Discussion Papers In Economics
And Business

Capacity Expansion in Markets with Intertemporal
Consumption Externalities
Hiroshi Kitamura

Discussion Paper 07-11

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANCapacity Expansion in Markets with Intertemporal
Consumption Externalities
Hiroshi Kitamura

Discussion Paper 07-11
April 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANCapacity Expansion in Markets with Intertemporal
Consumption Externalities∗
Hiroshi Kitamura†
Graduate School of Economics, Osaka University
April 2, 2007

Abstract
This paper analyzes market capacity expansion in the presence of intertemporal consumption externalities such as consumer learning, networks, or bandwagon eﬀects. The externality leads to an
endogenous shift of market demand that responds to past market capacity. Whereas market capacity
grows in waves, its magnitude depends on the degree of market concentration. The competitive environment contributes to S-shaped time patterns of market capacity expansion that is slow from thesocial viewpoint. On the other hand, using an introductory price, a monopolist plans an initially larger,
but eventually smaller, amount of market cultivation than a competitive market capacity expansion.

JEL Classifications Code: D11, L11, L14.
Keywords: Intertemporal consumption externalities; S-shaped diﬀusion; Market structure; Introductory
price.
∗In the process of this study, I talked to many people, and their knowledge and ideas contributed significantly to the analysis
in this paper. I especially thank Katsuya Takii, Junichiro Ishida, and Ryo Horii for helpful discussions and comments. I also
thank Tatsuhiko Nariu, Kazuo Mino, Tetsuya Shinkai, Keizo Mizuno, Tsuyoshi Toshimitsu, Masamichi Kawano, Noriyuki Doi,
Toshihiro Okada, Kohei Daido, Yasutomo Murasawa, Eiichi Miyagawa, Tatsuhiro Shichijo, Misato Sato, and the seminar participants at the Japanese Economic Association, Osaka University, Kwansei Gakuin University, and Osaka Prefecture University
for constructive comments. All errors are mine.
†
Address:Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan. Emailaddress:gge003kh@mail2.econ.osaka-u.ac.jp

1

Introduction

In a number of successful product or services markets, the level of market demand often increases overtime. One of the significant features of this growth is that it is gradual rather than instantaneous. Although
diﬀerent explanations are possible for this gradual expansion, there is an important but relatively neglected
reason in the literature: an intertemporal consumption externality where market demand is endogenously
determined as an increasing function of past market capacity, defined as the number of consumers who
buy the product. Several reasons exist for such an externality. First, there may be consumer learning.
An increase in past market capacity leads to an accumulation of product information among consumers
and to an updating of consumer preferences. Second, indirect network externalities may exist1 . The
variety of complementary products is an increasing function of the number of past product users (forexample, computers and software). Another reason, which is purely psychological, is the bandwagon
eﬀect. Consumers often wish to consume a popular product. They may regard past market capacity as asign of popularity2 .
When an intertemporal consumption externality exists, the market equilibrium contains dynamic aspects: market capacity increases in waves. More importantly, market capacity expansion may be highly
dependent on the degree of market concentration. Intuitively, market concentration enables firms to eﬃciently expand the market by internalizing the externality, but also earn a large, socially ineﬃcient level ofprofits. In contrast, the competitive environment restricts firm profits but makes the internalization of theexternality diﬃcult. It is apparently ambiguous how these diﬀerences aﬀect the level, time pattern, andproduct price of the market capacity expansion. The main aim of this paper is to theoretically examine therelation between market concentration and the properties of the market capacity expansion in the presenceof an intertemporal consumption externality.
In this paper, we develop a dynamic model of market capacity expansion with endogenous market
demand in an infinite-horizon framework. The market demand increases with increases in the previous
period’s market capacity because of an intertemporal consumption externality. We examine two types ofmarket capacity expansions: a competitive (decentralized) market capacity expansion and a monopoly
1

See Katz and Shapiro (1994).
This may be naive behavior, but is consistent with the existence of advertising and media reports containing information on
past product sales. See Monterio and Gonzalez (1999) who analyze the role of advertising past sales.
2

1

(centralized) market capacity expansion. In the competitive market, small and identical firms enter themarket responding to each period’s demand growth given the equilibrium price determined by the zeroprofit condition. In contrast to competitive firms, a monopolist has the ability to control price. This ability
enables the monopolist to control both the level of current market capacity and future demand growth
because of the internalization of the externality. By increasing current market capacity, the monopolist
generates a large amount of subsequent period demand. However, an increase in current market capacity
reduces current profits. The main diﬀerence between the two market capacity expansions is whether theintertemporal trade oﬀ between current revenue and future demand growth exists or not.
The two market capacity expansions diﬀer with respect to the levels, the time patterns, and the equilibrium prices, respectively. Assuming that the degree of demand growth (the benefit of an externality)
decreases with increases in the previous period’s market capacity, the analysis provides several interesting
results. First, the competitive market capacity expansion is initially slower and smaller than the monopoly
market capacity expansion. However, it gradually becomes faster and larger in the long run. Because of
the externality, the competitive market capacity expansion is slow and ineﬃcient from the social viewpoint. In contrast, the monopolist plans a profit structure in order to earn large profits from a large market
demand by sacrificing early profits. The monopolist has an incentive for a large amount of early market
cultivation by internalizing the externality. This leads to faster market capacity expansion than in the competitive market. However, the monopolist slows down the rate of market cultivation to yield large profits
by restricting market capacity.
Second, the initially slow competitive market capacity expansion has an S-shaped time pattern that isreported by a number of researchers who investigate the time patterns of a number of firms3 and products4 .
In contrast, monopoly market capacity expansion has diﬃculty in following the initial convex part of the
S-shaped time patterns. The intuitive logic for the diﬀerence in the initial time patterns is as follows. In
the competitive market, the attractiveness of a new entry is determined by the strength of its externality
3
See, for example, Gort and Klepper (1982) who investigate the time patterns of a number of firms in 46 product markets.
Note that S-shaped diﬀusion is not an isolated phenomenon. Empirical evidence shows that the time patterns of the intra-firm andinter-firm technology diﬀusion processes also tend to be S-shaped: see the seminal work of Griliches (1957), Mansfield (1968),
and the survey of technological diﬀusion by Stoneman (2002). For a discussion of the theoretical mechanism of adoption of newtechnology, see Jensen (1982) and Rengnum (1981).
4
S-shaped product diﬀusion is treated as a stylized fact in the marketing literature. It is observed in a number of markets suchas color televisions (Karshenas and Stoneman (1992)), fax machines (Economides (1995)), and clothes dryers (Krishnan, Bass,and Jain (1999)). This phenomenon is also observed in the services market in the presence of network externalities such as themobile telecommunications services market and the Digital Subscriber Line services market.

2

eﬀect. A strong externality eﬀect makes a current period entry more profitable than in the previous period
and it raises the attractiveness of the new entry. This leads to the initial convex part of the S-shaped time
pattern. On the other hand, a strong externality eﬀect also raises the monopolist’s benefit of internalizing
the externality. This provides the monopolist with a strong incentive for a large amount of early market
cultivation in order to generate a large amount of demand sooner. Therefore, the initial market capacity
expansion by the monopolist is less likely to increase, but more likely to maintain, the concave time
patterns.
Finally, the two market capacity expansions also diﬀer with respect to the equilibrium prices. Whereas
the competitive equilibrium price is constant over time, the monopolist has an incentive to initially oﬀer a
low introductory price5 . The initial low introductory price contributes to the large amount of early market
cultivation by the monopolist. The eventual high price induces the monopolist to generate large profits
with the larger demand.
This paper is related to a number of literatures. First, this paper is most relevant to the industrialorganization literature concerned with market capacity expansion in the presence of intertemporal externalities. The majority of previous studies on market capacity expansion are related to firm learning:learning by doing (Jovanovic and Lach (1989)), learning consumer demand (Rob (1991)), and two sided
learning between consumers and firms (Bergemann and Välimäki (1997) and Vettas (1998)). S-shaped
diﬀusion has been explained in these literatures. In contrast, this paper does not focus on the role of firm
learning, but focuses on the role of the consumption externality.
In the literature on firm learning, Vettas (2000) analyzes market capacity expansion with an intertemporal consumption externality. He shows that the competitive diﬀusion path becomes S-shaped and that it
is always slower than the optimal path by a planner or a monopolist. In his model, however, the benefitsof market concentration are overestimated because the monopolist does not have an incentive to restrict
output because of perfectly elastic demand: the demand curve is a horizontal straight line. In addition,
the perfectly elastic demand induces his model to require firm learning for gradual market diﬀusion6 and5

A number of studies state that introductory pricing is possible when network externalities exist (Rohlfs (1974), Katz andShapiro (1985, 1986), and Cabral, Salant, and Woroch (1999)). In addition, an introductory price has been theoretically observedin the optimal pricing of experience goods (Schmalensee (1982), Shapiro (1983), and Bergemann and Välimäki (2006)) and to
gradually raise prices. In the marketing literature, Krishnan, Bass, and Jain (1999) analyze the optimal pricing strategy for new
products based on the Bass model and show the role of introductory pricing.
6
If firm learning does not exist, market capacity expansion becomes instantaneous. The structure of firm learning in his modelbuilds on Rob (1991).

3

develops an unnatural relation between market capacity and prices: while monopoly market capacity islarger than for the competitive case, monopoly price is always higher7 . In contrast, the demand here has
a more common structure: the demand curve is a downward sloping line and leads to gradual diﬀusion
without firm learning, the ineﬃcient properties of monopoly diﬀusion, and the natural relation between
the price and the market capacity.
Furthermore, this paper is related to the literature concerned with consumption externalities. One of
the established literatures concerned with intertemporal consumption externalities is rational addiction
where a consumer’s utility is positively related to the volume of own past consumption (see Becker andMurphy (1988)). Frank (1989) studies an intertemporal eﬀect where consumers’ own past experienceaﬀects present consumption from a perspective of relative consumption. While these literatures do not
focus on the role of social learning on a consumer’s preferences, social learning also leads to intertemporalconsumption externalities (see Banerjee (1992), Bikhchandai, Hirsheifer, and Welch (1992), Ellison andFundenberg (1993), and MacFadden and Train (1996)). Becker (1991) studies restraint pricing where
consumer demand is positively related to market capacity, and Caminal and Vives (1996) analyze theimportance of past market share as a signal of product quality8 . The model in this paper builds on thesepoints of view9 . Assuming the existence of an intertemporal consumption externality, this paper explores
how it aﬀects market capacity expansion depending on market concentration.
The remainder of this paper is organized as follows. Section 2 sets up the model. Section 3 introduces
the concept of competitive equilibrium and analyzes its properties: the existence of S-shaped diﬀusion.
Section 4 sets up the social planner problem and shows that competitive market capacity expansion isslow from the social viewpoint. Section 5 sets up the monopoly problem and compares the competitivemarket capacity expansion with the monopoly case. Section 6 contains concluding remarks. The proofsof all results are provided in the Appendix.
7

In contrast to our model, an increase in current market capacity does not lower market price, but raises future prices in thecase of perfectly elastic demand. In addition, as stated by Rob (1991), monopoly market capacity is always larger than in thecompetitive case because of the informational externality generated by the firm’s learning. Therefore, a larger monopoly capacity
leads to higher market prices.
8
See also Doganoglu (2003) who examines dynamic price competition in a horizontally diﬀerentiated duopoly market.
9
In the marketing literature, the modeling of S-shaped product diﬀusion relies largely on so-called “epidemic” models established by Bass (1969). See Mahajan, Muller and Bass (1990, 1995). In such models, information of a new product is assumed
to spread from users to nonusers by personal contact. This social interaction may provide an explanation of intertemporal consumption externalities in this analysis. Therefore, this paper may be regarded as a complementary economic analysis of thismarketing literature.

4

2

Model

This section develops the model. We characterize the consumers’ behavior in 2.1 and the firms’ behavior
in the competitive market in 2.2. We assume that time is discrete and the horizon is infinite. It is also
assumed that the market in this paper is a perishable good market or a services market in which theservice 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 fora product. Let θ be the type of consumer, which is stationary for all periods and is uniformly distributedon the interval [0,1]. The market capacity, the number of consumers who purchase the product, at period
t is denoted by qt . The consumers’ willingness to pay depends on the previous period’s market capacity
because of the intertemporal consumption externality. We assume the following reservation price for type
θ consumer at periods t = 1, 2, .., vt (θ):
Assumption 1.
vt (θ) = V(θ, qt−1 ) = ρθ + σ(qt−1 )(1)where ρ > 0, σ′ (q) > 0, σ′′ (q) < 0, σ(0) = 0, limq→0 σ′ (q) = ∞, and limq→∞ σ′ (q) = 0, and where ρ is apreference parameter.
σ(qt−1 ) represents the intertemporal consumption externality, which has two properties. First, σ′ (qt−1 )> 0 implies that each consumer’s reservation price at period t is strictly increasing in the previous period’s
market capacity. Second, σ′′ (qt−1 ) < 0 implies that the increase in the reservation price, or equivalently
the benefit of the externality, is strictly decreasing in the previous period’s market capacity. Of course,
there may exist a locally increasing part especially at smaller market capacities but it is unrealistic that theincreasing part would be observed for larger market capacities. This assumption guarantees that there is aunique upper limit for market capacity for every market capacity expansion in this analysis10 .
A consumer of type θ pays pt for a product and enjoys consumer surplus of vt (θ) − pt . The consumer
is assumed to purchase the product if and only if consumer surplus is nonnegative, i.e., vt (θ) − pt ≥ 0.
10

If the benefit of the externality increases initially, but eventually decreases with increases in the previous period’s market
capacity, multiple steady states may exist.

5

Then, the inverse demand function at period t, P(qt−1 , qt ), becomes:ρ + σ(qt−1 ) − ρqt 0 ≤ qt ≤ 1,
P(qt−1 , qt ) = 0
qt > 1.

(2)

for all t = 1, 2,.., and 0 ≤ qt−1 ≤ 1. It is easy to see that the inverse demand function is strictly increasingin the previous period’s market capacity, but strictly decreasing in the current period’s market capacity.

2.2

Firms under Competitive Environment

Firms in the competitive market are identical, small and price takers. At the beginning of each period,
the set of firms is composed of two subsets associated with potential entrants and incumbents. There isassumed to be no asymmetry of information between the two subsets and to be no demand uncertainty. At
the beginning of each period, potential entrants decide whether to enter the market or not, and incumbents
decide whether to exit the market or not. Potential entrants enter the market with entry cost c > 0, whichis the initial investment in purchases such as machines. We assume that machines are durable and areoperated for multiple periods. Machines can be operated at any level between zero to one unit for eachperiod in an environment of constant return to scale. To simplify the analysis, we assume that the scrap
value of machines is zero and the marginal cost is zero.
Let xt be the number of incumbents at period t, yt the number of new firms entering the market at the
beginning of period t. Because the entry cost is not recoverable and the marginal cost is zero, incumbents
do not have an incentive to exit the market. Therefore, yt = xt − xt−1 ≥ 0 for all t = 1, 2,... Assuming that∑x0 = 0, we have xt = tτ=1 yτ . Let i > 0 be a constant interest rate and the discount factor is denoted byβ ≡ 1/(1 + i). For each period, firms maximize the discounted sum of future operation profits, which isdenoted by R(xt−1 , yt )11 , i.e.:R(xt−1 , yt ) = pt + βR(xt , yt+1 ),(3)

for all t = 1, 2,,.., where pt represents the direct operation profit (market price) at period t and βR(xt , yt+1 )represents the discounted future operation profits. Potential entrants enter the market if and only if thepresent value of net profits is positive, i.e., R(xt−1 , yt ) > c. If the demand is initially low and the fixed
cost or the discount factor is high, then entry may not occur in the first period. The following assumption
guarantees first period entry.
11

Because entrants and incumbents are symmetric and the horizon is finite, they have the same present value of their future
revenue streams.

6

Assumption 2.
ρ > (1 − β)c(4)Assumption 2 implies that entering the market is profitable in the first period. If ρ ≤ (1 − β)c, then first
period entry is not attractive and it does not occur. Because this condition holds for all following periods,
the market capacity expansion never occurs.

3

Analysis

This section provides the characterization of competitive equilibrium and explores the existence of Sshaped market diﬀusion. We first characterize the competitive equilibrium in 3.1. Then, the existence ofS-shaped market diﬀusion is examined.

3.1

Competitive Equilibrium

Each period’s equilibrium condition is determined by the market clearing condition and the zero profit
condition. Let rt be the number of new consumers who purchase the product at period t. Now, we define
the competitive equilibrium as follows:
Definition. The competitive equilibrium consists of three sequences {pct , rtc , yct } that simultaneously satisfy
the following conditions:
1. The market clears for all t = 1, 2,...
rtc = yct(⇔ qct = xtc ).

(5)

2. The market price is determined by the inverse demand of consumers for all t = 1, 2,...
cc
pct = P(xt−1, xt−1+ yct ).

(6)

3. New entry occurs until excess profits become zero for all t = 1, 2,...
cR(xt−1, yct ) ≤ c,with equality if yct > 0.

7

(7)

According to the above definition, the properties of the competitive market capacity expansion are
identified. From the market clearing condition, the number of new consumers, rtc is nondecreasing. In
addition, from the zero profit condition and equation (3), the equilibrium price in the competitive market
becomes:
pct = (1 − β)c,(8)

for all t ≥ 1 such that yct > 0. This implies that the equilibrium price is constant as long as the market isin the transition process and it does not depend on the externality. Given this equilibrium price, potential
entrants enter the market in response to demand growth in each period. By putting the inverse demand
function into equation (8), the competitive market capacity expansion is summarized as follows:ccρ + σ(xt−1) − ρ[xt−1+ yct ] + βc = c(9)

for all t ≥ 1 such that yct > 0. Equation (9) implies that the competitive market capacity expansion isrepresented by a first order diﬀerence equation with respect to xtc . Denote the steady state of equation (9)by xc . Then, the dynamical system of equation (9) is summarized in Figure 1. It is easy to see that the
c
c
c
c
c
12
sequences {xtc }∞0 satisfy xt ∈ [0, x ] for all t = 1, 2,.., and monotonicity, x0 = 0 and xt → x as t → ∞ .

3.2

S-shaped Market Diﬀusion

From now on, we examine the time pattern of the competitive market capacity expansion and show that it
becomes S-shaped (initially convex, but eventually concave) when the externality eﬀect is initially strong
enough. In terms of firm entry, S-shaped market diﬀusion implies that the level of new entry is initially
increasing, but eventually decreasing. We first explore the determinants of the amount of new entry at eachperiod. Then, the initial convexity and the eventual concavity of the market diﬀusion path are examined,
respectively.
Let ∆yct ≡ yct+1 − yct . The time path of the market capacity expansion becomes S-shaped if ∆ytc > 0
initially, but ∆yct < 0 eventually. We also define the potential firms’ profitability of entry at the beginning
12

By rearranging the terms of equation (9), we have:cc)/σ + [ρ − (1 − β)c]/ρ.
) = σ(xt−1xtc = X(xt−1

(9′ )

c
c
c →0 X ′ (xc ) = ∞, and) < 0, lim xt−1
) > 0, X ′′ (xt−1
From the properties of σ, we have X(0) = [ρ − (1 − β)c]/ρ > 0, X ′ (xt−1
t−1
c
c
c
′ c
c
lim xt−1 →∞ X (xt−1 ) = 0. Therefore, X(xt−1 ) crosses the xt = xt−1 line only once and there is a unique steady state, xc .

8

Figure 1: The dynamical system of competitive diﬀusion pathc ) ≡ R(xc , 0) − c, i.e.:of each period as Π(xt−1
t−1
c
c
c
Π(xt−1) = ρ + σ(xt−1) − ρxt−1− (1 − β)c,(10)

for all t = 1, 2,... Equation (10) implies that the net profit of new entry at the beginning of each period is afunction of the previous market capacity. It does not depend on the current period’s market capacity. Let∆πct ≡ πct+1 − πct . The following lemma shows the relation between the profitability of new entry and thelevel of new entry at each period.
Lemma 1. Suppose that x0c = 0, and σ(1) ≤ (1 − β)c so that xc ≤ 1. Then, for all t = 1, 2, .., ∆yct R 0 if
and only is ∆πct R 0.


Proof. See Appendix.

Lemma 1 implies that an increase or decrease in the amount of new entry is equivalent to an increaseor decrease in the profitability of entering the market at the beginning of each period. If the profitability
increases (decreases), then the amount of new entry increases (decreases). Therefore, we focus on thec ) more precisely in order to explore the time pattern of competitive market capacity
properties of Π(xt−1

expansion. The following lemma summarizes the properties of the profitability of new entry:c ) has a single peaked property; more precisely,
Lemma 2. Π(xt−191. Π(x) satisfies Π(x) > 0 for 0 ≤ x < xc , Π(0) = ρ − (1 − β)c, and Π(xc ) = 0.
2. There is a unique maximizer x̄ = σ′−1 (ρ) ∈ (0, xc ) such that Π( x̄) = π̄ > ρ − (1 − β)c.


Proof. See Appendix.

The single peaked property of the profitability of new entry is summarized in Figure 2. This property
c , xc ) = ρ + σ(xc ) − ρxc .
follows from the features of the inverse demand function with yct = 0, P(xt−1
t−1
t−1
t−1

An increase in the previous period’s market capacity leads to two independent eﬀects associated with theeﬀect of demand growth because of the externality (as captured by σ(x)) and with the eﬀect of declining
market price because of the downward sloping demand curve (as captured by ρx). First, market demand isincreasing in the previous period’s market capacity because of the intertemporal consumption externality.
The eﬀect of demand growth on the profitability of new entry is always positive. However, the downward
sloping demand curve leads to a decline in the market price as the previous period’s market capacity
increases. This eﬀect is always negative and it reduces the profitability of new entry. Therefore, the net
eﬀect on the profitability of new entry is dependant on the relative magnitude of the two eﬀects.
From the property of σ(·), the eﬀect of demand growth is initially stronger than that of a decliningcmarket price, and the profitability of new entry is increasing for all 0 ≤ xt−1
< x̄. However, as themarket capacity increases, the eﬀect of demand growth becomes weaker and the profitability of new entryc > x̄. Its value becomes zero at the steady state, xc . As a result, Π(xc ) has the single
decreases for all xt−1
t−1

peaked property.
The single peaked property of the profitability of new entry and lemma 1 imply that the amount ofc ≤ x̄ but decreases for x̄ < xc < xc . Because market capacity is strictly
new entry increases for 0 < xt−1
t−1

increasing for all periods, it is easy to see that the time path of competitive market capacity expansion
eventually becomes concave. Therefore, it becomes S-shaped if 0 < x1c ≤ x̄. This condition is equivalentto σ′ (x1c ) > ρ. This implies that the strong externality eﬀect contributes to the initial convexity of the time
path of competitive market capacity expansion. Moreover, the necessary and suﬃcient condition for the
initial convexity is Π(x1c ) > Π(x0c ), where x0c = 0. Let x∗ be x > 0 such that Π(x) = Π(x0c ). From Figure
2, it is easy to see that Π(x1c ) > Π(x0c ) if and only if x1c < x∗ . More precisely, we have the following
proposition:

10

Proposition 1. Suppose that x0c = 0. Then, the time pattern of competitive market capacity expansion
becomes S-shaped if and only if()ρ − (1 − β)cσ> ρ − (1 − β)c.
ρ

(11)


Proof. See Appendix.

One of the important properties of the competitive capacity expansion is that the first period entry,(ρ − (1 − β)c)/ρ, does not depend on the externality, σ(·), but the subsequent period entry does. Therefore,
the strong externality eﬀect does not lead to a larger market capacity expansion in the first period, but
it does in subsequent periods. From inequality (11) and the properties of σ(·), it is easy to see that Sshaped market diﬀusion is more likely to be observed under small values of initial profitability, Π(x0c ) =ρ − (1 − β)c, which follows from the low preference parameter, the low discount factor, and the high entry
cost. Therefore, we conclude that the S-shaped time pattern of competitive market capacity expansion isobserved for a strong externality eﬀect and low initial profitability.

Figure 2: Property of Π(·)

4

WelfareIn the previous section, we showed that the competitive market capacity expansion has an S-shaped time
pattern. We examine its welfare properties in this section and compare it with the monopoly market
11capacity expansion in the following section. We first set up a social planner problem in 4.1 and thenexamine the properties of a socially optimal market capacity expansion in 4.2.

4.1

The Planner’s Problem

A social planner expands market capacity with expansion cost c per unit and zero marginal operation cost.
The planner maximizes the discounted sum of future welfare. Let xto be the market capacity set by theplanner at period t and yot the amount of new capacity expansion at period t. Socially optimal planningsatisfies the following Bellman Equation:oV(xt−1) = maxoyt ≥0{∫ xo +yo
t
t−1

0

}
o
o
[ρ + σ(xt−1) − ρw − cyot ]dw + βV(xt−1+ yot ) .

(12)

The interpretation of the above equation is that the present value of the sum of future welfare iscurrent welfare plus the discounted next period value of the sum of future welfare. We now characterize
the feature of the social optimum market capacity expansion as follows:
Proposition 2. Let xo be the market size in the steady state of the social planner problem and η ≡[ρ − σ′′ (xo )xo ]/βσ′ (xo ) > 0. Suppose that σ(1) + βσ′ (1) ≤ (1 − β)c (so that the Euler equation does notjump), βη2 > 4 (so that the eigenvalues are real numbers), and βη > 1 + β (so that xo is saddle). Then,
there is a unique xo > 0, and for all t = 1, 2, .., there exists a unique optimal solution of the planner’s
problem yot > 0, which satisfies the following conditions:
1. For all t = 1, 2, .., the optimal solution yot satisfies the following second order diﬀerence equation:oooR(xt−1, yot ) + βσ′ (xt−1+ yot )[xt−1+ yot + yot+1 ] = c,(13)where R(xt−1 , yt ) = ρ + σ(xt−1 ) − ρ[xt−1 + yt ] + βc, and2. For all t = 1, 2, .., xto ∈ [0, xo ] with x0o = 0 and xto → xo = 0 as t → ∞.
Proof. See Appendix.

Note that there is a second order diﬀerence equation, but one initial condition, x0o = 0. Therefore,
another boundary condition is required. Proposition 2 shows that there exists a unique steady state xo
whose local property is saddle. It also shows that there exist unique sequences {xto }∞0 converging on
12

xo , which is determined by two boundary conditions, and satisfies monotonicity and xto ∈ [0, xo ] for allt = 1, 2, ...
Equation (13) shows that the marginal expansion cost is equal to the marginal social benefit at thesocially optimal plan. The right hand side is the marginal expansion cost. The left hand side is themarginal social benefit, which is composed of two elements. The first term represents the discounted sumof future revenue. The second term indicates the social benefit from the internalization of the externality;
increased current sales increase the subsequent period’s market demand.

4.2 Optimal Capacity Expansion
Note that the only diﬀerence between the competitive market capacity expansion and the socially optimal
market capacity expansion is whether to internalize the social benefit from the externality or not. Fromthe social viewpoint, it is more eﬃcient to design a market capacity expansion taking into consideration
an increase in the subsequent period’s demand as a result of increased current sales. On this point, thecompetitive market capacity expansion is socially ineﬃcient.
Let pot be the shadow price of the product in the planner’s problem. From equation (13), it is denotedby:opot = (1 − β)c − βσ′ (xto )xt+1
> 0,

(14)

for all t ≥ 113 . Note that the equilibrium price in the competitive market does not depend on the strengthof the externality eﬀect and it is constant for all periods, i.e., pct = (1 − β)c for all t = 1, 2, ... On theother hand, the shadow price in the planner’s planning depends on the strength of the externality eﬀect
and changes responding to the strength of the externality eﬀect. It is easy to see that it is lower than theequilibrium price in the competitive market. This property leads to the socially eﬃcient market capacity
expansion, which is faster than the competitive one:co , we have
Proposition 3. Suppose that x0i = 0, for all i = c, o and β ∈ (0, 1]. Then, for all xt−1
≤ xt−1

xtc < xto .
Proof. See Appendix.

13

The last inequality follows from the assumption σ(1) + σ′ (1) ≤ (1 − β)c.

13

Moreover, the two market capacity expansions diﬀer according to their time patterns. Whereas theexternality does not aﬀect the first period’s market capacity expansion in the competitive market, it does
under socially optimal planning. The strong externality eﬀect makes the competitive market capacity
expansion initially too slow from the social viewpoint and initially convex. On the other hand, it leads
to an initially large market capacity under optimal planning and makes the S-shaped time pattern morediﬃcult to obtain.

5Comparing Competitive Diﬀusion and Monopoly Diﬀusion

In this section, we examine how the market capacity expansion diﬀers depending on the degree of market
concentration. We first set up the monopoly market capacity expansion in 5.1. Then, we compare it withthe competitive one in 5.2.

5.1

The Monopolist’s Planning

A monopolist is assumed to expand market capacity with expansion cost c per unit and the zero marginal
operation costs. The monopolist maximizes the discounted sum of future profits. Let xtm be the market
capacity by the monopolist at period t and ymt the amount of new capacity expansion at period t. The profit
maximization problem is summarized by the following Bellman equation:
{
}
m
m
m
m
m
m
m
V(xt−1) = max
ρ
+
σ(x
)
−
ρ[x
+
y
]
−
cy
+
βV(x
+
y
)
.
t
t
t
t−1
t−1
t−1
m
yt ≥0

(15)

The interpretation of the above equation is that the present value of the sum of future profits is composedof the current profit and the discounted next period profit. The following proposition characterizes themonopolist’s planning.
Proposition 4. Let xm be the market size in the steady state under the monopolist’s plan and γ = [2ρ −σ′′ (xm )xm ]/βσ′ (xo ) > 0. Suppose that σ(1) + σ′ (1) ≤ ρ + (1 − β)c, and βγ2 > 4. Then, there exists a
unique xm > 0 and for all t = 1, 2, .., there exists a unique ymt > 0, which satisfies the following conditions:
1. For all t = 1, 2, .., the optimal solution ym
t > 0 satisfies the following second order diﬀerence
equation:
m
′ m
m
m
m
m
m
m
R(xt−1, ymt ) + βσ (xt−1 + yt )[xt−1 + yt + yt+1 ] − ρ[xt−1 + yt ] = c.

14

(16)

2. For all t = 1, 2, .., xtm ∈ [0, xm ] with x0m = 0 and xtm → xm as t → ∞.
Proof. See Appendix.

Equation (16) implies that the marginal expansion cost for the monopolist is equal to the marginal
benefit, which is composed of three elements. The first term on the right hand side of equation (16) is thepresent value of the sum of direct future revenues. The second term is the discounted value of the future
benefit in which an increased current market capacity raises subsequent period demand. The last term
represents a marginal loss of current revenue. This term is regarded as the current benefit, in which thedecreased current market capacity raises the current revenue.
The last two terms represent the intertemporal trade oﬀ with respect to the market cultivation strategiesof the monopolist. From the discounted future benefit, the monopolist has an incentive to lower the current
market price and increase current market capacity. This incentive coincides with the social planner’s
incentive to maximize social welfare. However, the monopolist has the incentive to raise the current
market price and to reduce output because of the current benefit. Therefore, the optimal planning by themonopolist is determined by the magnitude of both benefits.

5.2

Initial Eﬃciency and Eventual Ineﬃciency of Monopoly Diﬀusion

The comparison of both market capacity expansions starts from the initial market capacity expansions.
Then, the eventual market capacity expansions are examined. Note that the only diﬀerence between bothmarket capacity expansions is that the monopolist faces an intertemporal trade oﬀ.
Let pmt be the monopoly equilibrium price at period t. It is denoted by:
′ m m
m
pm
t = (1 − β)c − βσ (xt )xt+1 + ρxt ,(17)

for all t = 1, 2, ... It is easy to see that the monopoly equilibrium price is endogenously determined by
the intertemporal trade oﬀ between the discounted future benefit and the current benefit. By comparing(8) with (17), the monopoly equilibrium price at the current period is lower (higher) than the competitive
one if the net benefit of increasing current output is positive (negative). Because the inverse demand isstrictly decreasing in the current market capacity, the larger (smaller) monopoly capacity is observed aslong as the previous monopoly market capacity is at least as large (small) as the previous competitivemarket capacity.
15m ≶ ρxm . Then, for all xm Q xc , we
Proposition 5. Suppose that x0i = 0 for all i = c, m and βσ′ (xtm )xt+1
t
t−1
t−1

have xtm ≶ xtc .
Proof. See Appendix.

Note that the higher value of the discounted future benefit arises in the environment of a strong externality eﬀect, σ′ (xtm ), the large new market capacity expansion in the subsequent period, ym
t+1 , and the low
discount value, β. On the other hand, the degree of the current benefit, ρxtm is constant under the current
market capacity. Therefore, we conclude that the monopoly market capacity is larger than the competitive14
one in the early periods because the externality eﬀect is decreasing over time and ym
t+1 is initially large .

Next, we compare the eventual capacity expansions in the both markets. As the market capacity increases, the net benefit of increasing the current market capacity decreases and the monopoly equilibrium
price increases. The following lemma shows that the net benefit of increasing the current market capacity
becomes negative in the steady state.
Lemma 3. In the steady state, ρ > βσ′ (xm ).
Proof. See Appendix.

From lemma 3, it is easy to see that the monopoly equilibrium price in the steady state is higher
than the competitive one by comparing equations (8) and (17). This implies that the monopolist has
an incentive to slow down the market cultivation and to eventually earn positive profits. The following
proposition shows that the slow downed monopoly market capacity expansion leads to a smaller market
size in the steady state than for the competitive market capacity expansion.
Proposition 6. In the steady state, xc > xm .
Proof. See Appendix.

The characteristic monopoly equilibria15 may explain the low introductory price. In the presence of
the intertemporal consumption externality, it is optimal for the monopolist to reduce the initial profits
Whereas the concavity of σ(x) contributes this result, this result holds even if the externality eﬀect is constant, i.e., σ′ (x) =σ̄ < ρ. In this setting, the discounted sum of the future benefit in the first period is denoted by σ̄[x1m + ym2 ]. Because ym2 is positive,
the monopoly market capacity expansion is initially faster than the competitive one if ρ − σ̄ is close to zero.
15
The results here diﬀer from the results of Vettas (2000). The monopoly price in Vettas is dependent only on the previous
period’s market capacity because of perfectly elastic demand: an increase in the monopolist’s output does not lower price. In
the equilibrium, the large amount of market capacity in the previous period raises prices and revenues. In this environment, the14

16

by using the introductory price, and to start to earn larger profits with higher demand and a higher pricein the earlier periods. If the externality eﬀect is initially strong enough, the monopolist earns negative
profits in the initial periods, but soon obtains strong demand growth. It is interesting to recognize that themonopolist’s incentive to cultivate the market initially leads to a socially more eﬃcient output level thanthe competitive market under a strong externality eﬀect.
Furthermore, the above results provide some implications about the time pattern of the monopoly
market capacity expansion. As well as the socially optimal capacity expansion, the first period monopoly
market capacity expansion is influenced by the externality. In addition, it eventually leads to a smallermarket size than the competitive one. Therefore, the monopoly market capacity expansion in the first
period is relatively larger than the competitive one. It tends to be concave and has the greatest diﬃcultyof the three market capacity expansions in becoming S-shaped.

6

Conclusion

This paper presents a dynamic model of market capacity expansion where the market demand endogenously shifts in response to the previous period’s market capacity because of an intertemporal consumption externality. We explore how the degree of market diﬀusion depends on the market structures withrespect to its levels, time patterns, and prices. The major results reported here are summarized as follows.
First, the competitive environment leads to a constant equilibrium price, which is not influenced by theexternality. This contributes to the S-shaped time pattern of market capacity expansion. In addition, thecompetitive market capacity expansion is socially ineﬃcient and initially too slow because it does notinternalize the benefit of the externality through the market price. Therefore, subsidies for initial entrants
can be an eﬀective policy.
On the other hand, market concentration enables the firm to control the market price by internalizing
the eﬀect of the externality. The monopolist has an incentive to initially cultivate the market at a fast paceusing a low introductory price and eventually earn large profits with higher levels of demand. Whereas
this strategy leads to negative profits initially, it is optimal because the monopolist generates high levelsof market demand sooner.
monopolist has no incentive to restrict the current market capacity, but has an incentive to increase it instead. In addition, themonopoly market capacity is eﬃcient and larger than the competitive one because of firm learning. As a result, the monopoly
market capacity is eﬃcient and larger than the competitive one, while the monopoly price is higher.

17Whereas the model has ignored the impact of heterogeneous firms, demand uncertainty, technologicalprogress, and strategic behavior, which mainly impact the producer, these elements may be important
issues for market diﬀusion. However, our concern here is to examine how market diﬀusion diﬀers between
market structures when an intertemporal consumption externality exists, and to provide an alternative
explanation of S-shaped market diﬀusion and a low introductory price as simply as possible without theseelements. Therefore, this paper is to be regarded as a complement to these issues.
There are the several issues requiring future work. First, the empirical importance of the intertemporalconsumption externality and the relation between market capacity expansion and market concentration.
In addition, this paper ignores the aspect of oligopolistic market diﬀusion. My conjecture is that an
oligopolistic market capacity expansion is faster than the monopoly, but slower than the competitive market, and that its time pattern is likely to be S-shaped as the number of firms increase. Finally, there isconcern over how general our results are. While the analysis here is couched in terms of a parametricexample, the results may extend to more general settings. We hope this study helps researchers addressthese issues.

Appendix
Proof of Lemma 1
Note that for all t = 1, 2, .., new entry yct > 0 satisfies πct = ρyct . It is straightforward that πct is positively
related to yct .

Proof of Lemma 2
By diﬀerentiating Π(x) with respect to x, we have,Π′ (x) = σ′ (x) − ρ,(18)

Π′′ (x) = σ′′ (x).

(19)

and

By the definition of σ(), Π(x) is a strictly concave function for x ≥ 0 and has a unique maximizerx̄ = σ′−1 (ρ). It is easy to see that Π( x̄) > Π(0) = ρ − (1 − β)c and Π(xc ) = 0.

18

.

Proof of Proposition 1
Because the competitive diﬀusion path eventually becomes concave, it becomes S-shaped if and only if
πc1 > 0. This condition is equivalent to:σ(x1c ) > ρx1c .
Putting x1c = [ρ + (1 − β)c] into this inequality and rearranging terms, we have inequality (11).

(20)
Proof of Proposition 2
Note that there is a second order diﬀerence equation and one initial condition, x0o = 0. By using a phaseoodiagram, we show that there exist unique sequences, {xto }∞0 , which satisfy (i) for all t = 1, 2, .., xt ∈ [0, x ],and (ii) monotonicity, x0o = 0 and xto → xo as t → ∞.
We first show that there exists a unique xo > 0. On the steady state,ρxo − σ(xo ) − βσ′ (xo )xo = ρ − (1 − β)c.

(21)

Let T (x) = ρx − σ(x) − βσ′ (x). It is easy to see that T (x) is strictly increasing in x > 0 such that T (x) > 0.
Therefore, there exists a unique xo > 0.
To analyze the property of equation (13) further, we transform equation (13) into an equivalent systemo , xo ) space. Solving (13) with respect to xo , we have:in the (xt−1
t
t+1
o
xt+1
=

1
o
{ρxo − σ(xt−1) − [ρ − (1 − β)c]},′βσ (xto ) t(22)

for all t = 1, 2, ... Note that equation (22) is a second order diﬀerence equation with one variable, xo . Byo , we translate equation (22) to a simultaneous equation with two variables, xo and zo .
defining zot ≡ xt−1o ). Then, equation (20) becomes:
Let the right hand side of equation (20) be φ(xto , xt−1o = φ(xo , zo ), xt+1t tzo = xto .
t+1

for all t = 1, 2, ... By using Taylor’s formula, the linearized system around xo is denoted by:
] [
[ o
]
][
η − 1β xto − xo
xt+1 − xo=.
zot+1 − xo
1 0 zot − xo

(23)

(24)

for all t = 1, 2, ... Then, the characteristic equation becomes:(λo )2 − ηλo +
19

1
= 0.
β

(25)

Assuming βη2 > 4 and βη > 1 + β, we have eigenvalues λo1 , λo2 ∈ R2+ such that 0 < λo1 < 1 < λo2 andeigenvector corresponding to λo1 is:(eo1 , eo2 ) = (λo1 , 1).

(26)

Therefore, the local property of xo is saddle.
We finally draw the phase diagram. From equation (23), zot = zot+1 locus is zot = xto , that is, the 45
o locus is xo = φ(xo , zo ). By solving
degree line of the (zot , xto ) plane. On the other hand, the xto = xt+1t
t t

φ(xto , zot ) with respect to zot , we have:
zot = σ−1 ([ρ − βσ′ (xto )])xto − [ρ − (1 − β)c]),(27)

for all t = 1, 2, ... Because [ρ − βσ′ (xto )]xto is strictly increasing in xto and convex for [ρ − βσ′ (xto )]xto > 0,o locus is strictly increasing and convex in xo .
and σ−1 (x) is strictly increasing and convex, the xto = xt+1
t

We next examine whether zot and xto are increasing or decreasing above and below the phase-lines. It iseasy to check that for the points above the zot = zot+1 line, we have zot > zot+1 , and for the points below theo line, we have xo > xo , and forzot = zot+1 line, we have zot < zot+1 . Also, for the points above xto = xt+1t
t+1
o , we have xo < xo . The phase diagram is summarized in Figure 3 where thethe points below xto = xt+1
t
t+1

arrows show the direction of increase at each point.
We now characterize the optimal solution by using the above properties. It satisfies the following
o locus is increasing and (ii) xo ∈ [ x̄o , xo ] for all t = 1, 2, .., where:
conditions: (i) the xto = xt+1t{}x̄o = x ∈ R++ : [ρ − βσ′ (x)]x = ρ − (1 − β)c .

(28)

Because [ρ − βσ′ (x)]x is strictly increasing in x for all [ρ − βσ′ (x)]x > 0, there exists a unique x̄o > 0.
o locus intersects the xo = 0 axis once at x̄o > 0. Any xo < [ x̄o , xo ] does not
This implies that the xto = xt+1tt−1

become the optimal solution. Therefore, we conclude that the optimal solution that satisfies conditions(i) and (ii) is restricted in the shaded portion. Any points outside the shaded portion diverge and do not
become the optimal solution. For the points above the zot = zot+1 locus, xto becomes negative in finite
time but this violates a feasibility condition. On the other hand, for the points above zot+1 > zot but belowooooo , xo is increasing over time satisfying zo
xto = xt+1
t
t+1 > zt (xt > xt−1 ). However, for large xt such thato , the inverse demand function becomes P(xo , xo ) = 0. In these circumstances, Euler equation
xto > xt−1
t−1 t

20

(13) does not hold. It is easy to see that there exists a unique path that converges to xo and the optimal
solution consists of the points that belong to the stable manifold of xo .

Figure 3: The dynamical system of socially optimal diﬀusion path

Proof of Proposition 3
o , xo ) ≡ ρ + σ(xo ) − ρxo + βσ′ (xo )xo . Rearranging equation (9), the competitive diﬀusion
Let U(xt−1
t
t
t t+1
t−1

path is denoted by:ccU(xt−1, xtc ) − βσ′ (xtc )xt+1
= c.

(29)

c
o . Suppose in negation that xc ≥ xc . Then, by using the properties of U(, ), we have theLet xt−1≤ xt−1ttfollowing inequalities:oocU(xt−1, xto ) ≥ U(xt−1, xtc ) ≥ U(xt−1, xtc ),

(30)

for all t = 1, 2, .., where the first inequality follows from U2 (xt−1 , xt ) < 0 and the second inequalityc , xc ) ≤ c which
follows from U1 (xt−1 , xt ) > 0. Because U(xto , xto ) = c in the equilibrium, we have U(xt−1tc is positive.
contradicts equation (29) because βσ′ (xtc )xt+1

21

Proof of Proposition 4Note that there is a second order diﬀerence equation and one initial condition, x0m = 0. By using aphase diagram, we show that there exist unique sequences, {xtm }∞0 , which satisfy (i) for all t = 1, 2, ..,xtm ∈ [0, xm ], and (ii) monotonicity, x0m = 0 and xtm → xm as t → ∞.
We first show that there exists a unique xm > 0. In the steady state,2ρxm − σ(xm ) − βσ′ (xm )xm = ρ − (1 − β)c.

(31)

Let F(x) = 2ρx − σ(x) − βσ′ (x). It is easy to see that F(x) is strictly increasing in x > 0 such that F(x) > 0.
Therefore, there exists a unique xm > 0.
To analyze the properties of equation (16) further, we transform equation (16) to an equivalent system
m , xm ) space. Solving (16) with respect to xm , we have:in (xt−1
t
t+1
m
xt+1
=

1
m
{2ρxtm − σ(xt−1) − [ρ − (1 − β)c]},βσ′ (xtm )(32)

for all t = 1, 2, ... Note that equation (32) is a second order diﬀerence equation with one variable, xm . Bymm
defining zmt ≡ xt−1 , we translate equation (32) to a set of simultaneous equations with two variables, xm ). Then, equation (32) becomes:
and zm . Let the right hand side of equation (32) be ψ(xtm , xt−1
m = ψ(xm , zm ), xt+1ttzm = xtm .
t+1

(33)

for all t = 1, 2, ... By using Taylor’s formula, the linearized system around xm is denoted by:
] [
[ m
]
][
γ − 1β xtm − xm
xt+1 − xm
m =
m .
zm
1 0 zmt −x
t+1 − x(34)

for all t = 1, 2, ... Then, the characteristic equation becomes:(λm )2 − γλm +

1
= 0.
β

(35)

Assuming βγ2 > 4, it is easy to check that βγ > 1 + β because ρ > βσ′ (xm )16 . Therefore, we have
m
m
m
m
2
eigenvalues λm
1 , λ2 ∈ R+ such that 0 < λ1 < 1 < λ2 and an eigenvector corresponding to λ1 is:mm(em
1 , e2 ) = (λ1 , 1).
16This property is proved in Lemma 3.

22

(36)

Therefore, the local property of xm is saddle.
mmm
We finally draw the phase diagram. From equation (33), the zm
t = zt+1 locus is zt = xt , that is, the 45
m
m
m
m
m m
degree line of the (zmt , xt ) plane. On the other hand, the xt = xt+1 locus is xt = ψ(xt , zt ). By solvingmψ(xtm , zmt ) with respect to zt , we have:
−1
′ m
o
zm
t = σ ([2ρ − βσ (xt )])xt − [2ρ − (1 − β)c]),(37)

for all t = 1, 2, ... Because [2ρ − βσ′ (xtm )]xtm is strictly increasing in xtm and convex for [2ρ − βσ′ (xtm )]xtm >m locus is strictly increasing and convex in
0, and σ−1 (x) is strictly increasing and convex, the xtm = xt+1m
xtm . We next examine whether zmt and xt are increasing or decreasing above and below the phase-lines.
mmm
It is easy to check that for the points above zm
t = zt+1 line, we have zt > zt+1 , and for the points below
m
m
m
m
m
m
m
zm
t = zt+1 , we have zt < zt+1 . Also, for the points above xt = xt+1 line, we have xt > xt+1 , and for them , we have xm < xm . The phase diagram is presented in Figure 4 where the arrows
points below xtm = xt+1
t
t+1

show the direction of increase at each point.
We now characterize the optimal monopoly solution by using the above properties. It satisfies them locus is increasing and (ii) xm ∈ [ x̄m , xm ] for all t = 1, 2, .., where:
following conditions: (i) the xtm = xt+1t{}x̄m = x ∈ R++ : [2ρ − βσ′ (x)]x = ρ − (1 − β)c .

(38)

Because [2ρ − βσ′ (x)]x is strictly increasing in x for all [2ρ − βσ′ (x)]x > 0, there exists a unique x̄m > 0.
m locus intersects the xm = 0 axis at x̄m > 0. Any xm < [ x̄m , xm ] does not
This implies that the xtm = xt+1tt−1

become the optimal monopoly solution. Therefore, we conclude that the optimal monopoly solution thatsatisfies conditions (i) and (ii) is restricted to the shaded portion. Any points outside the shaded portionmdiverge and do not become the optimal monopoly solution. For the points above the zm
t = zt+1 locus,

xtm becomes negative in finite time, but this violates a feasibility condition. On the other hand, for the
m
m
m
m
m
m
m
m
points above zm
t+1 > zt , but below xt = xt+1 , xt is increasing over time satisfying zt+1 > zt (xt > xt−1 ).
m , the inverse demand function becomes P(xm , xm ) = 0. In
However, for large xtm such that xtm > xt−1t−1 tthese circumstances, Euler equation (16) does not hold. It is easy to see that there exists a unique paththat converges to xm and the optimal monopoly solution consists of the points that belong to the stable

manifold of xm .

23

Figure 4: The dynamical system of monopoly diﬀusion path

Proof of Proposition 5
c , xc ) ≡ R(xc , yc ). We prove the first case. Let βσ′ (xm )xm < ρxm and xm ≤ xc . Suppose in
Let L(xt−1
t
t
t
t−1 t
t+1
t−1
t−1

negation that xtm ≥ xtc . Then, using the property of L(, ), we have the following inequalities:ccmL(xt−1, xtc ) ≥ L(xt−1, xtm ) ≥ L(xt−1, xtm ),

(39)m ≤ xc , where the first inequality follows from L (x , xt ) < 0 and L (x , x ) >for all t ≥ 1 such that xt−1
2 t−1
1 t−1 tt−1
c , xc ) = c in the competitive equilibrium, we have L(xm , xm ) ≤ c. This is a contradiction
0. Because L(xt−1t
t−1 tm < ρxm . In the same way, we can prove the second case.
to equation (16) because βσ′ (xtm )xt+1t

Proof of Lemma 3
Suppose in negation that ρxm ≤ βσ′ (xm ). Then, we have the following inequalities:ρxm ≤ βσ′ (xm )xm < σ(xm ),(40)

Where the last inequality follows from the property of σ(). These inequalities imply that:2ρxm ≤ σ(xm ) + βσ′ (xm )xm .
This contradicts the steady state condition, 2ρxm > σ(xm ) + βσ′ (xm )xm .
24

(41)


Proof of Proposition 6
Suppose in negation that xc ≤ xm . This implies that Π(xm ) ≤ 0. But this contradicts the steady state
condition:
Π(xm ) = ρxm − βσ′ (xm )xm > 0.

(42)

Therefore, xc > xm .

References[1] Banerjee, Abhijit. 1992, ‘A Simple Model of Herd Behavior’, Quarterly Journal of Economics. 107,
pp797-817.
[2] Bass, Frank. 1969, ‘A New Product Growth Model for Consumer Durables’, Management Science,
15, pp215-227.
[3] Becker, Gary. 1991, ‘A Note on Restaurant Pricing and Other Examples of Social Influences on
Price’, Journal of Political Economy. 99, pp1109-1116.
[4] Becker, Gary. and Murphy, Kevin. 1988, ‘A Theory of Rational Addiction’, Journal of Political
Economy. 96, pp675-700.
[5] Bergemann, Dirk. and Välimäki, Juuso. 1997, ‘Market Diﬀusion with Two-sided Leaning’, Rand
Journal of Economics. 28, pp773-795.
[6] Bergemann, Dirk. and Välimäki, Juuso. 2006, ‘Dynamic Pricing of New Experience Goods’, Journalof Political Economy. 114, pp713-743.
[7] Bikhchandai, Sushi., Hirsheifer, David. and Welch, Ivo. 1992, ‘A Theory of Fads, Fashion, Custom,
and Cultural Change as Informational Cascades’, Journal of Political Economy, 100, pp992-1026.
[8] Cabral, Luis. Salant, David. and Woroch, Glenn. 1999, ‘Monopoly Pricing with Network Externalities’, International Journal of Industrial Organization, 17, pp199-214.
[9] Caminal, Ramon. and Vives, Xavier. 1996, ‘Why Market Shares matter: An Informational-Based
Theory’, Rand Journal of Economics. 27, pp221-239.
25

[10] Caves, Richard. 1998, ‘Industrial Organization and New Findings on the Turnover and Mobility ofFirms’, Journal of Economic literature. 36, pp1947-1982.
[11] Doganoglu, Toker. 2003, ‘Dynamic Price Competition with Consumption Externalities’, Netomics.
5, pp43-69.
[12] Economides, Nicholas. and Himmelberg, Charles. 1995,‘Critical Mass and Network Size with Application to the US Fax Market’, Discussion Paper, Stern School of Business, NYU, EC-95-11.
[13] Ellison, Glenn. and Fundenberg, Drew. 1993, ‘Rules of Thumb for Social Learning’, Journal of
Political Economy. 101, pp612-643.
[14] Frank, Robert. 1989, ‘Frames of Reference and the Quality of Life’, American Economic Review.
79, pp80-85.
[15] Gort, Michael. and Klepper, Steven. 1982, ‘Time Paths in the Diﬀusion of Product Innovations’,
Economic Journal. 92, pp630-652.
[16] Grliches, Zyi. 1957, ‘Hybrid Corn: An Exploration in the Economics of Technological Change’,
Econometrica. 25, pp501-522.
[17] Jensen, Richard. 1982, ‘Adoption and Diﬀusion of an Innovation of Uncertain Profitability’, Journalof Economic Theory. 27, pp182-193.
[18] Jovanovic, Boyan. and Lach, Saul. 1989, ‘Entry, Exit, and Diﬀusion with Learning by Doing’, American Economic Review. 79, pp690-699.
[19] Jovanovic, Boyan. 1998, ‘Michael Gort’s Contribution to Economics’, Review of Economic Dynamics. 1, pp327-337.
[20] Karshenas, Massoud. and Stoneman, Paul. 1992, ‘Flexible Model of Technological Diﬀusion Incorporating Economic Factors with an Application to the Spread of Colour Television Ownership in theUK’, Journal of Forecasting. 11, pp577-601.
[21] Katz, Michael. and Shapiro, Carl. 1985, ‘Network Externalities, Competition, and Compatibility’,
American Economic Review, 75, pp424-440.
26

[22] Katz, Michael. and Shapiro, Carl. 1986, ‘Technology Adoption in the Presence of Network Externalities’, Journal of Political Economy, 94,pp822-841.
[23] Katz, Michael. and Shapiro, Carl. 1994, ‘Systems Competition and Network Eﬀects’, Journal of
Economic Perspective, 8, pp93-115.
[24] Klepper, Steven. 1996, ‘Entry, Exit, Growth, and Innovation over the Product Life Cycle’, American
Economic Review, 86, pp562-583.
[25] Krishnan, Trichy., Bass, Frank., Jain, Dipack. 1999, ‘Optimal Pricing Strategy for New Products’,Marketing Science, 45, pp1650-1663.
[26] Mahajan, Vijay., Muller, Eitan., and Bass, Frank. 1990, ‘New Product Diﬀusion Models in Marketing: A Review and Directions for Research’, Journal of Marketing, 54, pp1-26.
[27] Mahajan, Vijay., Muller, Eitan., and Bass, Frank. 1995, ‘Diﬀusion of New Products: Empirical
Generalizations and Managerial Uses’, Marketing Science, 14, pp79-88.
[28] Mansfield, Edwin. 1968, ‘Industrial Research and Technological Innovation’, New York: W.W.
Norton, 1968.
[29] McFadden, Daniel. and Train, Kenneth. 1996, ‘Consumers’ Evaluation of New Products: Learningfrom Self and Others’, Journal of Political Economy, 104, pp683-703.
[30] Monteiro, Paulo. and Gonzalez, Jose. 1999, ‘”We Sold a Million Units” - The Role of Advertising
Past-Sales’, Econpapers no. 357.
[31] Reinganum, Jennifer. 1981,‘Market Structure and the Diﬀusion of New technology’, Bell Journal of
Economics, 12, pp618-624.
[32] Rob, Rafael. 1991, ‘Learning and Capacity Expansion under Demand Uncertainty’, Review of Economic studies, 58, pp.655-675.
[33] Rohlfs, Jeﬀrey. 1974, ‘A Theory of Interdependent Demand for a Communication Service’, Bell
Journal of Economics and Management Science, 5, pp16-37.

27

[34] Schmalensee, Richard. 1982, ‘Product Diﬀerentiation Advantages of Pioneering Brands’, American
Economic Review, 72, pp.349-365.
[35] Shapiro, Carl. 1983, ‘Optimal Pricing of Experience Goods’, Bell Journal of Economics, 14, pp497507.
[36] Stoneman, Paul. 2002, ‘The Economics of Technological Diﬀusion’, Blakwell.
[37] Sutton, John. 1997, ‘Gibrat’s Legacy’, Journal of Economic Literature, 35, pp40-59.
[38] Vettas, Nikolaos. 1998, ‘Demand and Supply in New Markets: Diﬀusion with Bilateral Learning’
Rand Journal of Economics. 29, pp215-233.
[39] Vettas, Nikolaos. 2000, ‘Investment Dynamics in Markets with Endogenous Demand’ Journal ofIndustrial Economics. 48, pp189-203.

28