GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.46

Market Diffusion with Consumer-Based Bilateral Learning

Hiroshi Kitamura and Akira Miyaoka

May 2009; revised September 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Market Diffusion with Consumer-Based
Bilateral Learning∗
Hiroshi Kitamura†

Akira Miyaoka‡

September 5, 2011

Abstract
This paper analyzes the market diffusion of a new product whose quality is uncertain. Consumers learn the product quality by observing the history of market outcomes.
Firms cannot observe how consumers evaluate the product quality; instead, they learn byobserving consumer behavior. New entry occurs gradually because of informational externalities. This dual uncertainty contributes to an S-shaped diffusion of the new productwith declining prices.

JEL Classifications Code: D11, L11, L14.
Keywords: Experience Goods; Quality Uncertainty; Bilateral Learning; S-shaped Diffusion.
∗

In the process of this study, we talked to many people, and their knowledge and ideas contributed significantly to the analysis in this paper. We especially thank Katsuya Takii and Junichiro Ishida for helpful
discussions and comments. We also thank Akira Yakita (the editor in chief), Kunio Urakawa (the editor), andtwo anonymous referees for constructive comments and suggestions. The first author gratefully acknowledges
financial support from JSPS Grant-in-Aid for Research Activity start-up No. 22830075 and for Scientific Research (A) No. 22243022 organized by Masaki Nakabayashi. The second author gratefully acknowledges
financial support from the Global COE program “Human Behavior and Socioeconomic Dynamics” of Osaka
University. The usual disclaimer applies.
†
Corresponding Author: Faculty of Economics, Sapporo Gakuin University, 11 Bunkyodai, Ebetsu,
Hokkaido, 069-8555, Japan. E-mail: kitamura@sgu.ac.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043,
Japan. E-mail: jge013ma@mail2.econ.osaka-u.ac.jp

1 Introduction
When a new product or service market opens, consumers are usually uncertain about its characteristics. An important source of product information is actual experience of the product.
Another important source is learning from others. Consumers usually learn about productcharacteristics through their observations of past market outcomes that reflect the behavior
of consumers who have experienced the product. The degree of their uncertainty depends on
the history of market outcomes and it decreases as the number of past observations increases.
At the same time, firms face demand uncertainty regarding how consumers evaluate thequality of their products. This causes them to be uncertain about the profitability of themarket. They learn the product quality by observing the behavior of consumers in the market.
Therefore, both demand and supply are endogenously determined by consumer learning.
The aim of this paper is to explore theoretically the time pattern of market diffusion where
both the market demand and the level of entry are endogenously determined by consumer
learning. The significant result reported here is that consumer-based bilateral learning explains the S-shaped diffusion with decreasing prices.
Previous studies that estimate the time pattern of market diffusion find that it is initially
too slow and tends to be S-shaped.1 In terms of the entry of firms, the S-shaped time patternof market diffusion implies that the level of new entries initially rises but eventually falls.
In addition to the time pattern of diffusion, it is commonly observed that the market priceof the product falls over time (Klepper and Graddy (1990)). In the theoretical literature, thephenomenon of S-shaped diffusion with a strictly declining market price has been viewed as
the result of a reduction in production cost (Jovanovic and Lach (1989)). In this paper, we
provide an alternative explanation to technological progress, instead examining the S-shaped
diffusion in terms of informational externalities.
In this paper, we define a dynamic model of market diffusion with consumer-based bilateral learning. The product involved is assumed to be what Shapiro (1983) calls an “experience
1See, for example, Gort and Klepper (1982), who investigate the time pattern of the number of firms in 46
product markets. Note that S-shaped diffusion is not an isolated phenomenon. Empirical evidence shows thatthe time pattern of innovation, whether in usage or ownership, of new product technologies by households tends
to be S-shaped. See the seminal work by Griliches (1957) and Mansfield (1968), together with the survey by
Stoneman (2002).

1

good”, which means that consumers immediately realize the product’s quality once they try it.
For all periods, there exist two types of consumer: experienced consumers who have alreadytried the product and inexperienced consumers who have not. Experienced consumers know
the product quality but inexperienced consumers do not. Inexperienced consumers learn the
product quality by observing the behavior of experienced consumers and update their beliefs
in a Bayesian fashion. As they continually update their beliefs about the product’s quality,
their expectations become more accurate in each period. At the same time, firms face uncertainty regarding how consumers evaluate the quality of their product. They learn the productquality in the same way that inexperienced consumers do. For each period, entry is determined by the zero expected profit condition. In the case of a high-quality product, bilaterallearning positively affects market diffusion and market capacity grows.
We also define a market diffusion process with unilateral learning in which firms know
the true quality of the product in the benchmark case. Then, we compare both market capacity expansions in terms of entry level, equilibrium price, and time pattern. The analysis
in the case of a high-quality product provides several interesting results. The main results
are as follows: (i) diffusion with bilateral learning leads to a lower entry level; (ii) although
unilateral learning leads to constant equilibrium prices, bilateral learning leads to declining
equilibrium prices over time; and (iii) bilateral learning is more likely to generate S-shaped
diffusion.
The intuitive logic for these results is as follows. The only difference between these twomarket diffusions is whether or not firms know the quality of their product. Under bilaterallearning, because of the presence of firm uncertainty about product quality, firms may earn
negative profits following overentry when they overestimate the product quality. In this case,the expected profitability of entering the market under bilateral learning is lower than that
under unilateral learning. Therefore, the level of entry is lower and the equilibrium price ishigher than for the unilateral learning case.
However, as the market capacity increases, the degree of uncertainty decreases. From thefirm’s viewpoint, this effect leads to two benefits. The first benefit is that this effect raises
the expected profitability associated with the demand change through consumer learning.
Note that this effect can be observed under both unilateral and bilateral learning. The second2benefit is that it reduces the expected loss from overestimation through firm learning. Because
unilateral learning does not have the second benefit, bilateral learning generates an additional
increase in the expected profitability. This leads to more new entries than in the unilateralcase. Therefore, bilateral learning speeds up market diffusion and reduces equilibrium prices.
S-shaped diffusion is more likely to be observed in the case of bilateral learning because
bilateral learning leads to fewer initial entries but more subsequent entries.
This paper relates to the literature on market diffusion. In the theoretical literature, market
diffusion is regarded as the result of firm learning (Rob (1991)) or intertemporal consumption
externalities (Vettas (2000) and Kitamura (2010)). One of the main differences between thispaper and those studies is that the others do not simultaneously explain two empirical facts:
S-shaped diffusion and strictly declining pricing.2
By extending the model in Rob (1991), Vettas (1998) analyzes the model of S-shaped
diffusion with bilateral learning.3 In the model in Vettas (1998), consumers are uncertain
about product quality and firms are uncertain about the number of consumers. The model in
Vettas (1998) and that in this paper differ in the demand structure; that is, the demand functionin Vettas (1998) is horizontal but the demand function in this paper is downward sloping. This
seemingly small difference leads to a crucial difference in the explanation of why market
expansion is gradual. In the model in Vettas (1998), the reason is firm learning. In contrast,in the model in this paper, the reason is the downward-sloping demand function. Whenthe demand function is horizontal, there is no cost of expansion. Therefore, Vettas (1998)
assumes sequential resolution of uncertainty for firms regarding the number of consumers toderive gradual adjustment. In contrast, when the demand function is downward sloping, a risein current output lowers current prices. This makes faster expansion more costly. Therefore,
the downward-sloping demand function plays the role of adjustment costs, leading to gradualmarket expansion.
In addition, the difference in demand structure leads to the difference in market prices. In
2

In a study of strictly declining prices without technological progress, Bagwell and Riordan (1991) showthat the prices of high-quality products are initially high and strictly decline as a result of strategic behavior. In
contrast, this paper shows strictly declining prices without strategic behavior but with firm learning.
3
Bergemann and Välimäki (1997) also analyze market diffusion with bilateral learning. In their model,
however, the number of firms is fixed exogenously.

3the model in Vettas (1998), market prices increase over time. When the demand function ishorizontal, updates on product quality directly increase the market price. In contrast, when
the demand function is downward sloping, an update on product quality leads to a demand
shift but does not necessarily lead to an increase in the market price.
The remainder of this paper is organized as follows. Section 2 defines the model. Section
3 introduces market diffusion with unilateral learning and market diffusion with bilaterallearning. Section 4 compares properties of the bilateral learning diffusion with that of theunilateral learning diffusion. Section 5 contains concluding remarks. The proofs of all results
are provided in Appendix.

2 ModelThis section defines the model. We assume that time is discrete and the time horizon is
infinite, and that a period consists of two stages: the first half and the second half. The
product here is perishable.

2.1 Consumers
There are a number of mass unit consumers in each period. Each consumer has a different
preference for a product and purchases a product in each half of every period if and only if
his/her reservation price is higher than the market price. Let θ be the type of consumer, whichis stationary for all periods and is uniformly distributed on the interval [0, 1]. The reservation
price of type θ consumer is assumed to depend on his/her type and the product quality, q:
V(θ, q) = ρθ + q,

(1)

where ρ > 0 is a preference parameter.4 We assume that the product is an experience good:
consumers do not know the product quality before they try it but once they try it, they perfectly realize its true quality. The following assumption provides the property of q.
Assumption 1. q is uniformly distributed on the interval [qL , qH ] ∈ R+ .
4

Assumption 1 implies that every consumer’s reservation price grows in the same way. When an increase
in reservation price differs among consumers, for example, V(θ, q) = ρ + θq, we have almost the same results,
which leads to a more S-shaped time pattern of market diffusion.

4

At the beginning of each period, the set of consumers is composed of experienced consumers who have tried the product and inexperienced consumers who have not. Experienced
consumers realize the actual product quality, q̃, and purchase the product if and only if theprice is less than or equal to V(θ, q̃). On the other hand, inexperienced consumers who do notknow the true product quality form a belief regarding its quality by observing the behavior ofexperienced consumers in a Bayesian fashion.
Assumption 2. At the beginning of (the first half of) each period, inexperienced consumers
update their beliefs on the product quality by observing the history of market outcomes, whichsummarize the behavior of experienced consumers.
The exact learning mechanism is characterized below. This assumption implies that thetransmission of information indirectly occurs from experienced consumers to inexperienced
consumers through the observation of past market outcomes.5 Although the direct information from experienced consumers to inexperienced consumers is important for learning about
product quality, we do not focus on this effect in this paper. Let qht be the highest quality that
inexperienced consumers expect in period t and qlt be the lowest quality in period t. It is easy
to see that qh1 = qH and ql1 = qL . The expected quality in period t is denoted by mt . Then, we
have the following lemma.
Lemma 1. For all t = 1, 2, ...,mt =qht + qlt.
2(2)

2.2 Firms
Firms are small and identical. At the beginning of each period, the set of firms is composedof two subsets: potential entrants and incumbents. There is no asymmetry of information
between the two subsets. Under bilateral learning, firms do not know how consumers evaluate
their product quality but they know its prior distribution and consumers’ updating rules. Firms
When the equilibrium price in the first half of period t is larger than that in period t − 1, inexperienced
consumers can update their beliefs at the beginning of the second half of period t. However, as mentioned below,
we focus on the equilibrium where the price is nonincreasing, and consumers cannot update their beliefs at the
beginning of the second half of the period along the equilibrium path. Therefore, allowing for the possibility of
belief updating in the second half of the period does not alter our main results. See also footnote 13.
55

update their beliefs about the product quality in the same way as inexperienced consumers.
On the other hand, under unilateral learning, firms know the consumers’ evaluations of thequality of their product. The entry decision for potential entrants is assumed to be made at
the beginning of the first half of each period.6 Potential entrants enter the market with entry
cost c > 0, which is the initial investment such as the purchase of machinery. We assume
that the machinery is durable and continues to work over time. Machines can be operated
at any level from zero to one marginal unit for each half of every period in an environmentof constant returns to scale. To simplify the analysis, we assume that the scrap value of themachinery and the marginal operating cost are zero.
Let xt and yt be the numbers of incumbents and new entrants in period t, respectively.
Because the entry cost is not recoverable and the marginal cost is zero, incumbents have no
incentive to exit the market and, therefore, yt = xt − xt−1 ≥ 0 for all t = 1, 2, .... Assuming thatP
x0 = 0, we have xt = tτ=1 yτ for all t = 1, 2, .... Let i > 0 be a constant interest rate. Then,the discount factor is denoted by β ≡ 1/(1 + i). Let Rt (xt−1 + yt ) be the discounted sum of
future expected profits in period t when there exist xt−1 incumbents and yt new entrants in themarket at the beginning of period t.7 The discounted sum of future expected operating profits
in period t is composed of the expected operating profits in both the first and the second halvesof period t and the discounted continuation operating profits. The aim of firms is to maximize
the discounted sum of future profits, which is denoted by Rt (xt ) − c. Potential entrants enter
the market when the present value of expected profits is positive, i.e., Rt (xt−1 ) > c.

2.3 Equilibrium
Each period’s entry level and equilibrium price are endogenously determined in this model.
Under both unilateral and bilateral learning, the equilibrium condition is determined by themarket-clearing condition and the zero expected profit condition. Let dt be the number ofconsumers who purchase the product in period t and ft be the number of new consumers in
6

If potential entrants could also enter the market in the second half of each period, they would have tochoose between entering in the second half of the period without updating their beliefs about the quality andentering in the first half of the following period after updating their beliefs. This makes the analysis considerablycomplicated.
7
Because entrants and incumbents are symmetric and the time horizon is infinite, they have the same present
value of expected operating profits.

6

period t. If d0 = 0, then dt =Pt

τ=1 fτ for all t = 1, 2, .... Now, we define the competitive

equilibrium as follows.
Definition. Under both unilateral and bilateral learning, the competitive equilibrium consists of three sequences {pt , ft , yt } that simultaneously satisfy the following conditions.
1. The market clears for all t = 1, 2, ...:
ft = yt .

(3)

2. The market price is determined by the inverse demand of consumers for all t = 1, 2, ...:
pt = Pt (xt−1 + yt ).

(4)

3. New entry satisfies the zero expected profit condition for all t = 1, 2, ...:Rt (xt−1 + yt ) ≤ c, with equality if yt > 0.

(5)

3 Analysis
In this section, we explain the demand evolution with consumer learning and derive the competitive equilibrium under both unilateral and bilateral learning. After examining consumers’learning process, we analyze the diffusion with unilateral learning, where firms know the
product quality, as a benchmark case. Then, we explore the diffusion with bilateral learningand show that there exists a unique equilibrium in which prices gradually decline.

3.1 Demand Evolution with Consumer Learning
In our model, inexperienced consumers learn the product quality by observing the historyof market outcomes such as past market prices and the number of past sales. To make thelearning process in this paper function well, we introduce two assumptions. First, we assume
that the market price in each period satisfies the following properties.
Assumption 3. The market price satisfies the following properties.

71. When inexperienced consumers and firms underestimate the product quality, mt ≤ q̃,
the equilibrium price is constant during the period.
2. When inexperienced consumers and firms overestimate the product quality, mt > q̃, theequilibrium price declines in the second half of the period.
The intuition behind this assumption is as follows. Firms sell their product at a constantprice during a period when they underestimate the product quality.8 However, when they
overestimate the product quality, they reduce the market price. If overestimation occurs,some experienced consumers do not repurchase the product in the second half of the period.
Therefore, the market price must fall so that the second half sales equal the first half sales.
Second, we assume the following inequalities.
Assumption 4.
2ρ +7(1 − β)qH + (25 − 9β)qL> (1 − β)c > 2qH .
8(2 − β)(6)The first inequality implies that in the second half of each period, at least some experienced consumers who purchased the product in the first half of the period repurchase theproduct. The importance of this assumption is explained below. In addition, this inequality
guarantees positive first period entry on the competitive equilibrium under both unilateral andbilateral learning. The second inequality implies that the diffusion rate is less than one forevery period and it guarantees that there exist at least some inexperienced consumers and thatlearning exists for all periods.9
The demand evolution based on consumer learning is characterized as follows. At the
beginning of the first period (first half of the first period), all consumers are inexperienced
consumers whose expectation about the quality is m1 = (qH +qL )/2. Then, the inverse demand
function is:ρ(1 − x1 ) + m1
P1 (x1 ) = 080 ≤ x1 ≤ 1,
x1 > 1,(7)

For example, if firms must pay the “menu costs” for price adjustment, they may be unwilling to change
prices.
9
The inequalities in (6) are derived by using the equilibrium capacity level under bilateral learning which
we obtain in Proposition 2. For the details of the derivation of these inequalities, see footnotes 11 and 18.

8and the equilibrium price in the first half of period 1 is p1f st = ρ(1 − x1∗ ) + m1 , given theequilibrium market capacity in period 1, x1∗ . At the beginning of the second half of period
1, consumers who purchased the product become experienced consumers and realize its true
quality, q̃. On the other hand, inexperienced consumers do not know the product quality.
At the beginning of period 2, inexperienced consumers observe experienced consumers’ repurchase behavior through the change in equilibrium price during period 1. There are two
possible cases.
Case 1 m1 > q̃When inexperienced consumers overestimated the product quality at the beginning of period
1, some experienced consumers did not repurchase the product at the same price in the second
half of the period. Therefore, the market price must decrease so that the second half sales
equal the first half sales. Let p1sec be the equilibrium price in the second half of period 1;
e1sec be the number of experienced consumers who repurchased the product; and u1sec be thenumber of inexperienced consumers. Then, the following three equations are simultaneously
satisfied in the market equilibrium in the second half of period 1:
p1sec = ρ(1 − e1sec ) + q̃,(8)

p1sec = ρ(1 − x1∗ − u1sec ) + m1 ,(9)x1∗ = e1sec + u1sec ,(10)

where e1sec > 0 and u1sec > 0. The first equation represents the inverse demand of experienced
consumers, the second equation is the inverse demand of inexperienced consumers, and thethird equation indicates the feasibility condition. Since inexperienced consumers know p1sec ,x1∗ , and m1 , then, from these simultaneous equations, they immediately know the true qualityof the product, q̃, and learning stops. For the subsequent periods t = 2, 3, ..., the inverse
demand function becomes:ρ(1 − xt ) + q̃Pt (xt ) = 00 ≤ xt ≤ 1,
xt > 1.

(11)

Note that, from equations (8)–(10), we have e1sec = x1∗ − (m1 − q̃)/2ρ. Therefore, when x1∗ issufficiently small or q̃ is sufficiently low, no experienced consumers repurchase the product9

in the second half, i.e., e1sec = 0. In this case, inexperienced consumers cannot realize thetrue quality of the product, q̃, and learning does not stop.10 To exclude this possibility, in ourmodel, the first inequality of (6) in Assumption 4 guarantees that x1∗ is sufficiently large sothat x1∗ > (m1 − qL )/2ρ holds.11 This means that, under Assumption 4, when overestimation
occurs, at least some experienced consumers necessarily repurchase the product and that
inexperienced consumers know the product quality regardless of the value of q̃.
Case 2 m1 ≤ q̃When inexperienced consumers underestimated the product quality at the beginning of period 1, the equilibrium price was constant during the period, by Assumption 3. Since all
experienced consumers repurchased the product at the same price, inexperienced consumers
believe that the true quality of the product is at least as high as m1 and update the lowest
quality ql2 = m1 . Then, the expected product quality in period 2 becomes m2 = (qH + m1 )/2 =(3qH + qL )/4. There are two cases. First, when inexperienced consumers overestimate theproduct quality in period 2, m2 > q̃, the inverse demand function at the beginning of period 2
becomes:12ρ(1 − x2 ) + q̃ρ(2−x1∗ −x2 )+m2 +q̃
2P2 (x2 ) = ρ(1 − x2 ) + m2





0

0 ≤ x2 ≤ x1∗ − (m2 − q̃)/ρ < 1,x1∗ − (m2 − q̃)/ρ < x2 < x1∗ + (m2 − q̃)/ρ < 1,
x1∗ + (m2 − q̃)/ρ ≤ x2 ≤ 1,
x2 > 1.

(12)This is because equation (8) vanishes when e1sec = 0.
By substituting the equilibrium capacity level under bilateral learning which we obtain in Proposition 2 intox1∗ > (m1 − qL )/2ρ and rearranging, we have the first inequality of (6) in Assumption 4.
12
When inexperienced consumers overestimate the product quality in period 2, the highest reservation priceamong inexperienced consumers, ρ(1 − x1∗ ) + m2 , can be higher than that among experienced consumers, ρ + q̃.
Therefore, with overestimation, there can be two types of inverse demand function depending on whether
ρ + q̃ > ρ(1 − x1∗ ) + m2 or ρ + q̃ ≤ ρ(1 − x1∗ ) + m2 . However, we can show that ρ + q̃ > ρ(1 − x1∗ ) + m2
always holds under Assumption 4 and, therefore, we have equation (12) as the inverse demand function
with overestimation in period 2. This argument also holds for all subsequent periods with overestimation. For details, see Section A in the separate technical appendix, available on the first author’s website:
http://www.geocities.jp/hiro4kitamura/index.html.
10
11

10

Second, when inexperienced consumers underestimate the product quality, m2 ≤ q̃, the inverse demand function becomes:ρ(1 − x2 ) + q̃P2 (x2 ) = 
ρ(1 − x2 ) + m2




0

0 ≤ x2 ≤ x1∗ < 1,
x1∗ < x2 ≤ 1,
x2 > 1.

(13)

Given the capacity x2∗ > x1∗ , the equilibrium price in the first half of period 2 can be either
p2f st = [ρ(2 − x1∗ − x2∗ ) + m2 + q̃]/2 or p2f st = ρ(1 − x2∗ ) + m2 with overestimation, while
p2f st = ρ(1 − x2∗ ) + m2 with underestimation. However, since our goal is to show the existenceof a competitive equilibrium with decreasing price under bilateral learning, we focus on thecase where the equilibrium price is nonincreasing, i.e., p1f st ≥ p1sec ≥ p2f st ≥ p2sec ≥ .... On
this equilibrium path, since all experienced consumers purchase the product, the equilibrium
price becomes p2f st = ρ(1 − x2∗ ) + m2 in the first half of period 2.13
When m2 > q̃, since inexperienced consumers observe at the beginning of period 3 thatthe equilibrium price declined during period 2, they realize the true quality of the product andlearning stops in the same way as in the case of m1 > q̃.14 On the other hand, when m2 ≤ q̃,
inexperienced consumers observe at the beginning of period 3 that the equilibrium price wasconstant during period 2. They realize that they underestimated the product quality in period
2 and believe that the true quality of the product is at least as high as the expected quality
in period 2. Then, they update their beliefs and their posterior distribution has the expectedvalue m3 = (qH + m2 )/2 = (7qH + qL )/8.
For all t = 3, 4, ..., the same learning process continues as long as the true quality of the∗
Of course, if the equilibrium price could be increasing, ptf st = [ρ(2 − xt−1
− xt∗ ) + mt + q̃]/2 could be realized
in the first half of period t = 2, 3, ... with overestimation. In Section B in the separate technical appendix, we∗
describe the learning process of consumers when the equilibrium price is ptf st = [ρ(2 − xt−1
− xt∗ ) + mt + q̃]/2. In
addition, if we allow for the possibility of belief updating in the second half of a period, when the equilibrium
price in the first half of period t is ptf st = ρ(1− xt∗ )+mt and it is larger than the price in period t −1, inexperienced
consumers can update their beliefs at the beginning of the second half of period t. In this case, the learning
process differs from what is explained here. This possibility is also discussed in Section B in the technicalappendix.
14
For t = 2, 3, ..., as in the case of t = 1, when overestimation occurs, experienced consumers repurchase theproduct in the second half and consumer learning necessarily stops if and only if xt∗ > (mt − q̃)/2ρ is satisfied.
Note that, since mt − q̃ has the largest value for mt = m1 and q̃ = qL , and since xt∗ is nondecreasing, it is easy to
see that xt∗ ≥ x1∗ and (m1 − qL )/2ρ > (mt − q̃)/2ρ for all t = 2, 3, .... Therefore, Assumption 4, x1∗ > (m1 − qL )/2ρ,
implies that xt∗ > (mt − q̃)/2ρ holds for any q̃ ∈ [qL , qH ], and guarantees that consumer learning stops once
overestimation occurs for all t = 2, 3, ....
13

11

product is higher than their expected quality: mt ≤ q̃. The expected quality in each period is:
mt = qH − 2−t (qH − qL ),(14)

for all t = 1, 2, ..., t˙, where t˙ is the first t ≥ 1 such that mt > q̃. Let ∆mt ≡ mt+1 − mt . Then, it
is easy to see that ∆mt > 0 for all t = 1, 2, ..., and that the value of ∆mt strictly decreases overtime and converges to zero. The inverse demand function at the beginning of each period for
all t = 2, 3, ..., t˙ − 1 becomes:ρ(1 − xt ) + q̃Pt (xt ) = 
ρ(1 − xt ) + mt




0

∗
0 ≤ xt ≤ xt−1
< 1,∗
xt−1 < xt ≤ 1,
xt > 1;(15)∗
0 ≤ xt ≤ xt−1
− (mt − q̃)/ρ < 1,∗∗
xt−1 − (mt − q̃)/ρ < xt < xt−1
+ (mt − q̃)/ρ < 1,∗
xt−1 + (mt − q̃)/ρ ≤ xt ≤ 1,
xt > 1;(16)

for t = t˙:15ρ(1 − xt ) + q̃
∗ −x )+m +q̃ρ(2−xt−1
t
t



2
Pt (xt ) = ρ(1 − xt ) + mt





0
and, finally, for t = t˙ + 1, t˙ + 2, ...:
ρ(1 − xt ) + q̃Pt (xt ) = 00 ≤ xt ≤ 1,
xt > 1.

(17)Inexperienced consumers observe the previous market outcome and update their beliefs atthe beginning of every period. More precisely, they observe whether or not new experienced
consumers in the first half of a previous period repurchase the product in the second half at
the same price.
The process of consumer learning in our model is different from that in Vettas (1998). In
Vettas (1998), since the demand function is horizontal, the market price is determined only bydemand of inexperienced consumers in each period. This leads to pt = mt for t = 2, 3, ..., t˙− 1
and pt increases as consumers update their beliefs upward. Therefore, by observing thatthe consumers who bought at pt−2 buy again at pt−1 , consumers can update their beliefs.
In contrast, in our model, the demand function is downward sloping and we focus on the15

See footnote 12.

12competitive equilibrium with nonincreasing price. Therefore, consumers cannot update theirbeliefs about product quality merely by observing that the consumers who bought at pt−2
buy again at pt−1 , since pt−1 is lower than or equal to pt−2 . Instead, by observing whether
consumers who bought at pt−1 in the first half of the period buy again at the same price pt−1
in the second half, where only incumbent firms are active, consumers revise their beliefs.

3.2 Benchmark: Diffusion with Unilateral Learning
In this subsection, we define diffusion with unilateral learning in which firms know the product quality and examine its properties for the benchmark case. Because our goal is to determine the properties of successful diffusion, we explore the case of a high-quality product andassume that q̃ = qH .
Suppose that firms know the product quality but consumers do not. Note that the only
difference from bilateral learning is that firms do not learn the product quality under unilaterallearning. Firms know that the actual quality of the product is qH , and that consumers always
underestimate the product quality for all t = 1, 2, .... Let Rut (xt−1 + yt ) be the discounted
sum of future operating profits in period t under unilateral learning. Since firms earn thesame operating profits ρ(1 − xt ) + mt for all periods, Rut (xt ) is composed of current profits,
2[ρ(1 − xt ) + mt ], and ongoing operating profits, βRut+1 (xt+1 ). Firms enter the market as longas Rut (xt−1 + yt ) > c and the level of new entries is determined by the zero profit condition.
The following proposition shows that there exists a unique competitive equilibrium underunilateral learning.
Proposition 1. Let xtu , yut , and put be capacity level, new entry level, and market price, respectively, at the competitive equilibrium with nonincreasing price under unilateral learning.
Suppose that x0u = 0. Then, there exists a unique competitive equilibrium under unilaterallearning that satisfies the following conditions.
1. For all t = 1, 2, ..., there exists unique yut > 0 that satisfies the following difference
equation:uc = 2[ρ(1 − [xt−1+ yut ]) + mt ] + βc, for all t = 1, 2, ....

13

(18)

2. {xtu , yut , put } satisfies the following equations:xtu = 1 −(1 − β)c − 2mt, for all t = 1, 2, ...;
2ρ

∆mt−1
, for all t = 2, 3, ...; andρ(1 − β)cput =, for all t = 1, 2, ....
2

yut =

(19)
(20)
(21)

3. The competitive equilibrium under unilateral learning always exists under Assumption
4. In addition, it exists even if Assumption 4 does not hold; more precisely, it exists if
and only if
2ρ + qH + qL > (1 − β)c > 2qH .

(22)

It is easy to see that the market capacity with unilateral learning, xtu , is increasing in the
expected value of product quality for inexperienced consumers, mt . There exists a uniquestrictly increasing sequence {xtu } such that xtu ∈ [0, xu ] for all t = 1, 2, ..., and x0u = 0 and
xtu → xu as t → ∞, where xu = 1 − [(1 − β)c − 2qH ]/2ρ. One of the important features of
unilateral learning is a constant equilibrium price. This implies that the benefit from eachperiod’s demand change because of consumer learning does not affect the equilibrium price.
Finally, inequality (22) is obtained from x1u > 0 and xu < 1. Compared with inequalities in (6)in Assumption 4, it is easy to see that 2ρ + qH + qL > 2ρ + [7(1 − β)qH + (25 − 9β)qL ]/8(2 − β).
Therefore, under unilateral learning, a competitive equilibrium exists even when Assumption
4 does not hold, i.e., 2ρ + qH + qL > (1 − β)c > 2ρ + [7(1 − β)qH + (25 − 9β)qL ]/8(2 − β).
In our model, the reason for gradual market expansion differs from that in Vettas (1998).
In the model in Vettas (1998), market expansion is gradual because of firm learning. In contrast, in our model, it is gradual because of the downward-sloping demand function. When the
demand function is horizontal, there is no cost of expansion and the market capacity expansion is instantaneous if there is no uncertainty. Therefore, Vettas (1998) assumes sequential
resolution of uncertainty about the number of consumers to derive gradual adjustment. In
contrast, when the demand function is downward sloping, a rise in the current market capacity lowers current prices. This makes faster expansion more costly and leads to gradualmarket expansion.
14

3.3 Diffusion with Bilateral Learning
In this subsection, we focus on the competitive equilibrium with decreasing price under bilateral learning and examine its existence. Let Rbt (xt−1 + yt ) be the discounted sum of expected
future operating profits in period t under bilateral learning. Then, from the definition, whenthe market is in the transition process yt > 0, c = Rbt (xt−1 + yt ) for all 0 < t ≤ t˙− 1. Rbt (xt−1 + yt )is composed of expected current period operating profits and the discounted continuation operating profits.
At the beginning of period t, firms know that they underestimate the product quality withprobability Pr{mt < q̃ | mt−1 < q̃}, and overestimate the product quality with probability
Pr{mt > q̃ | mt−1 < q̃}. When they underestimate the product quality in period t, the equilibrium price does not change during the period and they earn the operating profits ρ(1 − xt ) + mt
in both the first and the second halves of the period. In addition, since firms face the sameproblem in the following period, i.e., underestimation with Pr{mt+1 < q̃ | mt < q̃} and overestimation with Pr{mt+1 > q̃ | mt < q̃}, the discounted continuation operating profit becomes
βRbt+1 (xt+1 ).
On the other hand, when firms overestimate the product quality, the equilibrium pricedeclines during the period and learning stops. Under Assumption 4, they earn expected operating profit ρ(1 − xt ) + mt in the first half and ρ(1 − xt ) + (3mt + mt−1 )/4 in the second
half of the period.16 In the subsequent periods, since the expected quality for the overestimations at the beginning of period t is (mt + mt−1 )/2, firms earn the same operating profits ρ(1 − xt ) + (mt + mt−1 )/2. Thus, the discounted continuation operating profit becomes2β[ρ(1 − xt ) + (mt + mt−1 )/2]/(1 − β).
Therefore, under Assumption 4, the discounted sum of expected future operating profits
in period t, Rbt (xt ), is composed of expected current profits:
2[ρ(1 − xt ) + mt ] Pr{mt < q̃ | mt−1 < q̃}"#
3mt + mt−1
+ ρ(1 − xt ) + mt + ρ(1 − xt ) +Pr{mt > q̃ | mt−1 < q̃},
4

16

(23)

The equilibrium price in the second half with overestimation is derived in the proof of Proposition 2 inAppendix.

15and continuation operating profits:
βRbt+1 (xt+1 ) Pr{mt < q̃ | mt−1 < q̃}
mt + mt−1 
2β +
ρ(1 − xt+1 ) +Pr{mt > q̃ | mt−1 < q̃},
1−β
2

(24)

for all t = 1, 2, ..., t˙.17
The following lemma shows that the probabilities of underestimation and of overestimation are constant and are 1/2, respectively, for all periods.
Lemma 2. For all periods t = 1, 2, ..., t˙,
1
Pr{mt < q̃ | mt−1 < q̃} = Pr{mt > q̃ | mt−1 < q̃} = .
2(25)This result follows from the uniform distribution of q. From Lemma 2 and the zero
expected profit condition, Rbt (xt ) = c for all t = 1, 2, ..., t˙, we have the following proposition.
Proposition 2. Let xtb , ybt , and pbt be capacity level, new entry level, and market price, respectively, at the competitive equilibrium under bilateral learning. Suppose that x0b = 0 and
Assumption 4 hold. Then, there exists a unique competitive equilibrium with declining prices
under bilateral learning that satisfies the following conditions.
1. For all t = 1, 2, ..., t˙, there exists unique ybt > 0 that satisfies the following difference
equation:bc = {2[ρ(1 − [xt−1+ ybt ]) + mt ] + βc} Pr{mt < q̃ | mt−1 < q̃}
3mt + mt−1bb
+ {ρ(1 − [xt−1+ ybt ]) + mt + ρ(1 − [xt−1+ ybt ]) +
4
2β
m
+
m
t
t−1
b
+
[ρ(1 − [xt−1+ ybt ]) +]} Pr{mt > q̃ | mt−1 < q̃},
1−β
2

(26)

where m0 = qL .

For t = 2, 3, ..., if the equilibrium price in the first half of period t can be higher than that of period t − 1, the∗equilibrium price in the first half of period t with overestimation can be ptf st = [ρ(2 − xt−1
− xt∗ ) + mt + q̃]/2, notf st∗pt = ρ(1 − xt ) + mt . We assume that firms form the expectation that equilibrium price will be nonincreasing
and that they compute the present value of expected profits based on this expectation. As shown in Proposition
2, since equilibrium price is actually decreasing, this expectation is fulfilled in equilibrium.
1716

2. {xtb , ybt , pbt } satisfies the following equations:xtb = 1 −ybt =4c(1 − β)(2 − β) − (15 − 11β)mt − (1 + 3β)mt−1, for all t = 1, 2, ..., t˙; (27)8ρ(2 − β)(15 − 11β)mt − 14(1 − β)mt−1 − (1 + 3β)mt−2, for all t = 2, ..., t˙; and8ρ(2 − β)pbt =4(1 − β)(2 − β)c + (1 + 3β)∆mt−1, for all t = 1, 2, ..., t˙ − 1,8(2 − β)

(28)
(29)

where ∆mt ≡ mt+1 − mt .
It is easy to see that there exists a unique strictly increasing sequence of {xtb } such that xtb ∈[0, xb ] for all t = 1, 2, ..., and x0b = 0 and xtb → xb as t → ∞, where xb = 1−[(1−β)c−2qH ]/2ρ.
Furthermore, note that bilateral learning leads to the same entry level and equilibrium priceas unilateral learning in the steady state.18
Furthermore, it is easy to see that market prices decrease over time. This phenomenon
differs from that of Vettas (1998), where market prices increase as long as consumers underestimate the product quality. In Vettas (1998), since the demand function is assumed to be
horizontal, prices are demand driven. Therefore, as consumers revise their beliefs upward,
prices also increase. On the other hand, in our model, the demand function is downward
sloping. Given a downward-sloping demand function, market prices are determined not only
by demand but also by supply, because increases in outputs lead to lower prices. Therefore,in each period, although upward revision of consumers’ beliefs shifts the demand upward,
entry of new firms and increases in the output result in decreases in the price.

4 Comparison
In this section, we examine the properties of diffusion with bilateral learning by comparison
with the diffusion with unilateral learning. In particular, we focus on the size of market
capacity, the equilibrium price, and the shape of the diffusion. As in Subsection 3.2, we
assume that q̃ = qH .

18

From xb = 1 − [(1 − β)c − 2qH ]/2ρ < 1, we have the second inequality of (6) in Assumption 4.

17

4.1 Market Capacity and Equilibrium Price
We first compare both diffusions in terms of the size of market capacity and the equilibrium
price. From the above discussion, we know that both learning styles lead to the same market
capacity and equilibrium price in the long run. On the other hand, in the transition process,each diffusion has different market capacity and equilibrium price.
Proposition 3. Suppose that Assumption 4 holds. Compared with the diffusion with unilaterallearning, the diffusion with bilateral learning has the following properties:
1. Bilateral learning leads to a smaller market capacity and a higher equilibrium pricethan unilateral learning in every period: xtb < xtu and pbt > put for all t = 1, 2, ..., and
2. Bilateral learning accelerates the diffusion by reducing equilibrium prices: ybt > yut
and ∆pbt−1 < ∆put−1 = 0 for all t = 2, 3, ....
The intuitive logic for Proposition 3 is as follows. In the environment of bilateral learning,firms always face the risk of overestimation with probability 1/2. If overestimation occurs,
the equilibrium price falls during the period and firms earn low operating profits in the subsequent periods. Because of the risk of overestimating product quality, the firms’ discounted
sum of expected future profits in the bilateral learning case is smaller than that of future
profits in the unilateral learning case for all periods. Therefore, bilateral learning leads to asmaller market capacity than unilateral learning for all periods.
In addition, it is easy to see that the level of market capacity in the previous period does
not influence the change in inexperienced consumers’ demand in the current period, andthat the demand of inexperienced consumers is the same across both learning styles for all
periods. Because equilibrium prices in both learning styles are determined in the area of
demand of inexperienced consumers, bilateral learning generates a higher equilibrium pricethan unilateral learning for all periods.
The second property follows from a difference in the effect of a decrease in the degreeof uncertainty between unilateral learning and bilateral learning. In the environment of unilateral learning, a decrease in the degree of uncertainty in period t leads to a demand change
through consumer learning.
18

On the other hand, the decrease in the degree of uncertainty in period t in the environment of bilateral learning leads to two benefits associated with a demand change and with adecrease in the expected loss from overestimation. First, as for unilateral learning, a decreasein the degree of uncertainty in period t leads to a demand change through consumer learning.
Second, the decrease in the degree of uncertainty reduces the profit loss from overestimation
through firm learning. This benefit increases the incentives for potential entrants to enter themarket. Therefore, it speeds up diffusion and reduces the equilibrium price. Because of these
two benefits, the speed of market capacity expansion with bilateral learning is faster than thatwith unilateral learning after period 2.

4.2 S-shaped Diffusion
We now explore the existence of S-shaped diffusion. We first examine the case with unilaterallearning and then explore the case with bilateral learning. Let ∆yt = yt+1 − yt . S-shaped
diffusion implies that ∆yt is initially positive but is eventually negative.
The analysis of the shape of diffusion with unilateral learning starts from eventual concavity. We show that ∆yut < 0 for all t = 2, 3, .... Because we have ∆yut = (∆mt − ∆mt−1 )/ρ < 0,
the number of new entries decreases for all t = 2, 3, ....19 Therefore, the market diffusion pathwith unilateral learning eventually becomes concave. More importantly, it becomes S-shaped
if and only if ∆yu1 > 0. More precisely, we have the following proposition.
Proposition 4. Suppose that x0u = 0. Under Assumption 4, the diffusion with unilateral learning does not become S-shaped. However, under inequality (24), there exists an S-shaped
diffusion path if and only if:(1 − β)c > 2ρ +qH + 3qL.
2(30)

Inequality (30) implies that S-shaped diffusion with unilateral learning arises when the
profitability of entering the market is small enough in the first period. The profitability ofentry in the first period is yu1 = 1 − [(1 − β)c − (qH + qL )]/2ρ. On the other hand, the second
period profitability is yu2 = ∆m1 /ρ. By comparing yu1 and yu2 , it is easy to check that the first
period entry depends on the value of β and c but the second period entry does not. Therefore,
19

Note that the magnitude of the demand change is decreasing, i.e., ∆mt − ∆mt−1 < 0 for all t = 2, 3, ....

19

higher discount and a higher value of the entry cost make the first period entry level lower
but they do not make the second period entry level lower. This difference leads to the initialconvexity of the market diffusion path with unilateral learning.
Next, we examine the case with bilateral learning. Note that:∆ybt = −(qH − qL )(17 − 5β)< 0 for all t = 2, 3, ....
2t+4 ρ(2 − β)(31)

Therefore, the diffusion path under bilateral learning eventually becomes concave and it becomes S-shaped if and only if ∆yb1 > 0. More precisely, we have the following proposition.
Proposition 5. Suppose that Assumption 4 holds. Then, the market diffusion path with bilateral learning becomes S-shaped if and only if:(1 − β)c > 2ρ +(13 − 17β)qH + 3(17 − 5β)qL.
16(2 − β)(32)

Finally, we compare two diffusion paths. From Propositions 4 and 5, diffusion with unilateral learning does not become S-shaped under Assumption 4, but diffusion with bilaterallearning does so even under Assumption 4. Therefore, bilateral learning is more likely togenerate S-shaped diffusion than is unilateral learning. From Proposition 3, bilateral learning
leads to a lower first period entry than unilateral learning. However, the number of new entries becomes larger after period 2. Therefore, we have yb1 < yu1 and ybt > yut for all t = 2, 3, ....
This contributes to more S-shaped time patterns of diffusion with bilateral learning.
More importantly, S-shaped diffusion with bilateral learning is observed with declining
equilibrium prices, whereas unilateral learning leads to constant equilibrium prices. In theprevious studies, S-shaped diffusion with declining prices is viewed as a result of a reductionin production cost. This result implies that S-shaped diffusion with declining prices can be
explained by the dual uncertainty even in the absence of technological progress.

5 Conclusion
This paper presents a dynamic model of market diffusion in which consumers are uncertain
about the product quality and firms do not know how consumers evaluate it. We showed that
consumer-based bilateral learning leads to several interesting results by comparing it with20

unilateral learning. The main results are that bilateral learning (i) leads to a lower entry level;(ii) leads to a declining equilibrium price; and (iii) is more likely to generate S-shaped time
patterns.
Although a number of related studies explain S-shaped diffusion, they do not explain adeclining equilibrium price without technological improvement. The advantage of this paper
is that it shows that S-shaped diffusion with a declining price is possible without technologicalprogress. Rather, the diffusion arises because of informational externalities that increase themarket demand through consumer learning and increase the expected profitability of entryof firms through firm learning. Therefore, this paper establishes the theoretical link between
informational externalities and market diffusion.
There are several issues requiring future work. First, there is the empirical importanceof consumer-based bilateral learning. Second, there is concern about market diffusion withother means of consumer learning. For example, communication between consumers may
exist and be a major source of learning.
Finally, there is concern over the generality of our results. In particular, there is concern
about the robustness of our results with respect to the demand specification. Our model isrestricted to a linear demand structure, which implies that the type of consumer is uniformly
distributed. We predict that if we assume that the type of consumer follows a normal distribution, diffusion becomes more likely to be S-shaped because the number of high-type
consumers is small but that of intermediate consumer is large. We trust that this study willassist researchers in addressing these issues in the future.

Appendix
Proof of Lemma 1
Let f (q) be the probability density function of q and F(q) be the distribution function. Wheninexperienced consumers believe that q̃ ∈ [qlt , qht ], then the posterior density function becomes:f (q | qlt < q < qht ) =f (q)F(qht ) − F(qlt )

21

=

1
qht − qlt

for all t = 1, 2, ....

(33)

Therefore, the posterior density is uniform for all t = 1, 2, ..., and we have:
mt = E[q | qlt < q < qht ] =

Z qht
qlt

qht + qltqdq =for all t = 1, 2, ....
2qht − qlt

(34)


Proof of Proposition 1
In this proof, we derive condition (22) for the existence of the equilibrium. For the proof ofthe existence and uniqueness of the equilibrium, see Subsection C.1 in the separate technicalappendix. From equation (19), for the equilibrium to exist under unilateral learning, thefollowing two conditions must be satisfied:(1 − β)c − 2m1> 0,2ρ
(1 − β)c − 2qH< 1,
xu = 1 −2ρx1u = 1 −(35)

where xu = limt→∞ xtu . From these inequalities, we obtain condition (22).

Proof of Lemma 2
Note that:Pr{mt < q̃ | mt−1 < q̃} =1 − F(mt )q H − mt=.
1 − F(mt−1 ) qH − mt−1

(36)

Because mt = qH − 2−t (qH − qL ), then we have:Pr{mt < q̃ | mt−1 < q̃} =2−t1= for all t = 1, 2, ...,
1−t
2
2

(37)

and
Pr{mt > q̃ | mt−1 < q̃} = 1 − Pr{mt < q̃ | mt−1 < q̃} =1
for all t = 1, 2, ....
2

(38)


Proof of Proposition 2
In this proof, we derive the second half price in the overestimation case under bilateral learning. For the proof of the existence and uniqueness of the equilibrium, see Subsection C.2 in22

the separate technical appendix. Note that the firm’s expected quality with overestimation is(mt +mt−1 )/2 for each period. In this case, some experienced consumers do not repurchase theproduct in the second half. Let xtb be the number of sales and consumers in the first half, et be
the number of experienced consumers who repurchase the product in the second half, and ut
the number of inexperienced consumers who purchase the product in the second half. Then,the firm’s expected equilibrium price in the second half pb(sec)must satisfy the following threetequations:mt + mt−1,2= ρ(1 − xtb − utsec ) + mt ,pb(sec)= ρ(1 − etsec ) +t(39)

pb(sec)t(40)

xtb = etsec + utsec .

(41)

Note that Assumption 4 guarantees etsec > 0.20 Therefore, by solving the above equations, wehave pb(sec)= ρ(1 − xtb ) + (3mt + mt−1 )/4 for all t = 1, 2, ....
t

Proof of Proposition 3
We first prove the first property in Proposition 3. Note that:xtb − xtu = −∆mt−1 (1 + 3β)< 0.
8ρ(2 − β)(42)Therefore, xtu > xtb for all t = 1, 2, .... Because both learning styles lead to the same demandchange for all t = 1, 2, ..., we have put < pbt for all t = 1, 2, ....
We next prove the second property in Proposition 3. Note that:ybt − yut =(∆mt−2 − ∆mt−1 )(1 + 3β)>08ρ(2 − β)(43)∆pbt =(∆mt − ∆mt−1 )(1 + 3β)< 0.
8(2 − β)

(44)

and

From equation (43), it is easy to see that ybt > yut for all t = 2, 3, .... In addition, because∆put = 0, it is easy to see that ∆put > ∆pbt from equation (44).

From the three equations (39)–(41), we have etsec = xtb − [mt − (mt + mt−1 )/2]/2ρ. Because of (mt + mt−1 )/2 ∈[qL , qH ], Assumption 4 guarantees etsec = xtb − [mt − (mt + mt−1 )/2]/2ρ > 0 for all t = 1, 2, .... See also Subsection
20

3.1 and footnote 14.

23

Proof of Proposition 4
Because the diffusion path eventually becomes concave, S-shaped diffusion arises if and only
if ∆yu1 > 0. By equations (19) and (20), ∆yu1 can be rewritten as:∆yu1 = yu2 − yu1 = yu2 − x1u"#
qH − qL(1 − β)c − (qH + qL )
=
− 1−
.
4ρ
2ρ

(45)

Therefore, we have ∆yu1 > 0 if and only if inequality (30) holds. Since we have:qH + 3qL 7(1 − β)qH + (25 − 9β)qL (3β + 1)(qH − qL )
−
=
> 0,
2
8(2 − β)8(2 − β)(46)

inequality (30) and Assumption 4 do not hold simultaneously. On the other hand, because wehave:
qH + qL >qH + 3qL,
2(47)

there exist parameter values which satisfy inequalities (22) and (30) simultaneously. Therefore, unilateral learning does not lead to S-shaped diffusion under Assumption 4 but it canunder inequality (22).

Proof of Proposition 5
Because the diffusion path eventually becomes concave, the necessary and sufficient condition for S-shaped diffusion is ∆yb1 > 0. By equations (27) and (28), ∆yb1 can be rewritten
as:∆yb1 = yb2 − yb1 = yb2 − x1b"#(17 − 5β)(qH − qL )8c(1 − β)(2 − β) − (15 − 11β)qH − (17 − 5β)qL− 1−.
=
32ρ(2 − β)16ρ(2 − β)(48)

Therefore, we have ∆yb1 > 0 if and only if inequality (32) holds. Because we have:7(1 − β)qH + (25 − 9β)qL (13 − 17β)qH + 3(17 − 5β)qL (3β + 1)(qH − qL )−=> 0, (49)
8(2 − β)16(2 − β)16(2 − β)there exist parameter values which satisfy inequality (32) and Assumption 4 simultaneously.


24

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25