Equilibrium pricing and lead time decisions in a competitive industry

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Equilibrium pricing and lead time decisions in a competitive industry

I-Hsuan Hong

a,n,1

, Hsi-Mei Hsu

b,2

, Yi-Mu Wu

b

, Chun-Shao Yeh

b a

Institute of Industrial Engineering, National Taiwan University, Taipei 106, Taiwan

b

Department of Industrial Engineering & Management, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history:

Received 28 October 2009 Accepted 8 May 2012 Available online 7 June 2012 Keywords:

Price competition Lead time competition Nash equilibrium

a b s t r a c t

Pricing and lead time are two crucial decisions to a success in today’s competitive markets. This paper examines the equilibrium pricing and lead time decisions in a duopoly industry consisting of two large and several smaller firms with competition. We solve the sufficient Karush–Kuhn–Tucker (KKT) optimality conditions for the Nash equilibrium solution. We characterize the existence and uniqueness of the Nash equilibrium solution of pricing and lead time decisions for both homogenous and heterogeneous firms. Our case study provides important managerial insights about firms’ behavior under price and lead time competition in a semiconductor manufacturing industry.

1. Introduction

Observation of time-based marketplaces indicates that firms that provide more products and services in shorter lead times can charge a higher price and thus capture greater market share (Stalk, 1988;Stalk and Hout, 1990). In this paper we examine the equilibrium decision about the price and lead time in a duopoly consisting of two large firms and several smaller firms in a competitive market, where only the two large firms have domi-nant control over the market.

Capital-intensive industries, such as semiconductor manufac-turing and its related industries, are typical examples of a duopoly in the marketplace. For instance, Intel and AMD are the leading global producers of microprocessor chips. Taiwan Semiconductor Manufacturing Company (TSMC) and United Microelectronics Corporation (UMC) dominate the semiconductor foundry indus-try, accounting for more than 80% market share (The Register, 2003). Samsung Electronics and LG Philips in Korea and AUO and CMO in Taiwan dominate the thin-film transistor liquid crystal display (TFT-LCD) industry (Chang, 2005). In such oligopolistic markets, each individual firm has its own profit function and often is unwilling to reveal information. Decisions made by competing firms can be influenced by other firms’ behavior, especially in consumer product markets characterized by shor-tened product life-cycles.

A literature survey reveals that some researchers have exam-ined individual ﬁrms’ decisions about the equilibrium price or

lead time (Bertrand and van Ooijen, 2000;Das and Abdel-Malek, 2003;Kunnumkal and Topagloglu, 2008;Cai et al., 2011;Glock, 2012), while others (Atamer et al., 2011; Qian, 2011; Maihami and Kamalabadi, 2012) have focused on optimization within a single ﬁrm and neglected the competition among ﬁrms. The underlying concept is that pricing and lead time are trade-offs—a short lead time typically results in a high price.

Palaka et al. (1998) examine the lead time setting, capacity utilization, and pricing decisions facing a firm serving customers sensitive to quoted lead times.Hatoum and Chang (1997)present a model to determine the optimal demand level using a mechan-ism of quoted lead time and price.Ray and Jewkes (2004)present an analytical approach for a firm to maximize its profit by optimal selection of a lead time. ElHafsi (2000) develops a model that includes enough detail so that realistic price and day-to-day lead time can be achieved and quoted to the customer.So and Song (1998)study the impact of using delivery time guarantees as a competitive strategy in service industries where demands are sensitive to both price and delivery time. Ray (2005)develops analytical models that can assist a firm in deciding on its optimal pricing, stocking and investment values in varying operating environments.Pekgun et al. (2008)study a firm serving custo-mers sensitive to quoted price and lead time and analyze the inefficiencies created by the decentralization of the decisions, where pricing decisions are made by the marketing department and lead time decisions by the production department.

A growing body of literature on competitive models relies upon economic theory to analyze the behavior of independent ﬁrms in a market where no ﬁrm is better off by a unilateral change in its decision (seeGibbons, 1992). Several papers exam-ine competitive supply, particularly in a duopoly industry, of goods or services to time-sensitive customers. For example,Kalai et al. (1992)study competition in service rates without consid-eration of pricing competition.Chen and Wan (2003)consider the Contents lists available atSciVerse ScienceDirect

journal homepage:www.elsevier.com/locate/ijpe

Int. J. Production Economics

n

Corresponding autor. Tel.: þ 886 2 3366 9507; fax: þ 886 2 2362 5856. E-mail addresses: [email protected] (I-H. Hong), [email protected] (H.-M. Hsu).

1

The corresponding author holds a joint appointment with the Department of Mechanical Emgineering, National Taiwan University.

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duopoly price competition of make-to-order (MTO) firms. The results in Chen and Wan (2003) show that whenever market equilibrium exists and is unique, the firm with greater capacity, a higher service value, or a lower waiting cost can enjoy a price premium and a large market share.Li and Lee (1994)present a model of market competition in which a customer values cost, quality, and speed of delivery. Lederer and Li (1997)study the competition between firms that produce goods or services for customers sensitive to time delay where firms compete by setting prices and production rates for each type of customer and by choosing scheduling policies.So (2000)studies a similar issue of delivery time guarantees and pricing for service delivery. In reality, industries seldom consist of only two firms (a duopoly). Most research appears to ignore the effect of other smaller firms on pricing and lead time decisions in a duopoly industry. Thus our paper fills an important research void.

Our objective is to predict the equilibrium decision of the prices and lead times of the goods provided by all of the ﬁrms where none will be beneﬁted by unilaterally deviating from its current decision. The remainder of this paper is organized as follows. The model and its equilibrium outcomes are described in Section 2. InSection 3we conduct a case study in a semiconduc-tor foundry industry and produce several managerial insights about price and lead time decisions. Section 4 presents our concluding remarks and suggestions for further research.

2. The model

2.1. The duopoly market model

We model a duopoly market consisting of two large firms and several smaller firms in a duopoly market. Regardless of firm size, all customers are served on a first-come, first-served basis. The large firms compete non-cooperatively to provide a type of goods in an MTO fashion. Both are independent entities and are modeled as queues with exponential service times with a com-mon source of potential customer arrivals. Often, the decision variables for each firm will be influenced by the other’s behavior. We also consider the effects of the smaller firms’ decisions.

To begin, we denote the set of the two large firms by N ¼{X,Y} and the group of smaller firms by M. We assume that the customer arrival rate of firm iAN,

l

i, depends on ﬁrm i’s decision

of the price, pi, and lead time, ti. In a competitive market,

l

iis also

influenced by the decisions of its major competitor, firm jAN, jai, and of the smaller firms. In other words,

l

iis also a function of pj,

tj, pM, and tMin addition to piand ti. In our hypothetical model,

customers prefer lower prices and shorter lead times compared to the decisions offered by the other ﬁrms as shown in (1), where

l

i

is proportional to the differences of prices and lead times between the other ﬁrms and ﬁrm iAN.

l

ipðpjpiÞ

l

ipðpMpiÞ

l

ipðt_jt_iÞ

l

ipðtMtiÞ ð1Þ

Next, we elaborate on the precise relationships between prices and lead times in the market as shown in (2). The customer arrival rate

l

iof ﬁrm i is a function of the difference between ﬁrm

i’s decisions and the decisions of its competitors. Let

a

Mand

a

C

denote the preference factors accounting for the effect of the decision differences by the smaller ﬁrms, M, and the competitor, ﬁrm j. Similarly,

b

t and

b

p represent the preference factors for

explaining the effect of lead times and prices on the arrival rate. We assume that the competition effect is a convex combination between i’s competitor, ﬁrm j, and the smaller ﬁrms (

a

Mþ

a

C¼1,

a

M,

a

CZ0), and the decision effect is a convex combination of the price and lead time (

b

tþ

b

p¼1,

b

t,

b

pZ0). We now assume the arrival rate is

l

i¼

l

0mt

b

t½

a

MðtitMÞ þ

a

CðtitjÞ

mp

b

p½

a

MðpipMÞ þ

a

CðpipjÞ ð2Þ

where

l

0denotes the arrival rate when both prices and lead times

of all ﬁrms in the market are identical, and mtand mprepresent

the lead time sensitivity and price sensitivity of the arrival rate, respectively (

l

0, mt, mpZ0). A linear form of the arrival rate helps us obtain qualitative insights without much analytical complex-ity. It also has the desirable properties for approaching the equilibrium decisions of prices and lead times of the ﬁrms in the market. For illustrative purposes, we write

l

i as

l

i(pi,ti9

pj,tj,pM,tM); piand tiare decisions of ﬁrm i, and pj, tj, pM, tM are

decisions of other firms. We note the arrival rate shown in (2) does not limit itself to the situation where there are only two large firms and a group of smaller firms in a competitive industry. The form of (2) allows a decision maker to take into account the other major competitor and the group of all firms. A similar model appears in (Hatoum and Chang, 1997;Ray and Jewkes, 2004).

The objective of each firm is to maximize its own expected profit. Since the capacity is fixed, maximizing the expected profit is equivalent to maximizing the expected revenue. We assume an M/M/1 queuing system with mean service rate

m

ifor ﬁrm iAN. To

prevent quoting unrealistic lead times, we assume that all of the firms maintain a certain minimum service level, s, which can be set by each firm in response to competitiveness or to the industry in general. The probability that the total sojourn time in firm i is less than the quoted lead time is 1eðmiliÞti_{for an M/M/1 system}

(Kleinrock, 1975). Therefore, the requirement that the probability of meeting the quoted lead time for ﬁrm i must be at least s (e.g., 95%) can be represented in the following constraint as

1eðmiliÞtiZs or equivalently, ð

m

i

l

iÞtirlnð1sÞ

Since firm iAN is assumed to maximize its own profit per unit time, the maximization model for firm i can be written as Max pi,ti

p

iðpi,ti9pj,tj,pM,tMÞ ¼pi

l

iðpi,ti9pj,tj,pM,tMÞ ð3Þ s:t:ð

m

i

l

iÞtirlnð1sÞ ð4Þ

l

iðpi,ti9pj,tj,pM,tMÞ ¼

l

0mt

b

t½

a

MðtitMÞ þ

a

CðtitjÞ mp

b

p½

a

MðpipMÞ þ

a

CðpipjÞ ð5Þ pi,ti40: ð6Þ

Firm i maximizes its proﬁt function

p

iby quoting the price, pi

and lead time, ti. Clearly, ﬁrm i’s proﬁt function,

p

i, is a function of

piand ti, but it also depends on pj, tj, pM, and tM. Constraint (4)

ensures that ﬁrm i maintains a minimum service level, and (5) and (6) are the arrival rate deﬁnition and bounds for prices and lead times.

2.2. Establishing equilibrium price and lead time

In this section we elaborate upon our algorithm. In equili-brium, the ﬁrms have no incentive to deviate from the current quotation of their pricing and lead time decisions, given the other ﬁrms’ decisions.

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2.2.1. Preliminary

Consider a real single-valued scalar function f(x) deﬁned on some nonempty closed set X in the n-dimensional Euclidean space. Function f(x) is assumed to be twice continuously differ-entiable on X. Let

r

f(x) and

r

2_{f(x), respectively, denote the}

gradient and the Hessian matrix of f evaluated at x. A sufﬁcient condition for function f to be pseudo-convex is given inDeﬁnition 1. Let M(X,

b

) be the n n matrix and let T denote the transpose operator.

MðX,

b

Þ ¼

r

2f ðxÞ þ

b

r

f ðxÞ

r

f ðxÞT_, _ð7Þ

where

b

is a nonnegative real number.

Deﬁnition 1. (seeMereau and Paquet, 1974) A sufﬁcient condition for f(x) to be pseudo-convex on the convex set X is that there exists a real number

b

, 0r

b

o þN, such that M(X,

b

) is positive semi-deﬁnite.

2.2.2. The solution algorithm

Our goal is to solve for the equilibrium solution of the competing firms in our model. We demonstrate the Karush– Kuhn–Tucker (KKT) approach to find the Nash equilibrium solu-tion, defining the equilibrium as a set of decisions that satisfy each firm’s first-order conditions (KKT) for profit maximization. The solution satisfying those conditions possesses the property that no firm wants to alter its decision unilaterally and is known as the Nash equilibrium solution (Hobbs, 2001). However, we note that KKT conditions are necessary optimality conditions for the local optimum in general, not sufficient conditions for the optimum. Therefore, to satisfy the properties of the Nash equili-brium, we need to solve the globally sufficient KKT conditions simultaneously for the equilibrium instead of solving the general locally necessary KKT conditions; otherwise, we need to examine all possible KKT points for the equilibrium solution. This leads to Lemmas 1 and 2for the solution algorithm derivation.

Lemma 1. Proﬁt function (3) of ﬁrm iAN is pseudo-concave function.

Proof. See Appendix.

Lemma 2. The feasible region of constraints (4)–(6) is a convex set. Proof. See Appendix.

From Lemmas 1 and 2, we can conclude that the KKT optimality conditions to the problem (3)–(6) of firm iAN are both necessary and sufficient (Bazaraa et al., 1993). The KKT optimality conditions of firm iAN are stated as

O

i¼pi

l

ia1ðð

m

i

l

iÞtilnð1sÞÞ þ a2piþa3ti ð8Þ @

O

i @pi ¼0 ð9Þ @

O

i @ti ¼0 ð10Þ a1ðð

m

i

l

iÞtilnð1sÞÞ ¼ 0 ð11Þ a2pi¼0 ð12Þ a3ti¼0 ð13Þ ð

m

i

l

iÞtirlnð1sÞ ð14Þ pi40 ð15Þ ti40 ð16Þ a1, a2, a3Z0 ð17Þ

where a1, a2, and a3are dual variables to constraints (4) and (6).

Constraint (8) is the Lagrangian function definition for the purpose of notational simplicity. Constraints (9), (10) and (17) are corresponded to dual feasibility equalities, (11)–(13) are complementary slackness conditions, and (14)–(16) are primal feasibility equalities. Similarly, we also derive the KKT conditions of the other competing firm. The equilibrium solution of prices and lead times can be obtained by simultaneously solving the combined KKT conditions. Since the KKT optimality conditions of the model presented in this paper are sufficient, any solution simultaneously satisfying the combined KKT optimality condi-tions is optimal to each firm. In other words, this solution follows the definition of the Nash equilibrium where no firm wishes to alter its decision unilaterally.

Observation 1 The dual variables a2 and a3are equal to zero in

the KKT optimality conditions.

Proof. Since we only focus on a nontrivial solution, we assume that the equilibrium solution of prices and lead times is a positive value. This allows us to simplify the KKT conditions by letting a2

and a3be zero in (12) and (13). &

Observation 2 The dual variable a1 is non-zero in the KKT

optimality conditions.

Proof. From (10), we have @Oi

@ti ¼ mt

b

tpia1ð

m

iþ

l

imt

b

ttiÞ þ

a3¼0. Obviously, mt,

b

t, and piare non-zero, and a3¼0 due to

Observation 1. Thus, a1(

m

iþ

l

imt

b

tti) is non-zero as well and it

completes the proof. & 2.3. Demonstration

To illustrate the use of the KKT approach, we construct a simple and symmetric example of two leading ﬁrms, X and Y, and a group of smaller ﬁrms, M. We assume that the price and lead time of M are given: that is, pM¼10 and tM¼5. The customer

arrival rate,

l

0, is 3 when X and Y have the same prices and lead

times. The mean service rates of X and Y are

m

X¼

m

Y¼5. The

minimum service levels, s, for both ﬁrms are set at 95%. Other required parameters are

a

M¼.2,

a

C¼.8,

b

t¼.5,

b

p¼.5, and mt¼1,

mp¼.5. The KKT optimality conditions of ﬁrm X can be stated as

ð4:5pXþ:2pY:5tXþ:4tYÞ þ:25a1tXþa2¼0 :5pXa1ð1:25pXþ:2pYtXþ:4tYÞ þa3¼0 a1½ð1:25pXþ:2pY:5tXþ:4tYÞtXþ3 ¼ 0 a2pX¼0 a3tX¼0 ð1:25pXþ:2pY:5tXþ:4tYÞtXr3 pX,tX40 a1, a2, a3Z0:

The KKT optimality conditions of ﬁrm Y can be stated as ð4:5pYþ:2pX:5tYþ:4tXÞ þ:25a4tYþa5¼0 :5pYa4ð1:25pYþ:2pXtYþ:4tXÞ þa6¼0 a4½ð1:25pYþ:2pX:5tYþ:4tXÞtYþ3 ¼ 0 a5pY¼0 a6tY¼0 ð1:25pYþ:2pX:5tYþ:4tXÞtYr3

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pY,tY40

a4,a5,a6Z0:

Observation 1 allows us to simplify the combined KKT condi-tions by allowing a2, a3, a5, and a6 to be zero. We use the

commercial package, Mathematica 7 (Wolfram, 2009), to solve the combined KKT conditions of X and Y for the equilibrium price and lead time; the equilibrium solution is

pX,pY,tX,tY

¼ð16:67,16:67,1:51,1:51Þ,

and a1¼a4¼3.09, ai¼0, i ¼2, 3, 5, and 6. The corresponding proﬁts

of X and Y are 50.24, respectively. Intuitively, the solution and proﬁts are identical for the two ﬁrms in this symmetrical example.

2.4. Existence and uniqueness of equilibrium

Analyzing the required conditions for the existence and uniqueness of the equilibrium price and lead time of X and Y paves the way to solving the nonlinear system of KKT optimality conditions for the equilibrium solution. We first show a trivial observation for derivation purposes followed by two cases: homogeneous firms and heterogeneous firms.

Observation 3 Firm i determines the equilibrium price and lead time such that the equality of constraint (4) holds.

Proof. This result follows from Observation 2 and (11). & 2.4.1. Homogeneous ﬁrms

In the case of homogeneous ﬁrms, two competing ﬁrms have identical capacity, production technology, cost structure, quality, etc; in other words,

m

¼

m

X¼

m

Y. In addition, it is reasonable

assuming that these two ﬁrms return the identical equilibrium decision of prices and lead times. For notational simplicity, let p and t denote the equilibrium price and lead time of the two homogeneous ﬁrms without the subscript index. From Observa-tions 1 and 2, the variables of the combined KKT condiObserva-tions are reduced to only (t, p, a1) and the combined KKT conditions of the

ﬁrms can be further reduced using (43), simpliﬁed as u0

0u1

a

Mtu2ð1þ

a

MÞp þa1u2t ¼ 0 ð18Þ

u1pa1ðu00u1ð1 þ

a

MÞtu2

a

Mp

m

Þ ¼0 ð19Þ

ðu0

0u1

a

Mtu2

a

Mp

m

Þt þu5¼0 ð20Þ

p,t,a140 ð21Þ

Lemma 3. A unique positive solution exists to the equation ax3þbx þc¼0 when aco0 and b is a real number.

Proof. See Appendix. &

Proposition 1. If the two competing ﬁrms are homogenous, there exists a unique equilibrium solution of the price and lead time. Proof. There are only three unknowns in (18)–(21) and we only focus on nontrivial positive solutions of prices and lead times. Rearranging (20), we have p ¼ 1 u2aM ðu 0 0u1

a

Mt

m

Þ þ u5 t h i ð22Þ From (18) and (19), it is easy to have (23) without a1

u1u2tp ¼ ½u00u1

a

Mtu2ð1þ

a

MÞp

½u0

0u1ð1 þ

a

MÞtu2

a

Mp

m

ð23Þ

Substituting (22) into (23), we have u1

m

t þu5 u0 0 aMð1 þ 1 aMÞ

m

h i 1 tþ 1 þ 1 aM _u2 5 t2¼0: ð24Þ

Multiplying t2_{on both sides, (24) can be represented as}

At3_{Bt þ C ¼ 0} _ð25Þ where A ¼ u1

m

B ¼ u5 u0 0 aM 1 þ 1 aM

m

h i C ¼ 1 þ1 aM u52 A,C 4 0:

Since ( A)Co0 and fromLemma 3, a unique positive t exists such that (25) holds. In addition, (22) shows the one-to-one correspondence between t and p. As a result, a unique equilibrium price exists as well. &

2.4.2. Heterogeneous ﬁrms

In the case of heterogeneous ﬁrms, two competing ﬁrms do not have identical capacity, production technology, cost structure, quality, etc; in other words,

m

Xa

m

Yin this paper. With

Observa-tions 1, 2, and 3, the combined KKT optimality condiObserva-tions for X and Y can be rewritten as

@

O

X

@pX

¼u0

0u1tX2u2pXþu1

a

CtYþu2

a

CpYþa1u2tX¼0 ð26Þ

@

O

X

@tX

¼ u1pXa1ðu00

m

X2u1tXu2pXþu1

a

CtYþu2

a

CpYÞ ¼0

ð27Þ ðu0

0

m

Xu1tXu2pXþu1

a

CtYþu2

a

CpYÞtXþu5¼0 ð28Þ

@

O

Y

@pY

¼u0

0u1tY2u2pYþu1

a

CtXþu2

a

CpXþa4u2tY¼0 ð29Þ

@

O

Y

@tY

¼ u1pYa4ðu00

m

Y2u1tYu2pYþu1

a

CtXþu2

a

CpXÞ ¼0

ð30Þ ðu0

0

m

Yu1tYu2pYþu1

a

CtXþu2

a

CpXÞtYþu5¼0 ð31Þ

a1, a4, tX, tY, pX, pY40 ð32Þ

Rearranging (28) and (31), the prices can be represented by the functions of lead times

pX¼u2ð11a2CÞ u0 0ð1 þ

a

CÞ

m

X

a

C

m

Y u1tX u2 þ u5 u2ð1a2CÞtXþ aCu5 u2ð1a2CÞtY ð33Þ pY¼_u 1 2ð1a2CÞ u0 0ð1 þ

a

CÞ

m

Y

a

C

m

X u1tY u2 þ u5 u2ð1a2CÞtYþ aCu5 u2ð1a2CÞtX ð34Þ

By algebraic manipulations, X’s KKT optimality conditions, (26)–(28), can be represented as u2 5ð2

a

2CÞ t2 Xð1

a

2CÞ u1

m

XtXþ_t u5 XtYð1a2CÞ u5

a

C þ u5tY tXtYð1a2CÞ u0 0ð1 þ

a

CÞ

m

Xð2

a

2CÞ

a

C

m

Y ¼0 ð35Þ

Similarly, Y’s KKT optimality conditions, (29)–(31), can be represented as u2 5ð2

a

2CÞ t2 Yð1

a

2CÞ u1

m

YtYþ_t u5 XtYð1a2CÞ u5

a

C þ u5tX tXtYð1a2CÞ u0 0ð1 þ

a

CÞ

m

Yð2

a

2 CÞ

a

C

m

X ¼0: ð36Þ

For notational simplicity, (35) and (36) can be rewritten as EF1t3XþG

tX

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EF2t3YþG tY tXþH2tY¼0: ð38Þ where E ¼u25ð2a2CÞ 1a2 C F1¼u1

m

X F2¼u1

m

Y G ¼u25aC 1a2 C H1¼₁u5_a2 C u0 0ð1 þ

a

CÞ

m

Xð2

a

2CÞ

a

C

m

Y H2¼₁u5_a2 C u0 0ð1 þ

a

CÞ

m

Yð2

a

2CÞ

a

C

m

X :

Up to this point, the combined KKT optimality conditions of the two heterogeneous competing ﬁrms are represented as two equations with two unknowns, tXand tY, as shown in (37) and

(38). We can now analyze the equations plotted in a tXtY-plane

composed of horizontal axis tXand vertical axis tY. Rearranging

(37), we have tY¼

GtX

F1t3XH1tXE

ð39Þ From Lemma 3, there is a unique positive solution, tA

X, to

F1t3XH1tXE ¼ 0 since F1( E)o0. As tX approaches tAX, the

denominator of (39) approaches zero. Therefore, we have lim

tX-tAX

tY¼ lim tX-tAX

ðGtX=F1t3XH1tXEÞ ¼ 1. As tX approaches inﬁnity,

tYapproaches zero by L’Hospital Rule (Salas et al., 2003, pp. 615–

616). Thus line tX¼tAXand axis tXare asymptotic to (37) on tXtY

-plane. Similarly, line tY¼tAYand axis tYare asymptotic to (38) on

tXtY-plane where tAYis the solution to F2t3YH2tYE ¼ 0.

In addition, the ﬁrst derivative of (39) with respect to tXis

Gð2F1t3XþEÞ= F 1t3XH1tXE 2

, where G, F1, E, and tXare positive.

As a result, (qtY/qtX)o0 and it implies that (39) decreases in tX.

We next examine concavity or convexity of (39) on tXtY-plane.

Taking the second derivative of (39) with respect to tX, we have

@2tY @t2 X ¼2G½Eð6F1t 2 XH1Þ þF1tX3ð3F1t2XþH1Þ ðF1t3XH1tXEÞ3 ð40Þ Observation 4 F1t3XH1tXE 40 and F2t3YH2tYE4 0 when

tX40 and tY40.

Proof. See Appendix.

From Observation 4, the denominator of (40) is positive. With G 40, the sign of@2_t

Y

@t2 X

can be determined by inspecting the sign of E 6F1t2XH1

þF1t3X 3F1t2XþH1

. For notational simplicity, we let TðtXÞ ¼E 6F 1t2XH1þF1t3X3F1t2XþH1. We discuss the sign of

T(tX) in the following two disjunctive cases: H1Z0 and H₁o0. Proposition 2. As H1Z0, T(t_X) is positive for all t_X40. Proof. See Appendix.

Proposition 3. H1o0, T(tX) is positive for all tX40 when

1=6H1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=6F1Þ

p

þ4E 40, but T(tX) is negative for some tX40

when 1=6H1

p

þ4Eo0. Proof. See Appendix.

FromPropositions 2 and 3, @2_t

Y=@t2X is positive for all tX40

when H1Z0, or when H₁o0 and 1=6H₁pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH₁=6F₁Þþ4E 4 0. Thus, (37) on tXtY-plane is convex when H1Z0, or when H₁o0 and 1=6H1

p

þ4E 4 0. Similarly, (38) on tXtY

-plane is convex when H2Z0, or when H₂o0 and

1=6H2

p

þ4E4 0. We let SS denote E,F1,F2, G,H1,H2 !E 40,F140,F240,G 40, H140 [ H1o0, 16H1 ffiffiffiffiffiffiffi H1 6F1 q þ4E 40 n o , H240 [ H2o0, 16H2 ffiffiffiffiffiffiffi H2 6F2 q þ4E 40 n o 9 > > > = > > > ; 8 > > > < > > > : ð41Þ

Proposition 4. There exists a unique equilibrium solution of prices and lead time if the parameters, E,F1,F2,G,H1, and H2, of

two competing ﬁrms satisfy (41).

Proof. FromPropositions 2 and 3, (37) and (38) on tXtY-plane are

convex when (E,F1,F2,G,H1,H2)ASS. It is trivial to argue that the

slopes of (37) and (38) are different by inspecting the ﬁrst derivative. With the asymptotic lines of (37) and (38), the two equations can only cross once and the result follows. &

3. Sensitivity analysis: A case study

Our case study is designed to examine the behavior of the equilibrium price and lead time as predicted by our model when different ﬁrms compete under varying market conditions. We predict the required parameters in our model based on available public data for two leading semiconductor foundry manufacturers in Taiwan. Without implying identity, X and Y denote the two large ﬁrms.

3.1. Case study data

Our case study is based upon timely representative data for the semiconductor foundry manufacturing industry. We note that the data will differ for other industry sectors, geographic regions, and/or time epochs and the case study results may alter depend-ing on different data sets.

The average quarterly shipment for 200-mm wafer products per ﬁrm in 2007 is 722.5 thousand (K) (Science & Technology Policy Research and Information Center (STPI), 2008). This ball-park number allows us to predict the customer arrival rate,

l

0,

when both prices and lead times of all firms in the market are identical, since the demand of a duopolist is roughly close to the market average per firm due to the nearly half of weighting in the average for duopoly firms. Therefore, we let

l

0 equal 722.5K

pieces per quarter in the case study. From public financial information, we know that the quarterly shipments of X and Y in 2007 are about 2.001 million (M) and 1.077M pieces, respec-tively; this data allow us to estimate the ballpark number of firms’ service rates. We intentionally set an identical service rate of the duopoly firms to leave room for conducting the sensitivity analysis of the service rates. The service rate of the firms is quoted at the average quarterly shipments of X and Y, i.e.,

m

X¼

m

Y¼1.54M pieces per quarter. The total market share of X

and Y in Taiwan is 68% with smaller ﬁrms accounting for approximately 32% (IDC, 2008). We refer to the total market share of X and Y as the preference factor, i.e.,

a

C¼.68 and

consequently,

a

M¼.32, since a high market share ﬁrm potentially

has a larger impact on the decision difference for the demand change and vice versa for the smaller ﬁrms. We assume an equal preference factor of lead times and prices; in other words,

b

t¼

b

p¼.5. In addition, we assume that the two ﬁrms’ minimum

service levels sX and sY are both .95. The average selling price

(ASP) of ﬁnished wafer products is US $1000 (GS, 2008). The production lead time in the foundry industry ranges from 30 to 50 day and the number, 36-day, is chosen for prediction of mtand

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production lead times. The purpose of the data set presented in this section is to predict a reasonable data set of mtand mpfrom

the available public information for our later use in the sensitivity analysis. Our data set allows us to roughly predict and round mt¼180,000 and mp¼15,000 such that the available data satisfy

the KKT optimality conditions of (8)–(17) along with Observa-tions 1 and 2.

3.2. Impact of ﬁrm characteristics

To study the effects of each firm’s characteristics of service rate and service level on the equilibrium prices, lead times, and firm profits, we consider X and Y with the parameters proposed in Section 3.1, but only vary X’s service rate (

m

X) and service level

(sX), respectively. We compute the ratios of their corresponding

equilibrium prices, lead times, and proﬁts to observe how the equilibrium solution changes when only X adjusts its parameters. The results are given inFig. 1and the ballpark number of proﬁts can be found inFig. 2(b) andFig. 3(b).

InFig. 1(a), X’s service rate (

m

X) varies from 924, 1232, 1540,

1848, to 2156K pieces per quarter and Y’s service rate (

m

Y)

remains the same (

m

Y¼1540). In a similar setting in Fig. 1(b),

X’s service level (sX) varies from 0.9, 0.925, 0.95, 0.975, to 0.99

while ﬁrm Y’s service level (sY) is 0.95. We summarize the major

observations as follows:

(i) The ratios of the equilibrium prices remain almost the same; the ratios of the equilibrium lead times decrease; the ratios of the corresponding proﬁts increase as the ratios of the service rate increase. The trend of the ratios of the corresponding proﬁts is not surprising, but it is worth further examination of

the relationship between predicted equilibrium prices and lead times. An increase in the service rates implies an increase in capacity. This observation allows us to infer that a firm appears not to raise its price due to competitiveness as its capacity increases; otherwise, the firm may lose market share. Meanwhile, an increase in capacity enables a firm to reduce its lead time so that it can attract more demand to increase its corresponding profit.

(ii) The ratios of the corresponding profits and equilibrium prices remain almost the same; the ratios of the equilibrium lead times increase as the ratios of the service levels increase. An increase in service levels implies a more conservative per-spective in quoted lead times. In other words, management tends to quote a long lead time so the firm can easily satisfy the requirement of a high service level (defined as the probability that the total production cycle time is less than or equal to the quoted lead time). From the numerical results, an increase in lead times results in a decrease in the selling price. However, due to competitiveness, the other firm tends to reduce its selling price as well at the equilibrium. As a result, the ratios of both firms’ corresponding profits and equilibrium prices remain almost the same even though their profits and selling prices decrease as one firm’s service level increases. This shows that the ability to offer a higher service level does not benefit the firm with the higher service level. 3.3. Impact of preference factors of firms

We next study how the preference factors affect the equili-brium prices, lead times and proﬁts. We consider different values of

a

M,

a

C,

b

t, and

b

p such that

a

Mþ

a

C¼1 and

b

tþ

b

p¼1. Again,

Fig. 1. Impact of service rate (mX) and service level (sX).

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other parameters remain the same as proposed inSection 3.1; the results are shown inFigs. 2 and 3.

The factor,

a

M, can be interpreted as the preference measure

accounting for the effect of the decision difference by the smaller ﬁrms. As

a

M increases, the impact of the decision difference

between the large ﬁrm and the smaller ﬁrms on the customer arrival rate increases. Meanwhile, as

a

Mincreases, the impact of

the decision difference between the two large firms on the customer arrival rate decreases. We note this is a symmetric case where both X and Y share the same parameters; therefore, X and Y have identical equilibrium prices, lead times, and resulting profits. Fig. 2 presents the trend of the equilibrium price, lead time, and profit as

a

Mincreases. Only several discrete data sets

have been investigated in the case study so that the trend may behave in a non-smooth manner. Clearly, both ﬁrms’ equilibrium prices and lead times increase in

a

M and their corresponding

proﬁts decline after

a

M¼.2. A large value of

a

Mindicates a small

value of

a

C, which represents a smaller impact of the decision

difference between the two large ﬁrms on the customer arrival rate. Firms X and Y would not pay much attention to the competition between them; hence, both the equilibrium price and lead time increase as

a

Mincrease. However, the results shown

inFig. 2indicate that both an increase in the price and lead time do not necessarily reduce the proﬁtability of both ﬁrms depend-ing on the level of

a

M.

The factors,

b

t and

b

p, can be interpreted as the preference

measures accounting for the effect of the lead times and prices on the arrival rate, respectively. As

b

t increases, the impact of lead

time decisions on the customer arrival rate increases; meanwhile, the impact of price decisions on the customer arrival rate decreases. Similarly, X and Y have identical equilibrium prices, lead times, and resulting proﬁts in the symmetric case. Fig. 3 presents the trend of the equilibrium price, lead time, and proﬁt as

b

tincreases. Again, only several discrete data sets have been

investigated in the case study so that the trend may behave in a non-smooth manner. It is clear that the equilibrium lead times of X and Y decrease, but the equilibrium prices increase in

b

t. Thus, if

ﬁrms pay more attention to the lead time decision, but less attention to the price decision, the results are low equilibrium lead times and high prices. The corresponding proﬁts increase after

b

t¼.5; hence, the trend of proﬁtability is not determined

based on the value of

b

tor

b

p.

4. Conclusions

This paper has examined the equilibrium pricing and lead time decisions in a duopoly industry consisting of two leading firms and a group of smaller firms all competing to provide goods or services to customers. We consider each firm as a system that behaves as an M/M/1 queue. The objective function of each firm

depends on its own decision variables as well as on the decision variables of the competition. All firms attempt to maximize their profits by making decisions about the price and lead time subject to the constraints needed to satisfy the minimum required service level which is defined as the probability of meeting the promised lead time quotation.

We solve the combined KKT conditions for the equilibrium decision of the price and the lead time. In equilibrium, no firm wishes to deviate from its current decision given the others’ decisions. The equilibrium solution is obtained by simultaneously solving the sufficient KKT optimality conditions instead of the necessary conditions. Our model shows the sufficiency of the KKT optimality conditions so that the solution to the KKT conditions is indeed an equilibrium decision rather than an examination of all possible KKT points for the equilibrium. Thus, we characterize the analytical condition of the existence and uniqueness of the Nash equilibrium of price and lead time decisions for both homogenous and heterogeneous cases.

The case study of two leading semiconductor foundry manu-facturers and several smaller firms examines the behavior of the equilibrium price and lead time as the firms compete under varying market conditions. The results produce some helpful managerial insights. A unilateral increase in one firm’s capacity does not appear to raise its price due to competitiveness, but instead tends to reduce the lead time to attract more demands. The ability to offer a higher service level does not automatically guarantee a benefit to the firm with the higher service level. The equilibrium prices and lead times of the two large firms increase when they pay less attention to the competition between them, but it does not necessarily imply reduced profitability for either firm. Likewise, high prices and low lead times result when firms pay more attention to their lead time decisions, and less to the price decisions.

We suggest that three possible extensions to our prototypical duopoly model are worth investigation. First, since we base our model (and results) on a linear structure of the customer arrival function, it would be useful to extend the model to other structures of the customer arrival rate. Second, the service level and the preference measures are assumed as given and fixed in our model. Additional modeling should consider the service level or preference measures as decision variables that can also affect the customer arrival rate. Third, noting that unsatisfied orders may cause a significant loss in profits due to potential penalties, both penalty design and policy analysis are fruitful topics for further consideration.

Acknowledgement

This paper has beneﬁted from comments and suggestions on earlier drafts from an anonymous referee and the editorial

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services of Ann Stewart. This research was supported in part by the National Science Council, Taiwan, under Grants NSC96-2221-E-009–035, NSC97-2221-E-002–263-MY3, and NSC99-2221-E-002–151-MY3. We thank Ms. Pei-I Chen for her editing and computing assistance.

Appendix

Proof of Lemma 1. For notational simplicity, we rewrite the arrival rate

l

ias

l

iðpi,ti9pj,tj,pM,tMÞ ¼

l

0mt

b

t½

a

MðtitMÞ þ

a

CðtitjÞmp

b

p½

a

MðpipMÞ þ

a

CðpipjÞ ¼u0 00u1tiu2piþu3tjþu4pj ð42Þ where u0 0₀¼

l

0þ

a

M

b

tmttMþ

a

M

b

pmppM u1¼

b

tmt u2¼

b

pmp u3¼

a

C

b

tmt u4¼

a

C

b

pmp u5¼ lnð1sÞ u1, u2, u3, u4,u540: ð43Þ

By inserting (42) into (4) and rearranging it, we have ðu0u1tiu2piÞtiru5 ð44Þ

where

u0¼u00þu3tjþu4pj

m

i:

Proving that the proﬁt function of ﬁrm i,

p

i, is pseudo-concave

is equivalent to showing that

p

0

i¼

p

iis pseudo-convex. Since

p

0i

is twice continuously differentiable, the gradient,

r

ð

p

0 iÞ, and Hessian matrix,

r

2ð

p

0 iÞ, of

p

0i can be computed as

r

ð

p

0 iÞT¼ @p0 i @pi @p0 i @ti ¼

l

iþu2pi u1pi h i , and

r

2ð

p

0 iÞ ¼ @2_p0 i @p2 i @2_p0 i @pi@ti @2_p0 i @ti@pi @2_p0 i @t2 i 2 6 6 4 3 7 7 5 ¼ 2u2 u1 u1 0 " # : Rearranging (7), we have MðX,

b

Þ ¼ 2u2 u1 u1 0 " # þ

b

ð

l

iu2piÞ 2 _u 1pið

l

iu2piÞ u1pið

l

iu2piÞ ðu1piÞ2 " # ¼ 2u2þ

b

ð

l

iu2piÞ 2 _u 1

b

u1pið

l

iu2piÞ u1

b

u1pið

l

iu2piÞ

b

ðu1piÞ2 " # : The determinant of M(X,

b

) is u2 1 2

b

pi

l

i1 . Since we only focus on a nontrivial solution of the equilibrium price and lead time, it is reasonable assuming that lower and upper bounds exist for the prices and lead times of ﬁrm iAN. We let pi, pi, ti, and ti

denote these lower and upper bounds of prices and lead times, respectively. Let

b

¼1/2

f

where

j

¼pi

l

i and

l

i¼

u0u1tiu2piþu3tjþu4pj. It is obvious that

f

is a lower bound

of the profit function of firm i since the customer arrival rate has the highest value for the variable with negative coefficients and the lowest value for the variable with positive coefficients. As a result, the determinant of M(X,

b

) is positive for such

b

. In addition, the diagonal elements of M(X,

b

) are nonnegative. This gives the result that there exists a real number

b

, 0r

b

o þN,

such that M(X,

b

) is positive semi-deﬁnite. FollowingDeﬁnition 1,

p

0

i is a pseudo-convex function. &

Proof of Lemma 2. The feasible region C of constraints (4)–(6) can be represented as

C ¼ ðp i,tiÞ9pi40,ti40, andðu0u1tiu2piÞtiru5

Let z!1¼ ðp1,t1Þ AC and z!2¼ ðp2,t2Þ AC. Consider the point

z !

¼ ðp,tÞ ¼

a

!z1þ ð1

a

Þ!z2, 0r

a

r1. Because p140 and p240,

it is obvious that p¼

a

p1þ (1

a

)p240. Similarly, t 40. These

show that (6) holds for z!¼ ðp,tÞ.

Constraint (4) can be represented as (44). For z!1and z!2, we

have

u0u1t1u2p1ru5=t1 ð45Þ

and

u0u1t2u2p2ru5=t2: ð46Þ

Considering z!¼ ðp,tÞ, the left side of (44) can be rewritten as ðu0u1tu2pÞt ¼ ½

a

ðu0u1t1u2p1Þ

þ ð1

a

Þðu0u1t2u2p2Þð

a

t1þ ð1

a

Þt2Þ:

Because of (45) and (46), and the Cauchy inequality (seeBartle, 1976), we have ðu0u1tu2pÞtr

a

u5 t1 þ ð1

a

Þu5 t2 ð

a

t1þ ð1

a

Þt2Þ ¼ u5

a

2þ ð1

a

Þ2þ

a

ð1

a

Þ t2 t1 þt1 t2 ru5ð

a

2þ ð1

a

Þ2þ2

a

ð1

a

ÞÞ ¼ u5: Therefore

ðu0u1tu2pÞtru5: ð47Þ

Due to (47), z!¼ ðp,tÞ also satisﬁes (4) and (5). In summary, for z

!

1¼ ðp1,t1Þ AC and z!2¼ ðp2,t2Þ AC, we show that the point

z !

¼ ðp,tÞ ¼

a

!z1þ ð1

a

Þ!z2, 0r

a

r1 satisﬁes constraints (4)–(6).

It completes the proof.

Proof of Lemma 3. There are two situations such that aco0: ao0, c40 and a40, co0. We ﬁrst examine the former case when ao0 and c40. Let function f(x) be ax3_{þbxþc, which is}

obviously continuous as xZ0. The first and second derivatives of f are f0(x)¼3ax2þb and f00(x) ¼6ax. Function f is a concave function as x 40. If br0, f is a decreasing function when x is positive. In addition, we have f(0)¼c 40. As a result, f definitely intersects axis x at a sufficiently large value of x. In other words, there exists a unique positive solution to ax3_{þbx þc¼0 if b}_r0.

We then analyze the case when b40. There are two points such that the first derivative of f equals 0; that is, x ¼7pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb=3aÞ. Again, we have f(0)¼c 40. Function f increases within the range

0,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb=3aÞ

h i

, but decreases when x 4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb=3aÞ. With the con-cavity property, f intersects axis x at a sufficiently large value of x as well. A similar argument follows for the latter case when a 40, co0 and it completes the proof. &

Proof of Observation 4. Rearranging (37), we have tY¼

ðGtX=F1t3XH1tXEÞ 4 0. The inequality follows since G 40.

Simi-larly, F2t3YH2tYE 40 follows. &

Proof of Proposition 2. We discuss the sign of T(tX) in the

following two disjunctive cases: H1¼0 and H140. When H1¼0,

it is obvious that T(tX) is positive since all other terms are positive.

We then examine the case when H140. Let function B(tX) be

F1t3XH1tXE. One can easily distinguish the critical points (local

maximum or minimum) of B(tX) since B(tX) is a continuous

and twice differentiable function. As a result, B(tX) increases

as tXo ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=3F1Þ p , B(tX) decreases as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=3F1Þ p rtXo

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ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=3F1Þ p , and B(tX) increases as tXZ ffiffiffiffiffiffi H1 3F1 q . In addition, we have B(0)¼ Eo0, and B(tX) needs to be positive from Observation 4.

Combining these, we argue that the feasible region of tXis greater

than ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=3F1Þ

p

; namely, 3F1t2XH140, and leading to the result

that 6F1t2XH140. Under this inequality, we have T(tX)40 as

H140. &

Proof of Proposition 3. To inspect the locations of critical points of T(tX), we take the ﬁrst and second derivatives of T(tX) as

shown below T0

ðtXÞ ¼3F1tXð5F1t3XþH1tXþ4EÞ

T00_ðt

XÞ ¼6F1ð10F1t3XþH1tXþ2EÞ:

For notational simplicity, we let P(tX) denote 5F1t3XþH1tXþ4E.

One can easily distinguish the critical points of P(tX) since P(tX) is

a continuous and twice differentiable function. As a result, P(tX) increases as tXo ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ p , P(tX) decreases as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ p rtXo ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ p , and P(tX) increases as tXZ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ p

. We discuss the sign of T(tX) in the following

two disjunctive cases: (i) ð2=3ÞH1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ

p

þ4E 4 0, and (ii) ð2=3ÞH1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ p þ4Eo0. (i) 2 3H1 ffiffiffiffiffiffiffiffi H1 15F1 q þ4E 4 0 :

We haveP(0)¼4E 40. The minimum of P(tX) for all tX40 is at

tX¼

p

and with the value of Pð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ

p Þ, which is equal to ð2=3ÞH1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ p

þ4E. Trivially, under condition (i), P(tX) is positive for all tX40; namely, T’(tX) 40

for all tX40. Therefore, T(tX) increases as tX40. In addition,

we have T(0) ¼ EH140. Thus, T(tX) 40 for all tX40.

(ii) 2 3H1 ffiffiffiffiffiffiffiffi H1 15F1 q þ4Eo0 :

Under condition (ii), the local minimum of P(tX) at

tX¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ is negative. In addition, P(0) ¼4E40 and

P(t_{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}X) is an increasing and convex function as tX4

ðH1=15F1Þ

p

. One can argue that P(tX)¼0 for tX40 has two

roots denoted by tX1and tX2, where

0otX1o ffiffiffiffiffiffiffiffi H1 15F1 q ð48Þ tX24 ffiffiffiffiffiffiffiffi H1 15F1 q ð49Þ

At tX¼tX1 and tX¼tX2, T(tX) is in its local maximum and

minimum since T00 _t X1 o0 and T00 _t X2 40. Since TðtX2Þ is

the minimum point when tX40, thus, T(tX) is positive for

all tX40 if TðtX2Þ is positive. This allows us to ignore the

case that tX¼tX1. We then examine the sign of TðtX2Þin the

following by inspecting two disjunctive cases: (ii-a)

H1

6

p

þ4Eo0 and (ii-b) ðH1=6Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=6F1Þ p þ4E 40. (ii-a) H1 6 ffiffiffiffiffiffiffi H1 6F1 q þ4Eo0 :

Under case (ii-a), PðpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1ÞÞ ¼ ðH1=6ÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þþ

4Eo0. In addition, since PðtX2Þ ¼0, we have

Pð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þ p Þ PðtX2Þ. As mentioned earlier, P(tX) increases as tX4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ p

. From (49) and the obvious inequality pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þ4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ, both

tX2 and

p

are greater than ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ

p

.

Within this range, P(tX) is increasing in tX. Therefore

tX24 ffiffiffiffiffiffiffi H1 6F1 q ð50Þ Again, since PðtX2Þ ¼0, we have

5F1t3X2þH1tX2þ4E ¼ 0 ð51Þ

Substituting (51) into TðtX2Þ, we have

TðtX2Þ ¼

t_X2

4ðH1þ6F1t2X2ÞðH13F1t

2

X2Þ: ð52Þ

Since H1o0, H13F1t2X2o0. Since tX24

p

and F1 is positive, we have H1þ6F1t2X240. Combining the

above inequalities, therefore, T tX2

_{o0. Thus, T(t}

X) is

negative for some tX40 when ð2=3ÞH1

p

þ 4Eo0 and ð1=6ÞH1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þþ4Eo0.

(ii-b) H1 6 ffiffiffiffiffiffiffi H1 6F1 q þ4E 40 :

Under case (ii-b), Pð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þ

p Þ ¼H1 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=6F1Þ p þ 4E4 0. In addition, since PðtX2Þ ¼0, we have

Pð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þ

p

Þ4 PðtX2Þ. As mentioned earlier, P(tX)

increases as tX4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þ. It is similar to case

(a), where both tX2 and

p

are greater than ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðH1=15F1Þ

p

. Within this range, P(tX) is increasing in

tX. Therefore tX2o ffiffiffiffiffiffiffi H1 6F1 q ð53Þ Since tX2o ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=6F1Þ p

and F1 is positive, we have

H1þ6F1t2X2o0. Again, H13F1t

2

X2o0 since H1o0.

Therefore, the sign of T tX2

is positive. Thus, T(tX) is

positive for all tX40 when ð2=3ÞH1

p

þ 4Eo0 and ð1=6ÞH1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=6F1Þþ4E 40.

To summarize the above proof for clariﬁcation purposes, from (i), we have shown that T(tX)40 for all tX40 when

ð2=3ÞH1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH1=15F1Þþ4E 40. There are two disjunctive cases in

(ii) in which we state that T(tX) is negative for some tX40 when

ð2=3ÞH1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ p þ4Eo0 and ð1=6ÞH1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=6F1Þ p þ4Eo0, but T(tX) is positive for all tX40 when

ð2=3ÞH1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=15F1Þ p þ4Eo0 and ð1=6ÞH1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH1=6F1Þ p þ4E 40. In addition, the range of ð2=3ÞH1

p

þ4E 4 0 is con-tained in the range of ð1=6ÞH1

p

þ4E4 0. Hence, it completes the proof.

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