qt =
.
N
→∞
∂Nt
b
t

(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

(12)

3. Equilibrium prices (operating 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
8

(13)

One of the significant properties of the post-entry equilibrium is that the entry generates
an externality effect called a business-stealing effect; i.e., the new entry decreases the incumbents’ output levels. As proved in Mankiw and Whinston (1986) and Suzumura and Kiyono(1987), 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 todifferent welfare implications for market diffusion.

3.2 Free-Entry Diffusion
In this subsection, we characterize the free-entry diffusion given the post-entry equilibriumoutcome derived in 3.1. Let net be the free-entry equilibrium number of new entrants atperiod t, and let Nte be the free-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
operating profits at period t, which is composed of the direct operating profits at period t andthe discounted future operating profits,16 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:17
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.:ef ≥ R(Nt−1, net ), with equality if net > 0.
16

(15)

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.
17
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,
if the horizon is finite, incumbents and entrants are not symmetric. To hold the zero-profit condition, the entrants
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 each period.

9

According to the above definition, the properties of the free-entry equilibrium under thetransition process are identified. From the zero-profit condition (15), the operating 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:18
Proposition 1. Let N e be the steady-state number of firms in the free-entry equilibrium thatesatisfies Nte = Nt+1
= N e . Suppose that N0e = 0. Then, for all t = 1, 2, ..., there exists a unique

net > 0 that satisfies (15), while the free-entry equilibrium output per firm, qet , and the numberof firms satisfy the following conditions:
1. The output per firm is constant over time:r(1 − β) f
qet =, for all t = 1, 2, ....
b(16)

2. The number of firms is an increasing function of the number in the previous period:
q


f
e
−c
a + σ Nt−1 (1−β)
b− 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 → ∞.
The dynamical 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.19 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.
[Figure 1 about here.]18

In this paper, we assume that firms choose their output levels to maximize their profits for the period.
However, even if we assumed that firms maximize the discounted sum of future operating profits, our resultsin the free-entry equilibrium are unchanged. This is because the free-entry equilibrium continuation profit isalways equal to f due to the zero-profit condition and this does not depend on today’s output
p level. √19eee(1 − β) f / b)/b.
Note that by differentiating (17) with respect to Nt−1
, we have ∂Nte /∂Nt−1= σ0 (Nt−110The constant values of the equilibrium output, the profits per firm, and the equilibrium
price have several implications. First, the time pattern of the free-entry equilibrium numberof 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 hasbasically the same properties as in the previous literature on market diffusion.

3.3 Socially Optimal Diffusion
In this subsection, we derive the socially optimal diffusion. We consider a “second-best”
social planner, who can control the number of firms entering the market, but not their postentry output or price levels. Therefore, a social planner sets the number of firms to maximize
the social welfare, given the post-entry equilibrium outcome derived in 3.1.20 Let not be thenumber of new entrants and Nto be the number of firms set by the planner at period t, andlet qot be the output per firm under socially optimal planning at period t. Now, we define thesocial optimum as follows:
Definition 3. Given the post-entry equilibrium outcome, socially optimal planning satisfies
the following Bellman equation:(Z (Nt−1 +nt )qt
V(Nt−1 ) = max[a + σ(Nt−1 qt−1 ) − bQ]dQ− (Nt−1 + nt )qt c − nt f + βV(Nt−1 + nt ) ,
nt ≥0

0

(18)

subject to equation (8).
The interpretation of equation (18) 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.
We now characterize the socially optimal diffusion as follows:20

If firms were assumed to choose their output levels to maximize the discounted sum of future operating
profits, instead of current profits as we assume in this paper, then the social planner’s problem would be considerably complicated. Because forward-looking firms take into account how their output levels affect the levelof new entry set by the social planner in the future periods, the firms’ equilibrium outputs in each period would
be different from the stage game Nash equilibrium represented by equation (8). To keep the model tractableand facilitate a clear comparison between free-entry and socially optimal diffusion, we assume that firms act
myopically in the quantity-setting stage.

11

Proposition 2. Suppose that N0o = 0. Then, the optimal diffusion path satisfies the following
second-order difference equation:of =R(Nt−1, not )

∂qot∂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

+ [pot − c]Nto(19)

where R(Nt−1 , nt ) = πt + β f .
Equation (19) 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(19) is the present value of future operating profits. The second term is the welfare loss fromthe business-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 thesocial welfare. The third term is the future benefit from the demand-shift effect, in which anincrease in the number of firms directly raises the 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 demandshift 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.21
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 theprevious literature concerned with market diffusion. In past works, the business-stealing and21

In socially optimal diffusion, in contrast to free-entry diffusion, the number of firms in the first perioddepends 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.

12

business-creating effects are not considered, although there exists a benefit from internalizing
intertemporal externalities that are engines of the market growth.22

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. We first explore thesocial 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 (19). 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 (19) 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
Ntµ(Nt−1 , Nt ) =π(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 ) =

(20)

(21)
(22)
(23)

e
, Nte ) for all t = 1, 2, .... ThereNote that the free-entry diffusion satisfies (1 − β) f = π(Nt−1fore, the difference in the number of firms depends on the magnitude of the current loss,ooo), which depends on the, Nto , Nt+1
, Nto ), and the magnitude of the future benefit, λ(Nt−1µ(Nt−1demand-shift and business-creating effects.
22
Because 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 and business-creating effects in equation (20) areabsent, but the future benefit from the demand-shift effect still exists.

13

Furthermore, the future benefit at period t is determined by the degree of discounting, β,oqot+1 −
the degree of consumption externality, σ0 (Nto qot ), the growth rate of the market size, [Nt+1o. By comparing equations (22) andNto qot ]/Nto qot , and the number of firms at period t + 1, Nt+1(23), 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 3. Suppose that µ(Nt−1, Nto ) ≷ λ(Nt−1, Nto , Nt+1), i.e.:1≷ooqot+1 Nt+1+2
βσ0 (Nto qot ) Nt+1
.
o o
o
b
Nt qt Nt+1 + 1

(24)

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

Note that we have N0e = N0o = 0. Therefore, Proposition 3 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 3 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;23 (b) the growth rate of the market size is high;24 and (c) the number of firms is small. These elements increase the value ofthe right-hand side of inequality (24).

4.2 Social Inefficiency in the Mature Phase
Next, we turn to the analysis of the mature phase. Let N o (qo ) denote the steady-state numberoof firms (output per firm) in the socially optimal diffusion that satisfies Nto = Nt+1
= No

(qot = qot+1 = qo ). Then, in the steady state, we can rewrite equations (20)–(23) as follows:(1 − β) f = π(N o ) − µ(N o ) + λ(N o ),where
π(N) =(a + σ(Nq) − c)2,(N + 1)2 b

(25)

(26)

Note that the right-hand side of (24) 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.
23

24

14

N
π(N),
(27)
N+1
βσ0 (Nq) N + 2
λ(N) =
µ(N).
(28)
b
N+1
Note that (1 − β) f = π(N e ) holds in free-entry diffusion. Thus, as in the analysis of theµ(N) =

growing phase, the difference in the number of firms depends on the magnitudes of the current
loss, µ(N o ), and the future benefit, λ(N o ). As the market becomes mature, the degree ofconsumption externality and the growth rate of the market size become lower, and the numberof firms operating in the market increases. Compared with the growing phase, these changes
in the market environment lower the magnitude of the future benefit relative to the currentoooloss, i.e., λ(Nt−1, Nto , Nt+1)/µ(Nt−1, Nto ) ≥ λ(N o )/µ(N o ).25 More importantly, the following

lemma shows 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(30)

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 free-entry equilibrium number of firms is socially excessive:
Proposition 4. In the mature phase (steady state), the free-entry equilibrium number of firms
is larger than the socially optimal number of firms, i.e., N e > N o .
From Propositions 3 and 4, we conclude that the free-entry equilibrium number of firms
tends to be initially insufficient but eventually excessive.26 This is in contrast to the results of25

By comparing equations (23) and (28), 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 )(29)obecause σ00 (·) < 0, ∂Nt qt /∂Nt > 0, and Nto ≤ Nt+1
≤ No.
26
If we assumed a “first-best” social planner, who could control both the number of firms and their output
or price levels, the socially optimal number of firms would be one because of economies of scale. In contrast,
the equilibrium number of firms under free entry is larger than one by Assumption 2. Therefore, the “first-best
excess-entry theorem” in Suzumura and Kiyono (1987) still holds in our dynamic setting.

15

previous studies, where the number of firms in the free-entry diffusion is socially insufficient
over time. The second and third columns of Table 1 and Figure 2 present a numerical example
in which the number of firms under free entry is initially insufficient (from Period 1 to Period3) but eventually excessive (from Period 4 onward).27[Table 1 about here.][Figure 2 about here.]

4.3 Time Pattern and Social Inefficiency
In this subsection, we first examine the time pattern of free-entry diffusion and derive thecondition under which S-shaped diffusion arises under free entry. Then we discuss the relationship between the nature of inefficiency and time pattern of free-entry diffusion.
First, we show that the time pattern of free-entry diffusion becomes S-shaped (initiallyconvex and eventually concave) when the externality effect is initially strong enough. Notethat 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, and thetime 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 under free entry, the numbers of new entrants in the first and second periodspp
are ne1 = N1e = [a − c]/ (1 − β)b f − 1 and ne2 = N2e − N1e = σ(N1e qe1 )/ (1 − β)b f , respectively. While the former does not depend on the consumption externality, the latter does. Thisimplies that the strong consumption externality effect 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 5. 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.
(31)b
The mechanism of the initial convexity of free-entry diffusion here is the same as that in
Vettas (2000) and Kitamura (2010), in which firms are small atomistic price takers. Therefore,
27

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

16

our result establishes that the fundamental mechanism of S-shaped diffusion clarified in thosepapers is robust to considerations of oligopolistic interaction among firms.
From inequality (31) and the properties of σ(·), it is easy to see that low initial market size
contributes to the initial convexity of free-entry diffusion. More importantly, from Figure 1,e
we have σ0 (Nt−1
qet−1 )/b > net+1 /net for all t = 1, 2, .... This implies that the free-entry diffusion

up to Period t + 1 has a convex time pattern only if a degree of consumption externality isesufficiently strong such that σ0 (Nt−1
qet−1 )/b > 1.28

Next, we explore the relationship between time pattern and social inefficiency of freeentry diffusion. Proposition 3 shows that the number of firms under free entry in the initialperiods is socially insufficient for a strong degree of consumption externality. In addition,
Proposition 5 shows that the initial convexity of free-entry diffusion arises for a strong degreeof consumption externality. The following proposition summarizes the relationship between
the convex time pattern and insufficient entry in the initial phase of free-entry diffusion:
Proposition 6. If the time pattern of free-entry diffusion up to Period 3 is sufficiently convex
so that:

ne3 1
≥ ,
ne2 β

(32)

the number of firms under free entry in the first period is socially insufficient, i.e., N1e < N1o .
Note that the condition (31) alone (i.e., the convexity from Period 0 to Period 2) does notguarantee the insufficient entry in the initial phase of free-entry diffusion. This is because
the large value of σ(N1e qe1 ) does not necessarily imply the large value of σ0 (N1e qe1 )/b. Recalling that we eventually observe concave time pattern and excessive entry under free-entry
diffusion, the result of Proposition 6 implies that S-shaped diffusion can be a signal that the
number of firms under free entry is initially insufficient, but eventually excessive.29
e
From Figure 1, it can also be found that net /net−1 > σ0 (Nt−1
qet−1 )/b for all t = 2, 3, .... This implies that theefree-entry diffusion from Period t − 2 to Period t has a convex time pattern if σ0 (Nt−1
qet−1 )/b ≥ 1.
29
There is also a similar relationship between consumer surplus at each period and S-shaped diffusion. In ourmodel, the consumer surplus at each period is calculated as b(Nt qt )2 /2. Because the consumer surplus at eachperiod is increasing in the number of firms at each period, the consumer surplus under free-entry diffusion tends
to be smaller (larger) than under socially optimal diffusion if the number of firms under free entry is sociallyinsufficient (excessive). Therefore, the S-shaped diffusion can also be a signal that the consumer surplus underfree-entry diffusion is initially smaller, but eventually larger than under socially optimal diffusion.
28

17

The results in this section imply that oligopolistic interaction is an important factor forthe discussion of entry regulation policies in new industries where the S-shaped diffusion has
already been observed in other countries. The entry regulation policies in such industries
should be changed depending on the phase of market growth: entry should be initially encouraged, but eventually discouraged. Therefore, the regulatory authority can improve social
welfare by subsidizing early entry and taxing late entry. For the numerical example used
above, the optimal entry subsidy/tax schedule is computed as shown in the fourth column ofTable 1 and Figure 3.30 According to them, we can find that while a subsidy should be given
in Periods 1 and 2 for entry promotion, a tax should be imposed from Period 3 onward forentry restriction.31[Figure 3 about here.]

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 the
output per firm is endogenously determined by the oligopolistic interaction. The major result
reported here is that the nature of the inefficiency under free entry can depend on the degreeof market maturity and that an S-shaped time pattern can be a signal that the number of firms
under free entry is initially insufficient but eventually excessive.
Our result provides important policy implications for entry regulations in new industries
where S-shaped diffusion has already been observed in other countries. Our result implies
that entry should be initially encouraged but eventually discouraged. It may be possible to30

The optimal entry subsidy or tax is equal to the difference between the right-hand side of equation (15) andthe right-hand side of equation (19), i.e., equal to the last three terms of equation (19). If these terms are positive(negative), then a subsidy (tax) should be given (imposed).
31
From Table 1, it can be seen that, in Period 3, although the number of firms under free entry is sociallyinsufficient, a tax is imposed on new entrants under an optimal subsidy/tax schedule. Note that, under the
optimal subsidy/tax schedule, when potential entrants decide whether to enter the market at the beginning of
Period 3, the number of incumbent firms is not N2e , but N2o . Given N2 = N2o , the free-entry equilibrium numberof firms in Period 3 becomes 72.11, which is larger than N3o = 71.59. Therefore, under the optimal subsidy/tax
scheme, a social planner has an incentive to impose an entry tax to discourage new entry in Period 3.

18

improve social welfare by giving subsidies to early entrants but taxing late entrants without
violating intertemporally balanced budget constraints.
The result of this paper might be important for the Japanese generic drug market.32 In
Japan, the use of generic drugs is substantially lower than in other developed countries, andthe Japanese government has recently taken various policy measures for promoting the use ofgeneric drugs.33 Our result suggests that, while the government’s recent policy is reasonable
for the moment, it should be cautious of the possibility for excess entry in the future.
There are several issues requiring future research. First, as the number of firms becomesstable, there is a possibility of market restructuring. An example of market restructuring
is horizontal mergers. If there exist cost synergies, market maturity may lead to horizontal
mergers that reduce the number of firms and improve welfare (Davidson and Mukherjee,2007). This may explain the shakeout of firms corresponding to Stage 3 to Stage 5 in Gort andKlepper (1982).34 Second, there is concern about other intertemporal externalities. Although
we use only intertemporal consumption externalities here, we predict that our results would
hold even under other intertemporal externalities.
Finally, we assume that firms act myopically in the quantity-setting stage; i.e., firms
choose their output levels to maximize their period profits. This assumption makes the model
very tractable and enables us to provide a clear comparison between free-entry and socially
optimal diffusion. Although our approach seems like a natural first step to analyze market
diffusion with oligopolistic interaction among firms, it would be very important to allow for
forward-looking firms that decide their output levels to maximize the discounted sum of fu32

In the (generic) drug markets of the United States, several features are observed that are consistent with thefocus of this paper. First, an S-shaped time pattern of the number of generic entrants is actually observed (Ching,2010). Second, empirical evidence is found for the importance of intertemporal consumption externalities(Berndt, Pindyck, and Azoulay, 2003). Finally, some papers point out the significance of entry costs and thepossible existence of the “business-stealing effect” in this market. Scott Morton (1999) mentions that the costsof obtaining entry approval are significant for generic entrants. In addition, using the data of drugs that went offpatent during 1976–1987, Caves, Whinston, and Hurwitz (1991) empirically show the relatively small increases
in the overall generic market share achieved as the number of generic competitors increases, and mention thepossibility of excess entry in this market.
33
In 2009, the quantity-based share of generic drugs in Japan was only 20.3% (Japan Generic Medicines
Association, 2012), while those for the United States, Canada, the United Kingdom, and Germany were above
60% (based on analysis of the IMS Health MIDAS Market Segmentation data by Sawai Pharmaceutical (2011)).
34
Several papers develop theoretical models that explain the shakeout of firms. See, e.g., Jovanovic andMacdonald (1994), Horvath, Schivardi, and Woywode (2001), and Hanazono and Yang (2009).

19ture operating profits. We hope that our study helps researchers to address these issues.

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

ea + σ(Nt−1
qet−1 ) − c− 1,p(1 − β)b f(33)

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,
0
e
e
0
e
e
e
e →0 N (NN 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
Differentiating the right-hand side of (18) with respect to nt and rearranging, we have thefollowing first-order condition:

"

#
∂qt
f = [pt − c] qt + Nt
+ βV 0 (Nt ).
∂nt

(34)

Using the envelope theorem, we obtain:

#
"
∂qt−1
V (Nt−1 ) =σ (Nt−1 qt−1 ) qt−1 + Nt−1
Nt qt
∂Nt−1
"
#
∂qt
+ [pt − c] qt + Nt
+ βV 0 (Nt ).
∂Nt−1
0

0

Then, from equations (34) and (35), we can express V 0 (Nt−1 ) as follows:
"
#
∂qt−1
0
0
V (Nt−1 ) =σ (Nt−1 qt−1 ) qt−1 + Nt−1
N t qt
∂Nt−1
"
#
∂qt
∂qt
+ [pt − c]Nt−
+ f.
∂Nt−1 ∂nt
20

(35)

(36)

Taking equation (36) one period forward, substituting into the right-hand side of the firstorder condition (34), and rearranging terms, we finally obtain the equation (19).
Q.E.D.

Proof of Proposition 3
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:eeoπ(Nt−1, Nte ) ≥ π(Nt−1, Nto ) ≥ π(Nt−1, Nto ).

(37)e
Because π(Nt−1, Nte ) = (1 − β) f in the free-entry equilibrium, inequalities (37) imply thatooπ(Nt−1, Nto ) ≤ (1 − β) f . This is a contradiction to equation (20) because µ(Nt−1, Nto ) >ooλ(Nt−1, Nto , Nt+1). In the same way, we can prove the second case.

Q.E.D.

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≤

N o qo ≤βσ0 (N o qo )N o qo N o + 2 σ(N o qo ) N o + 2<,
bNo + 1b
No + 1(38)(39)

where the last inequality follows from the properties of σ(·). By substituting qo = (a +σ(N o qo ) − c)/(b(N o + 1)) into inequality (39), we have"#o oσ(N o qo ) N o + 2σ(N o qo ) N o + 2
o oo a + σ(N q ) − c
⇔
N
<
N q <
b
No + 1b(N o + 1)b
No + 1
N o [a − c]⇔< σ(N o qo ).
2

(40)
(41)

By using inequality (41), we have:oN [a−c](a + σ(N o qo ) − c)2 (a + 2 − c)2>π(N ) =b(N o + 1)2b(N o + 1)2" o#2
N + 2 (a − c)2 (a − c)2
= o
>
.
N +1
4b
4b
o

21

(42)

Note that inequality (38) implies that µ(N o ) ≤ λ(N o ). Then, together with inequality (42) andAssumption 2 (inequality (3)), we have:π(N o ) − µ(N o ) + λ(N o ) >(a − c)2> (1 − β) f.
4b

(43)

However, this contradicts the equilibrium condition (25). Therefore, inequality (30) holds.
Q.E.D.

Proof of Proposition 4
Let q(N) be the steady-state output per firm in the post-entry equilibrium given N. As mentioned in Subsection 3.2, we have σ0 (N e qe )/b < 1 in the steady state of free-entry diffusion,
where qe is the equilibrium level of steady-state output per firm in the free-entry diffusion.
Hence, we have N e > N̂, where N̂ is such that σ0 (N̂q(N̂))/b = 1. To begin the proof of
Proposition 4, we first prove the following lemma:
Lemma 3. Let Q(N) = Nq(N). 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.
Proof of Lemma 3
1. By differentiating Q(N) with respect to N, we have:
Q0 (N) =N[1 − N+1]q(N)0N σ (Q(N))
1 − N+1
b

22

> 0,

(44)

for N ∈ (N̂, ∞). It is easy to see that Q0 (N) → 0 as N → ∞. Then, by using L’Hôpital’srule, we obtain:N→∞n a + σ(Q(N)) − c

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) = lim

= lim

+

nh a + σ(Q(N)) − c ihbN→∞n a + σ(Q(N)) − c o
.
N→∞
b

N+1
b
N
N io
σ (Q(N)) [1 − N+1 ] N+1
1+
N σ0 (Q(N))
b
1 − N+1
b
0

(45)

= lim

Since σ(·) satisfies the Inada condition, and a > b, there exists a unique m ∈ (0, ∞)such 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:
π0 (N) = −2π(N)[b − σ0 (Q(N))]< 0,
N[b − σ0 (Q(N))] + b(46)

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 4. 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 ).

(47)

Since N e satisfies π(N e ) = (1−β) f , we have π(N o ) ≤ (1−β) f . However, this is a contradiction
to (25) because µ(N o ) > λ(N o ) by Lemma 2. Therefore, we have N e > N o .
Q.E.D.

23

Proof of Proposition 5
From Figure 1, the time pattern of free entry eventually becomes concave. Therefore, it
becomes S-shaped if and only if ne1 < ne2 . From equation (17), we obtain inequality (31).
Q.E.D.

Proof of Proposition 6
Note that, from Figure 1 and the strict concavity of σ(·), we havee
σ0 Nt−1
qet−1
net+1
> e
b
nt

(48)

for all t = 1, 2, .... Then, if the time pattern of free-entry diffusion up to Period 3 is sufficiently
convex so that inequality (32) holds, we obtain
σ0 N1e qe1
ne 1> e3 ≥ .
b
n2 β

(49)

Now, we show that N1e < N1o using a proof by contradiction. Suppose in negation that N1e ≥N1o . Then, because N1 q1 is increasing in N1 from Lemma 1 and N0e = N0o = 0 by assumption,
we have N1e qe1 ≥ N1o qo1 . Using the concavity of σ(·) and inequality (49), this implies that

σ0 N1o qo1
σ0 N1e qe1
1
≥
> .
(50)
b
b
β
From this inequality, and the fact that both N2o qo2 /N1o qo1 and (N2o + 2)/(N2o + 1) are larger than1, we have
βσ0 N1o qo1 N2o qo2 N2o + 2b

N1o qo1 N2o + 1

> 1.

(51)

Then, together with N0e = N0o = 0, Proposition 3 implies that N1e < N1o . However, this is acontradiction. Therefore, we have N1e < N1o .
Q.E.D.

24

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Figure 1: Dynamical system of free-entry diffusion

1

0

29

1

Figure 2: The free-entry and the socially optimal paths for a numerical example
160

Free-Entry Equilibrium Path
140

Number of firms

120
100
80

Socially Optimal Path

60
40
20
0
0

1

2

3

4

5

Period

30

6

7

8

9

10

Figure 3: Optimal subsidy/tax schedule for a numerical example
0.6

0.4

Tax/Subsidy

0.2

0
1

2

3

4

5

6

Period
-0.2

-0.4

-0.6

31

7

8

9

10

Table 1: Number of firms and optimal subsidy/tax schedule for a numerical example
Period
0
1
2
3
4
5
6
7
8
9
10

Free-entry Path Optimal Path Subsidy/Taxa
0
4.77
30.49
69.77
103.10124.29136.01142.05145.07146.55147.27

0
20.68
48.58
71.59
86.02
93.94
98.00
100.02101.02101.50101.73

–
0.501
0.295
-0.007-0.257-0.412-0.498-0.541-0.563-0.574-0.579Note: For a √= 8, b = 0.5, c = 5, f = 18, β = 0.97, andσ(Qt−1 ) = 6 Qt−1 .
a Positive (negative) figures indicate the level of subsidy (tax).

32