Embed Size (px)
Citation preview
ARTICLE IN PRESS
Contents lists available at ScienceDirect
Int. J. Production Economics
Int. J. Production Economics 119 (2009) 284–297
0925-52
doi:10.1
� Cor
E-m
(Y.H. Jin
journal homepage: www.elsevier.com/locate/ijpe
Racing to market leadership: Product launch and upgrade decisions
Ying Li a,�, Yanhong H. Jin b
a Department of Information and Operations Management, Texas A&M University, College Station, TX 77843, USAb Department of Agricultural, Food and Resource Economics, Rutgers, The State University of New Jersey, New Brunswick, NJ 08901-8520, USA
a r t i c l e i n f o
Article history:
Received 30 April 2008
Accepted 17 March 2009Available online 5 April 2009
Keywords:
Marketing
Product launch
Product upgrade
Market leadership
Incumbent-vs-entrant
73/$ - see front matter & 2009 Elsevier B.V. A
016/j.ijpe.2009.03.004
responding author. Tel.: +1979 4584787; fax:
ail addresses: [email protected] (Y. Li), yji
).
a b s t r a c t
Firms might not launch a next generation product as soon as a more efficient technology
or a better product design is ready. We use a stylized model to analyze firms’ product
launch and upgrade decisions in an incumbent-vs-entrant setting. We find that large
profit margins from the next generation product alone do not provide the entrant
sufficient incentives to launch the next generation product, although small profit
margins will deter the entrant from joining the competition. If the entrant intends to
enter the market at an earlier time, it should consider process improvements that lower
the firm’s launch costs of the current generation product. In addition, the incumbent
must respond strategically to the entrant’s arrival. In particular, when anticipating a
late arrival of the entrant, the incumbent should upgrade to the next generation
earlier. The incumbent also has a cost advantage in the race to launch the next
generation product.
& 2009 Elsevier B.V. All rights reserved.
1. Introduction
A product upgrade often requires the availability ofeither a new technology that enables more efficientproduction or a better product design that generates morerevenue. Should a firm release a next generation product assoon as a more efficient technology (or a better productdesign) is available?
Empirical examples show that some firms appear to betechnology chasers. Apple Inc.’s launch of the iPod Nanogenerated a great deal of buzz in September 2005. The sleekdesign of the Nano, undoubtedly, contributed to thepublicity. On the other hand, the fact that the Nano was areplacement of the iPod Mini took the public completely bysurprise and prompted much media coverage. On the sameday the Nano was launched, Apple Inc. discontinued thesales of the Mini, despite the fact that the Mini was popularand had been on the market for only a year and eight
ll rights reserved.
+1979 8455653.
months. Many industry analysts wondered whether AppleInc. upgraded a cash-cow current generation product to thenext generation too soon, given the company’s marketposition as a powerful incumbent.
Microsoft Corporation took a different approach whenlaunching its XBox series. The remarkable success of theSony PlayStation worried Microsoft in the late 1990s.Microsoft, however, waited and launched its XBox in 2001shortly after Sony’s PlayStation 2 appeared. As an entrant,Microsoft chose to join the gaming console market not witha current generation product (equivalent to PlayStation) butwith a next generation product (equivalent to PlayStation 2).This raises the question, is it always wise for an entrant toleapfrog?
This research aims to identify the driving forces behind afirm’s product launch/upgrade decision, provided that thefirm is technologically ready to produce the next generationproduct. We solve the optimal timing decisions for both theentrant and the incumbent, and then discuss marketleadership (the first one to launch the next generationproduct) based on the optimal timings.
When launching the next generation product, a firmusually incurs a smaller launch cost if it has already
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 285
launched the current generation product, because the firmcan learn from its past experience. But if the benefit ofbuilding upon the current generation is outweighed by thesuperior profitability of the next generation product, thenthe entrant chooses to leapfrog (like Microsoft did withXBox) instead of entering with the current generation andthen upgrading later. We also find that higher profitabilityof the next generation product provides the entrant with anincentive to launch the current generation product earlier ifthe firm prefers not to leapfrog.
The incumbent’s monopolist status ends upon thearrival of the entrant. However, the incumbent may stillfind it optimal to upgrade as if it remained the monopolistthroughout the planning horizon. If we interpret a latearrival of the entrant to the market as weak competition forthe incumbent, then our results show that weakercompetition leads the incumbent to an earlier upgrade.Therefore, the fact that Apple Inc. was far ahead of itscompetitors provides an incentive (not a disincentive) toupgrade as soon as possible. We also demonstrate theadvantage of the incumbent in the race to launch the nextgeneration product.
The rest of this paper unfolds as follows: Section 2reviews relevant literature; Section 3 presents our modeland explains the rationales behind our assumptions;Sections 4 and 5 analyze the firms’ timing decisions;Section 6 discusses the incumbent’s advantage on marketleadership; Section 7 summarizes the implications of ourfindings and concludes the paper with future researchdirections.
2. Relevant literature
We analyze how a potential entrant firm determines itsarrival time at the market, and this aspect links our modelto market entry literature. The literature on market entrycan be categorized into two groups: one group investigatesthe order-of-entry effect given that one firm moves first andothers are later entrants, and the other group examines thetiming decision regarding a firm’s market entry.
‘‘First-to-market’’ could lead to a market share advan-tage (see Lieberman and Montgomery, 1998 for an over-view). However, the cost disadvantages of pioneer firms arealso documented (Lieberman and Montgomery, 1998;Boulding and Christen, 2003). Bohlmann et al. (2002)provided a review of the literature on pioneer advantagesand disadvantages, and constructed a game-theoreticalmodel that incorporates a pioneer’s preemption advantageand possible vintage effect, namely the phenomenon thatlater entrants can have lower costs and higher quality byutilizing improved technology. Bohlmann et al. (2002)demonstrated that the vintage effect can overcome thepioneer’s first mover advantage. Theoretical models study-ing the order-of-entry effects focus on firms’ entries to anew market. Those models often grant one firm the statusof the first mover and characterize others later entrants,but they do not consider additional product generations.Our incumbent firm is not, automatically, the first one tolaunch the next generation product. Our entrant firm cancompete with the incumbent by launching either the
current generation product, or the next generation, or bothsequentially. In such a setting, we analyze the incumbent’sadvantage in the race to launch the next generationproduct.
A firm’s entry decision can be modeled as a discrete-time or continuous-time variable. Narasimhan and Zhang(2000) investigated two competing firms’ entry decisions ina three-stage setting where firms face binary options atdecision epochs. Firms are technologically ready to enterthe market, but demand uncertainty is not resolved untilthe second stage. Narasimhan and Zhang (2000) showedthat both the benefit of being the first and the disadvantageof lagging behind motivate a firm to enter a new market.Lin and Saggi (2002) studied how two firms choose theirentry times (continuous variables) assuming that there isno technological or market uncertainty. Their analysisshows that when the initial entry generates positiveexternalities for the subsequent entry, a higher fixed costof entry could benefit the first entrant by delaying the entrytime of the later entrant. Like Lin and Saggi (2002), weadopt a continuous-time approach and let firms makesimultaneous decisions at time zero. But our entrant’stiming decision involves two generations of a product. Ourfinding that a higher launch cost of the current generationproduct delays the potential entrant’s arrival is consistentwith Lin and Saggi (2002)’s result, although networkexternalities do not play a role in our model.
The development of the next generation product couldbe driven by the adoption of a new technology. There is arich literature on technology adoption. Hoppe (2002)provided a comprehensive survey of the literature on thetiming of new technology adoption. A significant part of thetechnology adoption literature analyzes situations in whichfirms are uncertain about either the arrival time or theprofitability of new technology. The uncertainty drivesfirms to wait and/or buy information (e.g. Jensen, 1982;McCardle, 1985; Mamer and McCardle, 1987; Jensen, 1988).Algorithms are devised for firms to determine the optimaltiming of technology adoption under uncertainty(e.g. Chung and Tsou, 1998; Farzin et al., 1998; Doraszelski,2004). Although uncertainties can affect a firm’s productlaunch and upgrade decisions, we choose to disregarduncertainties (like Reinganum, 1981; Fudenberg and Tirole,1985) and highlight other elements (e.g. competition andentry barrier) that also play a role in a firm’s decision. Ourmodel differs from that in Reinganum (1981) and Fuden-berg and Tirole (1985) because they considered firms thathave a single active choice, whereas our entrant choosesbetween two generations of the product.
Ofek and Sarvary (2003) and Souza (2004) studied therepeated introduction of technologically advanced nextgeneration products in multi-period settings with uncer-tainty. A firm’s single period decision is binary (to introduceor not) in Souza (2004), and the decision is an investmentlevel concerning a single generation of a product in Ofekand Sarvary (2003). Again, our model is different becauseour entrant firm can choose between two active options.Rahman and Loulou (2001) and Huisman and Kort (2003)modeled two generations of the technology, but the newertechnology is available only in the second period. The firmsmake adoption decisions in two periods based on the
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297286
different availabilities of technologies. By allowing twogenerations of a product to be available at time zero, weanalyze different timing decisions. In addition, our firms’decisions are continuous variables instead of discrete ones.
We contribute to the market entry and technologyadoption literature by analyzing how firms choose productlaunch and/or upgrade times when they are technologicallyready to release the next generation product. Suchdecisions are different from binary choices at each periodand are needed as firms plan for the future.
3. The model
We consider a setting where two firms might becomecompetitors. At time zero, Firm 1 is a monopolist incum-bent producing the current generation product while Firm2 is a potential entrant. Both firms have the capability torelease the next generation of the product. The firms,however, will not do so until the right moment.
3.1. Decision variables
As an incumbent firm, Firm 1 determines whether andwhen to upgrade its current product to the next generation.We denote the incumbent’s upgrade time by tn
1, which cantake any value from the interval ½0;1Þ. Although such asetting implies that the firm’s planning horizon is infinite,our results apply to situations where a firm’s planninghorizon is finite with minimal modifications. We also allowtn
1 to take a value of _. _ is not a real number, but a notationwhich indicates that the incumbent chooses never tolaunch the next generation product.
As an entrant firm, Firm 2 can compete with theincumbent by launching the current generation productor the next generation product. The potential entrant (Firm2), therefore, needs to determine not only whether andwhen to launch the next generation product, but alsowhether and when to launch the current generationproduct. We capture the entrant’s decisions by tc
2 (the timeto launch the current generation product) and tn
2 (thetime to launch the next generation product). Similar tothe specification of tn
1, tk2 can take any value from ½0;1Þ [
f_g for k ¼ c;n. Since firms rarely launch an older version ofa product after releasing the newer version, we require thattc
2ptn2 when neither decision variable takes the value of _.
Our focus is on the evolution of generations of the sameproduct. We assume that once a firm has launched a newergeneration of a product, the firm discontinues olderversions of the same product. We also assume that thefirms make their timing decisions simultaneously at timezero. The decisions are affected by the firms’ coststructures, profit margin functions and the anticipatedgame play.
3.2. Launch costs
The launch or upgrade of a product is often associatedwith one-time fixed costs, for example, the cost of installingand implementing the necessary technology, and promo-tion cost. We refer to such fixed costs as ‘‘launch costs’’. It is
worth noting that marketing and promotion costs, aimingto stimulate and improve awareness, positive perceptionand adoption of the product among potential consumers,could be a significant part of launch costs. For example,pharmaceutical companies spent about $8 billion on salesand marketing, along with distributed samples that costan additional $7.95 billion in the US market in 2000alone (Kyle, 2006). A typical razor costs Gillette Sensor$200 million in research, engineering, and tooling, and anadditional $110 million in first-year television and printadvertising (Hammonds, 1990).
The incumbent’s launch costs of the current generationproduct are sunk at time zero when the firm determineswhether and when to upgrade. The launch costs the incum-bent should take into consideration are therefore those directlyassociated with the next generation product. We denote byFn
1ðtn1Þ the launch costs the incumbent incurs at time tn
1.We denote by Fc
2ðtc2Þ the entrant’s launch costs of the
current generation product at time tc2. For the entrant’s
launch costs of the next generation product, we distinguishtwo situations: (i) if the entrant skips the current generationproduct and launches the next generation at time tn
2, then theentrant incurs Fn
2ðtn2Þ at time tn
2; (ii) if the entrant launches thecurrent generation product at time tc
2 and upgrades it to thenext generation at time tn
2, then the entrant incurs Fd2ðt
n2 � tc
2Þ
at time tn2. In the latter case, we assume that when the entrant
launches the next generation product, it builds upon its pastexperience of launching the current generation product, andthus incurs lower launch costs given the same launch time ofthe next generation product in both situations.
Note that if a firm has no plan to launch the kthgeneration of the product, any launch cost associated withthe generation is zero. Launch costs, in general, must benon-negative. Since we expect that the installation andimplementation costs decrease over time as technologiesmature and the marketing/promotion may cost less asconsumers become increasingly aware of the product, weassume that launch costs decrease over time. Furthermore,we assume that the marginal decrease in launch costsdeclines over time, namely the cost reduction is diminish-ing as the launch time is postponed because technologyimprovements become more and more marginal and thelearning effects decline as well.
The next generation product is supposed to be moretechnologically challenging for a firm than the currentgeneration product. Because of this, if the entrant decidesto launch only one generation of the product and thelaunch time is t, then the firm incurs a higher launch cost attime t if the product is of the next generation. That is,Fn
2ðtÞ4Fc2ðtÞ. Furthermore, we set Fd
2ð0Þ to the differencebetween Fn
2ðtÞ and Fc2ðtÞ. The rationale is that if the entrant
launches the current generation product and upgrades it tothe next generation immediately, then the firm, effectively,launches only the next generation product.
In summary, we make the following assumptions aboutlaunch costs throughout this paper:
�
(Assumption 1a): Fki is non-negative, continuouslydifferentiable, strictly decreasing, and strictly convexin the launch time of the kth generation product.
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 287
�
TabThe
(Inc
The
(Assumption 1b): Fd2 is non-negative, continuously
differentiable, strictly decreasing, and strictly convexin the time lapse between the two launches.
� (Assumption 2): Fd2ð0Þ ¼ Fn2ðtÞ � Fc
2ðtÞ.
3.3. Gross profit margins
Instead of modeling the formation of market pricesthrough demand functions, we consider the impact of pricesand revenues through a firm’s gross profit margin. A firm’sgross profit margin is its revenues minus variable costs.
A firm’s gross profit margin is generated continuously aslong as the firm has a product on the market. Similar toFudenberg and Tirole (1985)’s approach, we assumethat the state of the competition affects the firm’s grossprofit margins. Monopoly profit margins need not be thesame as duopoly profit margins. Specifically, we denote byDk
i Firm i’s (instantaneous) current gross profit margin ifFirm i offers the kth generation of the product in a duopolymarket, and by Mk the incumbent’s (instantaneous) currentgross profit margin if the incumbent offers the kthgeneration of the product in a monopoly market. Theentrant never earns monopoly profits because the incum-bent is always a player on the market.
Given any point of time t, the generation of the producta firm has on the market can be inferred from the firms’timing decisions. Depending upon whether the entrant’sproduct has arrived at the market by time t, the state of thecompetition at time t is either a duopoly or a monopoly.Table 1 summarizes the possible gross profit margins foreach firm at time t and the conditions under which eachpossibility materializes.
Our modeling of the profit margins implies that in aduopoly market, a firm’s gross profit margin does notchange as the firm’s competitor replaces one generation ofthe product with a newer generation. This is a strongassumption, but could be plausible. When brand name isthe dominant factor driving consumers’ choices, succes-sions of product generations at competing firms do notaffect a firm’s profit margin much. For example, in the PCmarket, people who prefer Apple’s Macintosh are not likelyto purchase computers from other brand names regardlessof the generation of the product others offer.
3.4. Profits
The firms make their launch and update decisions attime zero. While gross profit margins will be reaped in a
le 1firms’ current gross profit margins at time t.
umbent, entrant)
entrant’s product at time t None
The current generation
The next generation
continuous stream, the firms’ launch costs will occur asone-time fixed costs. We convert future revenues and coststo present values given discount rate r 2 ð0;1Þ. A rationalfirm will prefer launch times that result in a higher presentvalue of total profit. Depending on the firms’ timingdecisions, their gross profit margins and launch costs takedifferent forms. We discuss the specifics later in Sections 4and 5 when studying the entrant and the incumbent’sdecisions, respectively.
We recognize that our model focuses on the impact offixed launch/upgrade costs among all the possible costs afirm might incur. But we do take into consideration a firm’svariable/component costs by defining gross profit marginas the difference between revenue and variable costs. Sincegross profit margins depreciate over time, as reflected bythe discount rate r, we indirectly model variable costs thatare decreasing over time.
4. The entrant’s timing decisions
From the entrant’s perspective, the state of the competi-tion is always duopoly regardless of the incumbent’s timingdecision. As a result, the entrant’s gross profit margin attime s depends only upon the generation of the product theentrant is offering at the moment. The present value of theentrant’s profits is, hence, a function of the vector ðtc
2; tn2Þ.
Any well-defined vector ðtc2; t
n2Þ must belong to one of the
following three categories:
(i)
The entrant considers the launch of the currentgeneration product with no plan to upgrade it to thenext generation (i.e., tc2 2 ½0;1Þ [ f_g and tn2 ¼ _).
(ii)
The entrant skips the current generation product andconsiders only the launch of the next generation(i.e., tc2 ¼ _ and tn2 2 ½0;1Þ [ f_g).
(iii)
The entrant launches the current generation productand upgrades it to the next generation at a later time(i.e., tc2 2 ½0;1Þ and tn2 2 ½t
c2;1Þ).
Note that the option ðtc2; t
n2Þ ¼ ð_; _Þ is in both of the first two
categories. Excluding this option, the three categories are non-overlapping. We look for the entrant’s optimal timing decisionsby examining options in each of these three categories.
4.1. The entrant considers only the current generation product
If the entrant decides not to launch the next generationproduct, tn
2 is fixed at _. The entrant’s decision is reduced to
The incumbent’s product at time t
The current generation The next generation
ðMc ;0Þ ðMn ;0Þ
ðDc1 ;D
c2Þ ðDn
1;Dc2Þ
ðDc1 ;D
n2Þ ðDn
1;Dn2Þ
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297288
whether and when to launch the current generationproduct. For any tc
2 2 ½0;1Þ, the present value of theentrant’s total profits is
P2ðtc2; t
n2Þ ¼ P2ðt
c2; _Þ ¼
Z 1tc
2
Dc2 e�rs ds� Fc
2ðtc2Þ e�rtc
2 . (1)
Note that if tc2 takes the extreme value of _, then the
entrant earns zero profits, and we have P2ð_; _Þ ¼0 ¼ limt!1P2ðt; _Þ.
The first derivative of the entrant’s profit with respectto tc
2 is
d
dtc2
P2ðtc2; _Þ ¼ e�rtc
2 �Dc2 þ rFc
2ðtc2Þ �
d
dtc2
Fc2ðt
c2Þ
!. (2)
The sign of Eq. (2) determines whether or not theentrant’s profit is increasing at a certain value of tc
2. Lemma1 tracks the sign of Eq. (2) and derives the optimal time forthe entrant to launch the current generation product, giventhat the entrant has no plan to upgrade the product. (Theproof of Lemma 1, together with proofs of other technicalstatements, is included in the appendix.)
Lemma 1. Suppose the entrant rules out the possibility of
launching the next generation product. Define cc2ðtÞ ¼ rFc
2ðtÞ�
ðd=dtÞFc2ðtÞ.
(i)
If limt!1cc2ðtÞoDc2, the entrant’s optimal launch time of
the current generation product is tc_2 ¼ minftjcc
2ðtÞpDc2g.
In particular, cc2ðtc_
2 Þ ¼ Dc2 if Dc
2pcc2ð0Þ, and tc_
2 ¼ 0 if
cc2ð0ÞpDc
2.
(ii) If limt!1cc2ðtÞXDc
2, it is optimal for the entrant never to
launch the current generation product.
The intuition behind Lemma 1 is, essentially, a balancingact. The delay of launching the current generation productleads to savings in launch costs, but at the expense ofdelays in generating gross profit margins. The savings inlaunch costs come from two sources. First, due to thematuration of technologies over time, the firm’s currentlaunch cost Fc
2ðtÞ decreases over time. Second, taking apresent value perspective means that the later the launch,the smaller the present value of the launch cost. The latterimpact is measured by the discount factor r. From a presentvalue perspective, the marginal cost of the launch ise�rt ½rFc
2ðtÞ � ðd=dtÞFc2ðtÞ� ¼ e�rtcc
2ðtÞ at time t. Meanwhile,the launch of the current generation product producesgross profit margins. The marginal benefit of the launch ise�rtDc
2 at time t from a present value perspective. Theentrant must balance these two counteracting incentives.Lemma 1 above outlines the rule of thumb for the entrant:if there is a point in time at which the marginal cost of thelaunch equals the marginal benefit, then the entrant shouldlaunch the current generation product at that point in time;if such a time does not exist, then either the marginal costalways dominates the marginal benefit, or vice versa. If themarginal cost is indeed greater than the marginal benefit,the entrant always benefits from postponing the launchand, hence, prefers not to launch at all. Otherwise, theentrant should launch the current generation product asearly as possible, and hence takes action at time zero.
4.2. The entrant considers only the next generation product
The entrant can skip the launch of the current genera-tion product (tc
2 ¼ _) and enter the market with the nextgeneration at tn
2. In this case, for any tn2 2 ½0;1Þ, the present
value of the entrant’s total profits is
P2ðtc2; t
n2Þ ¼ P2ð_; t
n2Þ ¼
Z 1tn
2
Dn2 e�rs ds� Fn
2ðtn2Þ e�rtn
2 .
The first derivative of P2ð_; tn2Þ with respect to tn
2 is
d
dtn2
P2ð_; tn2Þ ¼ e�rtn
2 �Dn2 þ rFn
2ðtÞ �d
dtn2
Fn2ðtÞ
!.
The analysis here is similar to that in Section 4.1 in the sensethat the entrant launches, at most, one generation of theproduct. Hence, the derivation of the entrant’s optimal launchtime of the next generation product in Lemma 2 is similar tothe derivation in Lemma 1, which focuses on the entrant’soptimal launch time of the current generation product.
Lemma 2. Suppose the entrant skips the launch of the current
generation product. Define cn2ðtÞ ¼ rFn
2ðtÞ � ðd=dtÞFn2ðtÞ.
(i)
If limt!1cn2ðtÞoDn2, the entrant’s optimal launch time of
the next generation product is t_n2 ¼ minftjcn
2ðtÞpDn2g. In
particular, cn2ðt_n
2 Þ ¼ Dn2 if Dn
2pcn2ð0Þ, and t_n
2 ¼ 0 if
cn2ð0ÞpDn
2.
(ii) If limt!1cn2ðtÞXDn
2, it is optimal for the entrant never to
launch the next generation product.
4.3. The entrant considers both generations
If the entrant launches the current generation product attc
2a_ and upgrades the product to the next generation laterat tn
2a_, then D ¼ tn2 � tc
2 (i.e., the lapse of time between thelaunches of two generations of the product) is well definedand D 2 ½0;1Þ. In this case, finding the optimal launch timesis equivalent to finding a vector ðtc
2;DÞ such that ðtc2; t
c2 þ DÞ
maximizes the entrant’s profit. For any tc2 2 ½0;1Þ and any
D 2 ½0;1Þ, the present value of the entrant’s total profits is
P2ðtc2; t
n2Þ ¼ P2ðt
c2; t
c2 þDÞ
¼
Z tc2þD
tc2
Dc2e�rs dsþ
Z 1tc
2þDDn
2 e�rs ds� Fc2ðt
c2Þ e�rtc
2
� Fd2ðDÞ e
�rðtc2þDÞ. (3)
We first fix the launch time of the current generationproduct and examine how the entrant’s profit changes withthe lapse of time between the two launches. The firstderivative of P2ðt
c2; t
c2 þ DÞ with respect to D is
d
dDP2ðt
c2; t
c2 þ DÞ ¼ e�rðtc
2þDÞ Dc2 � Dn
2 þ rFd2ðDÞ �
d
dDFd
2ðDÞ� �
.
(4)
Since e�rðtc2þDÞ is always non-negative, the sign of Eq. (4) is
not dependent upon the value of tc2. Instead, whether or not
the entrant benefits from extending the lapse of timebetween the two launches at time tc
2 þ D is determined bythe value of Dc
2 � Dn2 þ rFd
2ðDÞ � ðd=dDÞFd2ðDÞ. Lemma 3
focuses on this value and derives the optimal time lapsebetween the two launches.
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 289
Lemma 3. Suppose the entrant launches the current genera-
tion product at tc2 2 ½0;1Þ. Define cd
2ðDÞ ¼ rFd2ðDÞ�
ðd=dDÞFd2ðDÞ.
(i)
If limD!1cd2ðDÞoDn2 � Dc2, it is optimal for the entrant to
upgrade to the next generation at tc2 þ d�, where
d� ¼ minfDjcd2ðDÞpDn
2 � Dc2g. In particular, d� ¼ 0 if
cd2ð0ÞpDn
2 � Dc2, and cd
2ðd�Þ ¼ Dn
2 � Dc2 if cd
2ð0Þ4Dn2 � Dc
2.
(ii) If limD!1cd2ðDÞXDn
2 � Dc2, it is optimal for the entrant not
to upgrade.
Lemma 3 above describes how the entrant balances theincentive to spend less and the inventive to earn more.Given the launch of the current generation product at tc
2,upgrading to the next generation product at time tc
2 þ Dcosts the entrant a fixed cost of e�rðtc
2þDÞFd2ðDÞ from a present
value perspective. Delaying the upgrade will lower thiscost. The marginal cost of the upgrade is e�rðtc
2þDÞcd2ðDÞ,
which quantifies the entrant’s incentive to spend less bydelaying the upgrade. As the entrant replaces the currentgeneration product with the next generation, the rate atwhich the entrant earns gross profit margins changes frome�rðtc
2þDÞDc2 to e�rðtc
2þDÞDn2. The marginal benefit of the
upgrade is e�rðtc2þDÞðDn
2 � Dc2Þ. A positive Dn
2 � Dc2 allows
the entrant to earn more and provides an incentive for theentrant to upgrade. Comparing cd
2ðDÞ to Dn2 � Dc
2, therefore,sheds light on how the entrant should strike a balance.
If cd2ðDÞ4Dn
2 � Dc2 for any D 2 ½0;1Þ (i.e., limD!1c
d2ðDÞX
Dn2 � Dc
2), then an upgrade to the next generation is neverjustified. In other words, the entrant’s decision to launchboth generations of the product is dominated by its optimaldecision when considering only the current generationproduct. The latter case was analyzed earlier in Section 4.1.
Now, we consider the case in which limD!1cd2ðDÞo
Dn2 � Dc
2. In this case, there exists a D such that the marginalbenefit of the upgrade outweighs the marginal cost at timetc
2 þ D. If, however, the marginal benefit is large enough tocover the marginal cost at any point of time tc
2 þD, thenany delay of the upgrade lowers the entrant’s profit and theoptimal time lapse between the two launches (d�) shouldbe zero. This action is equivalent to launching the nextgeneration product at time tc
2. Therefore, when d� ¼ 0, theentrant’s optimization problem is reduced to finding theoptimal time to launch the next generation product, whichis a case studied earlier in Section 4.2.
If the marginal benefit of the upgrade equates themarginal cost for some D40, such a D is the optimal timelapse between the two launches. That is, Dn
2 � Dc2 ¼ cd
2ðd�Þ.
We can fix the time lapse between the two launches at d�,and examine how the entrant’s profit is affected by itslaunch time of the current generation product. With d�40,the first order derivative of the entrant’s profit with respectto tc
2 on the interval ½0;1Þ is
d
dtc2
P2ðtc2; t
c2 þ d�Þ
¼ e�rtc2 ð�Dc
2 þcc2ðt
c2ÞÞ þ e�rðtc
2þd�ÞðDc
2 � Dn2 þ cd
2ðd�Þ
þd
Fd2ðd�ÞÞ
dD
¼ e�rtc2 ð�Dc
2 þcc2ðt
c2ÞÞ þ e�rðtc
2þd�Þ d
dDFd
2ðd�Þ
ðthe above equality follows because cd2ðd�Þ ¼ Dn
2 � Dc2
for any d� 2 ð0;1ÞÞ
¼ e�rtc2 ð�Dc
2 þcc2ðt
c2Þ þ e�rd� d
dDFd
2ðd�ÞÞ. (5)
We have learned from Section 4.1 that e�rtc2cc
2ðtc2Þ is the
entrant’s marginal cost of launching the current generationproduct at time tc
2. It measures the entrant’s incentive todelay its launch of the current generation product. Fromtime tc
2 to time tc2 þ d�, the entrant earns an instantaneous
gross profit margin of Dc2 at any moment. Beyond time
tc2 þ d�, the entrant earns a higher instantaneous gross
profit margin of Dn2. Taking into consideration the higher
profit margins in the future, the entrant can treat e�rtc2 ½Dc
2 �
e�rd� ðd=dDÞFd2ðd�Þ� as the marginal benefit of launching the
current generation product at time tc2. Therefore, a
comparison of cc2ðt
c2Þ and Dc
2 � e�rd� ðd=dDÞFd2ðd�Þ determines
whether and when the entrant should launch the currentgeneration product given the optimal time lapse betweenthe two launches. These intuitions are formalized inLemma 4.
Lemma 4. Suppose limD!1cd2ðDÞoDn
2 � Dc2. The entrant
considers launching the current generation product at tc2 and
upgrading the product to the next generation at tc2 þ d�.
(i)
If d� ¼ 0, any decision to launch both generations of theproduct sequentially is dominated by the maximizer of
P2ð_; tn2Þ over the region: tn
2 2 ½0;1Þ [ f_g.�
(ii)
If d�40 and limt!1cc2ðtÞXDc2 � e�rd ðd=dDÞFd2ðd�Þ, any
decision to launch both generations of the product
sequentially is dominated by ðtc2; t
n2Þ ¼ ð_; _Þ.
�
(iii)
If d�40 and limt!1cc2ðtÞoDc2 � e�rd ðd=dDÞFd2ðd�Þ, the
entrant’s optimal launch time of the current generation
product is tcd2 ¼ minftjcc
2ðtÞpDc2 � e�rd� ðd=dDÞFd
2ðd�Þg. In
particular, tcd2 ptc_
2 if limt!1cc2ðtÞoDc
2.
Lemma 4 points out that the entrant’s optimal launchtime of the current generation product would be earlierwhen an upgrade at a later time is guaranteed than whenthere is no plan to upgrade (i.e., tcd
2 ptc_2 ). This is because an
upgrade brings in more gross profit margins, and hencestrengthens the entrant’s incentive to launch the currentgeneration product. Recall that when an upgrade occurs d�
time units after the launch of the current generationproduct, the marginal benefit of the launch at time tc
2 ise�rtc
2 ½Dc2 � e�rd� ðd=dDÞFd
2ðd�Þ�4e�rtc
2 Dc2. The latter is the mar-
ginal benefit of launching the current generation productabsent any plan to upgrade.
4.4. The entrant’s overall decisions
We denote by ðtc2; t
n2Þ
opt the vector that represents theentrant’s optimal launch times. Based on our analysis inSections 4.1–4.3, ðtc
2; tn2Þ
opt must take one of these fourvalues: ð_; _Þ, ðtc_
2 ; _Þ, ð_; t_n2 Þ, or ðtcd
2 ; tcd2 þ d�Þ. Our next step
is to find out the necessary and sufficient conditions under
ARTICLE IN PRESS
Fig. 1. The entrant’s optimal timing decisions.
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297290
which each of these vectors becomes the entrant’s optimaldecision.
We find that how the entrant’s marginal benefit of theupgrade fares against its marginal cost of the upgrade playsan important role in determining the firm’s overall productlaunch strategies. If the marginal benefit always dominatesthe marginal cost, then the lure of the next generationproduct is large enough for the entrant to skip the currentgeneration product. If the marginal cost always dominatesthe marginal benefit, then it is not worthwhile for theentrant to pursue the next generation product at all. Forcases in between, the entrant should consider launchingboth generations of the product sequentially. Proposition 1combines this intuition with Lemmas 1–4, and formallystates the entrant’s optimal timing decisions.
Proposition 1. The entrant’s optimal timing decisions can be
characterized as follows:
(i)
ðtc2; tn2Þ
opt¼ ð_; t_n
2 Þ if and only if cd2ð0ÞpDn
2 � Dc2 and
limt!1cn2ðtÞoDn
2.
(ii) ðtc2; tn2Þ
opt¼ ðtcd
2 ; tcd2 þ d�Þ if and only if limD!1c
d2ðDÞo
Dn2 � Dc
2ocd2ð0Þ and limt!1c
c2ðtÞoDc
2 � e�rd� ddD Fd
2ðd�Þ.
(iii)
ðtc2; tn2Þ
opt¼ ðtc_
2 ; _Þ if and only if Dn2 � Dc
2plimD!1cd2ðDÞ
and limt!1cc2ðtÞoDc
2.
(iv) If none of the above sets of conditions are satisfied, thenðtc2; t
n2Þ
opt¼ ð_; _Þ.
We try to visualize the entrant’s optimal timing decisionsas shown in Fig. 1. In the figure, we use Dc
2 and Dn2 � Dc
2 ashorizontal and vertical axes, respectively, and comparethem to different thresholds relating to the entrant’s launchcosts. For any given Dn
2, all the feasible combinations ofðDc
2;Dn2 � Dc
2Þ are represented by a 451 line (diagonal) thatlinks the vertical and the horizontal axes. Each point on thediagonal presents a unique combination of (Dc
2;Dn2 � Dc
2)but all the points share the same Dn
2. As Dn2 gets smaller
(larger), the position of the corresponding diagonal movestoward (away from) the origin.
There are four regions of different shades shown inFig. 1, each representing a potential optimal decision for theentrant. Depending on the value of Dn
2, the correspondingdiagonal intersects with different regions. For any pair of Dc
2
and Dn2, the entrant’s optimal decision can be found by
drawing the diagonal that represents Dn2 and locating
the point ðDc2;D
n2 � Dc
2Þ on the diagonal. By identifying theregion that hosts the point ðDc
2;Dn2 � Dc
2Þ we find theentrant’s optimal decision.
The exact shape and size of each region depend uponhow the values of the cost thresholds rank against eachother and the gross profit margins. What we show in Fig. 1is simply one possibility. For example, if limt!1c
c2ðtÞ gets
smaller, then both the line that separates Region (III) fromRegion (IV) and the line that separates Region (II) fromRegion (IV) will move toward the vertical axis. The move-ments will result in a smaller Region (IV).
If we fix the cost thresholds and move the 451 line(diagonal) that represents the next generation product’sgross profit margin ðDn
2Þ, we can see how gross profitmargins affect the entrant’s timing decisions. Here are a
few observations. (a) If Dn2 is not sufficiently large, the
entrant will not launch any generation of the product. (b) Itis also interesting to note that a large Dn
2 does not guaranteethe entrant’s launch of the next generation product. Whenboth Dn
2 and Dc2 are large, we can have a small difference
between the gross profit margins for both generations ofthe product, which can spoil the launch of the nextgeneration product. (c) The entrant is motivated to launchboth generations of the product in a sequential manneronly if the firm earns enough profit margins from thecurrent generation product and can improve its profitmargins sufficiently by upgrading to the next generation.
If we experiment with the cost thresholds, it is easy tosee that the existence of the four regions illustrated in Fig. 1is not automatic. At one extreme, if limt!1c
c2ðtÞ ¼ 0, then
Region (IV) no longer exists, which implies that the entrantwill always join the competition. In this sense, limt!1c
c2ðtÞ
can be thought of as an entry barrier: the biggerlimt!1c
c2ðtÞ is, the more lucrative the next generation
product (i.e., the bigger the Dn2) has to be in order to justify
the entry.The entry–barrier effect of limt!1c
c2ðtÞ can also be
verified by noticing that a sufficient condition to guaranteethe entrant’s arrival at the market is limt!1c
c2ðtÞo
minfDc2;D
n2 � rFd
2ð0Þg. This condition follows from Proposi-tion 1. The proposition indicates that if limt!1c
c2ðtÞoDc
2,limt!1c
n2ðtÞoDn
2, and limt!1cc2ðtÞoDc
2 � e�rd� ðd=dDÞFd2ðd�Þ
all hold, then the entrant will enter the market. Meanwhile,0o� e�rd� ðd=dDÞFd
2ðd�Þ and cn
2ðtÞ ¼ rFd2ð0Þ þ cc
2ðtÞ, wherethe second equality follows from Assumption 2:Fd
2ð0Þ ¼ Fn2ðtÞ � Fc
2ðtÞ. Thus, the sufficient condition de-scribed at the beginning of this paragraph emerges. Usingsimilar logic, we find that if the entrant can reduce itslaunch cost cc
2ðtÞ consistently through some processimprovement, then the entrant’s possible entry times, tcd
2 ,tc_
2 , and t_n2 , will be advanced. This result is formally
established in Corollary 1.
Corollary 1. If the entrant can lower its launch costs of the
current generation product Fc2ðtÞ by �40 for any t 2 ½0;1Þ,
then the entrant is more likely to enter the market, and with
an earlier entry time.
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 291
Corollary 1 confirms that small launch costs of thecurrent generation product are a competitive advantage.
5. The incumbent’s timing decisions
Whether the incumbent (Firm 1) upgrades its product tothe next generation depends upon its anticipation as towhether and when the entrant (Firm 2) joins the competi-tion. If the entrant is expected to stay out of the market, theincumbent’s upgrade decision must fit its status as amonopolist. Otherwise, the incumbent’s status will bechanged from a monopolist to a duopolist upon the arrivalof the entrant. We analyze, in this section, the incumbent’soptimal response given the entrant’s product launch andupgrade strategies.
5.1. The incumbent’s decision when facing no competitors
When the entrant stays out of the market (i.e.,tc
2 ¼ tn2 ¼ _), the incumbent remains a monopolist produ-
cing either the current or the next generation product. Werefer to such an incumbent as a pure monopolist. Being apure monopolist, the incumbent’s decision of upgrading attn
1 2 ½0;1Þ results in the present value:
P1ðtn1; t
c2; t
n2Þ
¼ P1ðtn1; _; _Þ ¼
Z tn1
0Mc e�rs dsþ
Z 1tn
1
Mn e�rs ds� Fn1ðt
n1Þ e�rtn
1 .
(6)
If the incumbent chooses not to upgrade, the presentvalue of its total profits becomes
R10 Mc e�rs ds ¼
limt!1P1ðt; _; _Þ. The first derivative of the profit withrespect to tn
1 is
d
dtn1
P1ðtn1; _; _Þ ¼ e�rtn
1 Mc�Mn
þ rFn1ðt
n1Þ �
d
dtn1
Fn1ðt
n1Þ
!. (7)
Like the entrant, the incumbent wants to balance twocounteracting forces: the marginal benefit of upgrading tothe next generation product and the marginal cost of doingso. Proposition 2 characterizes the optimal upgrade time forthe incumbent when it is a pure monopolist.
Proposition 2. Suppose the entrant stays out of the market.
Define cn1ðtÞ ¼ rFn
1ðtÞ � ðd=dtÞFn1ðtÞ.
(i)
If limt!1cn1ðtÞXMn�Mc , the incumbent is better off
never to upgrade.
(ii)
If limt!1cn1ðtÞoMn�Mc , the optimal time for the
incumbent to upgrade its product to the next generation
is tm1 ¼minftjcn
1ðtÞpMn�Mc
g.
Proposition 2 suggests that the incumbent’s launch of thenext generation product is not automatic even if the firm isthe monopolist. Ultimately, it is the comparison betweenthe marginal benefit and cost of upgrading that determinesthe course of action.
5.2. The incumbent’s decision in anticipation of competition
The incumbent becomes a duopolist upon the entrant’sarrival. We denote the entrant’s arrival time by ~t2. If theentrant plans to launch the current generation product, itslaunch time of the current generation product is ~t2.Otherwise, the entrant’s launch time of the next generationproduct is ~t2. That is,
~t2 ¼tc
2 if tc2 2 ½0;1Þ;
tn2 if tc
2 ¼ _ and tn2 2 ½0;1Þ:
((8)
~t2 is well defined as long as the entrant competes withthe incumbent, namely ðtc
2; tn2Það_; _Þ. An extreme case is
that ~t2 ¼ 0. In this case, the entrant rushes to the marketand competes with the incumbent from time zero. Hence,the incumbent becomes a duopolist since time zero. Werefer to such an incumbent as a pure duopolist. Similar tothe definition of tm
1 in Proposition 2, we definetd
1 ¼min ftjcn1ðtÞpDn
1 � Dc1g. Note that td
1 is well definedonly if limt!1c
n1ðtÞoDn
1 � Dc1. If the incumbent is a pure
duopolist, then the optimal time for the incumbent toupgrade would be td
1 provided that the incumbent wasbetter off with an upgrade (i.e, limt!1c
n1ðtÞoDn
1 � Dc1). The
derivations are the same as those for Proposition 2 withminimal adaptations.
We find that a pure monopolist’s optimal upgrade timeis earlier than a pure duopolist’s as long as Dn
1 � Dc1o
Mn�Mc . The inequality implies that a monopolist enjoys a
larger increase in gross profit margins from a productupgrade than a duopolist does, which is a plausiblescenario. This result is formalized in Lemma 5.
Lemma 5. If limt!1cn1ðtÞoDn
1 � Dc1oMn
�Mc then tm1 ptd
1.In particular, if td
140 then tm1 otd
1.
It is perhaps more often the case that ~t240. Now theincumbent is neither a pure monopolist, nor a pure duopolist.For any tn
1 2 ½0;1Þ, with the expectation of becoming aduopolist from being a monopolist, the incumbent reapsboth monopoly gross profit margins from the currentgeneration product and duopoly gross profit margins fromthe next generation product, although during differentperiods of time. The incumbent, however, is able to enjoymonopoly gross profit margins from the next generationproduct only if its upgrade precedes the entrant’s arrival. Onthe other hand, the incumbent will earn duopoly gross profitmargins from the current generation product only if itupgrades after the entrant’s arrival. Therefore, if the incum-bent upgrades at some tn
1 2 ½0;1Þ, then the present value ofthe firm’s total profits is
P1ðtn1; t
c2; t
n2Þ ¼
R tn1
0 Mc e�rs dsþR ~t2
tn1
Mn e�rs ds
þR1~t2
Dn1 e�rs ds� Fn
1ðtn1Þ e�rtn
1 for tn1 2 ½0; ~t2�;R ~t2
0 Mc e�rs dsþR tn
1~t2
Dc1 e�rs ds
þR1
tn1
Dn1 e�rs ds� Fn
1ðtn1Þ e�rtn
1 for tn1 2 ½
~t2;1Þ:
8>>>>>>><>>>>>>>:
If the incumbent never upgrades, the present value of itstotal profits is P1ð_; t
c2; t
n2Þ ¼
R tn2
0 Mc e�rs dsþR1
tn2
Dc1 e�rs ds,
which equals limt!1P1ðt; tc2; t
n2Þ.
ARTICLE IN PRESS
Fig. 2. The incumbent’s optimal time to upgrade.
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297292
The first derivative of the incumbent’s profit withrespect to tn
1 is
d
dtn1
P1ðtn1; _; t
n2Þ ¼
e�rtn1 ðMc
�Mnþ cn
1ðtn1ÞÞ if tn
1 2 ½0; ~t2Þ;
e�rtn1 ðDc
1 � Dn1 þ cn
1ðtn1ÞÞ if tn
1 2 ½~t2;1Þ:
(
(9)
When ~t240, the incumbent starts off as a monopolist butbecomes a duopolist upon the arrival of the entrant. Wehave derived the optimal upgrade time for a pure monopo-list and a pure duopolist, respectively. Should the incum-bent choose its upgrade time as if it were a puremonopolist, or a pure duopolist? Will there be circum-stances in which the incumbent is better off never toupgrade? Proposition 3 below attempts to answer thesequestions.
We assume that Dn1 � Dc
1oMn�Mc . The incumbent,
therefore, has a stronger incentive to upgrade as a puremonopolist than as a pure duopolist. From Proposition 2 wehave learned that a pure monopolist prefers not to upgradeif Mn
�Mcplimt!1cn1ðtÞ. Hence, a pure duopolist will not
have any incentive to upgrade if Mn�Mcplimt!1c
n1ðtÞ.
Consequently, in this case, the incumbent, whose statuschanges from a monopolist to a duopolist, is better offnever to upgrade.
As Mn�Mc surpasses limt!1c
n1ðtÞ, a pure monopolist is
willing to upgrade while a pure duopolist prefers not to doso as long as Dn
1 � Dc1 is below limt!1c
n1ðtÞ. We find that if a
pure monopolist’s optimal upgrade time (i.e., tm1 ) arrives
after the incumbent has actually become a duopolist(i.e., ~t2), the incumbent is better off acting like a duopolist(instead of a monopolist) and prefers not to upgrade at all.
When Dn1 � Dc
1 is sufficiently large to justify the upgradefor a pure duopolist, the incumbent will upgrade. Thequestion now is whether the incumbent should upgrade atthe time that is best for a pure monopolist (i.e., tm
1 ) or at thetime that is best for a pure duopolist (i.e., td
1). From Lemma5 we know that tm
1 ptd1. We find that if td
1 is earlier than theentrant’s arrival (namely when the incumbent is stilla monopolist), then adopting td
1 is dominated by thedecision to go with tm
1 . Similarly, we also conclude that iftm
1 arrives after the incumbent becomes a duopolist, thenthe incumbent should act like a duopolist, as opposed to amonopolist.
The above results are formally established inProposition 3.
Proposition 3. Suppose the incumbent anticipates a compe-
tition with the entrant starting at ~t2. Let Dn1 � Dc
1oMn�Mc.
(i)
When Mn�Mcplimt!1cn1ðtÞ, the incumbent is better
off never to upgrade.
(ii) When Dn1 � Dc1plimt!1c
n1ðtÞoMn
�Mc , the incum-
bent’s optimal decision is either upgrading the product
at tm1 or never upgrading, whichever results in a larger
profit. In particular, if ~t2ptm1 , the incumbent is better off
never upgrading.
(iii) When limt!1cn1ðtÞoDn
1 � Dc1, the incumbent’s optimal
decision is to upgrade the product at either tm1 or td
1,whichever results in a larger profit. In particular, if~t2ptm
1 , the incumbent’s profit is maximized by tn1 ¼ td
1; if
td1p~t2, the incumbent’s profit is maximized by tn
1 ¼ tm1 .
Fig. 2 illustrates the incumbent’s optimal upgrade timing.We fix the values of Mn
�Mc and Dn1 � Dc
1 in the figure.Since the values of tm
1 and td1 are affected by the function
cn1, the values of tm
1 and td1 will change as the value
of limt!1cn1ðtÞ changes. Thus, the straight lines shown in
Fig. 2 that represent the values of tm1 and td
1 are not parallelto the horizontal axis that represents limt!1c
n1ðtÞ. These
two straight lines are upward sloping as limt!1cn1ðtÞ
increases because cn1 decreases with time.
A few observations follow from Fig. 2. As ~t2 getssufficiently large, the incumbent’s optimal upgrade timebecomes either tm
1 or _. This is the same outcome as whenthe entrant stays out of the market (see Section 5.1).
Although the entrant’s arrival time ð~t2Þ is a key elementin the incumbent’s upgrade decision, ~t2 does not alwaysaffect whether the incumbent upgrades or not. Regions (I),(II) and (IV) shown in Fig. 2 cover all the possible events aslong as limt!1c
n1ðtÞoDn
1 � Dc1. The incumbent may choose
different times to upgrade but will upgrade eventually in allthree regions. At the other end of the spectrum whereMn�Mcplimt!1c
n1ðtÞ, the incumbent finds it optimal
never to upgrade regardless of the value of ~t2 (see Region(V)). Therefore, the value of limt!1c
n1ðtÞ in relation to Dn
1 � Dc1
and Mn1 �Mc
1 is the determinant of the incumbent’s upgrade.When it is optimal for the incumbent to upgrade regardless
of ~t2 (see Regions (I), (II) and (IV)), the value of ~t2 affects theincumbent’s choice of the time to upgrade. ~t2 takes smallervalues in Region (IV) than in Region (I). The incumbent’soptimal time to upgrade, however, is earlier in Region (I) thanin Region (IV). Does an earlier arrival of the entrant prompt theincumbent to delay its upgrade? In order to answer thisquestion, we need to analyze how the incumbent’s preferenceof tm
1 changes with ~t2 in Regions (II) and (III).In Region (II) the incumbent’s optimal upgrade time is
either tm1 or td
1, whichever results in a higher present value ofthe incumbent’s total profits. The incumbent’s preferenceof tm
1 increases with the difference: P1ðtm1 ; t
c2; t
n2Þ�
P1ðtd1; t
c2; t
n2Þ. We show in Corollary 2 that this difference
can be expressed as a monotonically increasing function of~t2. Therefore, the anticipated impact of ~t2 can be establishedfor Region (II). A similar pattern can be shown for Region (III).
Corollary 2. Let Dn1 � Dc
1oMn�Mc . In both Regions (II) and
(III) shown in Fig. 2, the incumbent’s preference of tm1 as its
optimal upgrade time increases with ~t2.
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 293
If the incumbent upgrades its product to the nextgeneration, there are only two choices for the optimalupgrade time: tm
1 and td1. Recall from Lemma 5 that tm
1 otd1.
Corollary 2 suggests that an earlier arrival time of theentrant will delay the incumbent’s upgrade time.
Combining Corollary 2 and what has been shown inFig. 2, we conclude that as long as limt!1c
n1ðtÞoDn
1 � Dc1p
Mn�Mc , the incumbent will upgrade its product to the
next generation, and the optimal upgrade time is post-poned from tm
1 to td1 as the arrival time of the entrant
becomes earlier. When Dn1 � Dc
1plimt!1cn1ðtÞpMn
�Mc ,the entrant’s arrival time can determine whether theincumbent should upgrade at all. When the entrantarrives early enough the incumbent will be deterred fromupgrading.
6. Discussions on market leadership
Market leadership refers to a firm’s ability to lead thelaunch of a new generation of a product. In our setting, theincumbent leads the launch of the current generationproduct. Will the incumbent also be the first one to launchthe next generation product? We explore this questionbased on structural results obtained earlier.
In order to understand whether the position of anincumbent provides a firm any edge over its entrantcompetitor, let us consider for the moment firms withequivalent cost and profit margin structures. The incum-bent has already launched the current generation productat time zero. To put the incumbent and the entrant on thesame footing in terms of launch costs, the incumbent’slaunch costs of the next generation product should beregarded as upgrade costs. Equivalent launch costs of thenext generation product imply that Fn
1ðtÞ ¼ Fd2ðDÞ for any
t ¼ D 2 ½0;1Þ. Similarly, we would like to establish anequivalence between the two firms’ gross profit margins.When replacing the current generation product with thenext generation, the entrant’s gross profit margin improvesfrom Dc
2 to Dn2, and the incumbent’s gross profit margin
improves from Dc1 to Dn
1, or from Mc1 to Mn
1. We assume thatDn
2 � Dc2 ¼ Dn
1 � Dc1oMn
�Mc .If the entrant finds it optimal not to launch the current
generation product at all, the incumbent can never be thelaggard in terms of the next generation product. We are,however, more interested in situations where the entrantchooses to launch the next generation product. There aretwo possibilities, either the entrant launches only the nextgeneration product or the entrant launches the currentgeneration and then upgrades to the next generation at alater time.
If the entrant launches the current generation product attcd
2 and upgrades to the next generation d� time units later,Proposition 1 tells us that the optimal time lapse betweenthe two launches (d�) must equate the marginal cost of theupgrade to the marginal benefit for the entrant (i.e.,e�rðtc
2þd�Þcd
2ðd�Þ ¼ e�rðtc
2þd�ÞðDn
2 � Dc2)). Since the two firms
are equivalent in terms of costs and gross profit margins,we know that from the incumbent’s perspective themarginal cost of the upgrade at time d� (i.e., e�rd�cn
1ðd�Þ)
equals the marginal benefit a pure duopolist earns (i.e.,
e�rd� ðDn1 � Dc
1Þ). From the definition of td1, we know that
td1 ¼ d�. The incumbent’s optimal launch time of the next
generation product need not be td1. When both tm
1 and td1
are well defined the incumbent’s optimal launch time iseither tm
1 ptd1 or td
1 (from Proposition 3). Since the entrant’slaunch time of the next generation product is d� ¼ td
1 timeunits after its launch of the current generation product, weconclude that the entrant cannot launch the next genera-tion product earlier than the incumbent does.
If the entrant skips the launch of the current generationproduct and launches the next generation product at t_n
2 ,Proposition 1 tells us that we must have cd
2ð0ÞpDn2 � Dc
2.Given the cost and profit margin equivalence, we know thatcn
1ð0ÞpDn1 � Dc
1, which implies that tm1 ¼ td
1 ¼ 0. Again, theentrant cannot launch the next generation product earlierthan the incumbent does.
Proposition 4 demonstrates the incumbent’s advantagein the race to market leadership under a more generalcondition ðcn
1ðtÞpcd2ðDÞÞ. Note that Fn
1ðtÞ ¼ Fd2ðDÞ is an
extreme case of cn1ðtÞ ¼ cd
2ðDÞ.
Proposition 4. Assume cn1ðtÞpcd
2ðDÞ for any t ¼ D, and
Dn2 � Dc
2 ¼ Dn1 � Dc
1oMn�Mc. If it is optimal for the entrant
to set tn2a_, then the incumbent’s next generation product
appears on the market no later than the entrant’s counterpart.
Proposition 4 shows that for competing firms withequivalent launch costs of the next generation product andgross profit margins, the position of incumbent provides anadvantage as far as market leadership is concerned. In orderto be the incumbent, a firm must have launched the currentgeneration product by time zero. The launch costs of thecurrent generation product are sunk before decisions aboutthe next generation are made at time zero. Sunk costs are nolonger part of the firm’s cost–benefit analysis. Hence, theincumbent has a stronger incentive to launch the nextgeneration product than the entrant, even if both firms haveequivalent cost functions and gross profit margins.
It is likely that the incumbent can launch the nextgeneration product more efficiently than the entrant doesdue to the learning effect. It is also plausible that theincumbent earns more from the product upgrade than theentrant due to consumer loyalty to an enduring brand. Ourresearch implies that if any of the above scenarios occur, theincumbent’s market leadership still holds and is actuallystrengthened. From the entrant’s perspective, market leader-ship is not attainable unless the entrant is superior, either atlaunching products efficiently, or at generating more grossprofit margins. For an established firm with dominatingbrand powers, the latter is a possible and realistic way to takeover market leadership from the incumbent.
7. Conclusion and future research
We analyze firms’ product launch and upgrade timings inan incumbent-vs-entrant setting. Using a stylized model, wecharacterize the optimal time for the incumbent to upgradethe current generation product and the optimal times for theentrant to launch and upgrade a competing product.
Although the entrant firm should not base its entrydecision solely on gross profit margins, we find that small
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297294
profit margins generated by the next generation product arelikely to deter the entrant from joining the competition.Microsoft skipped launching an equivalent product to Sony’sPlayStation and leapfrogged to the next generation product.This is because Microsoft was capable of extracting enoughprofit margins from the next generation product to outweighthe benefit of learning. In general, the entrant need notleapfrog to the launch of the next generation product. As amatter of fact, large profit margins generated by the nextgeneration product do not guarantee the optimality oflaunching the next generation product. If an entrant wouldlike to advance its entry time to the market, it shouldconsider process improvements that can lower the firm’slaunch costs of the current generation product.
The incumbent firm must respond strategically to thearrival of the entrant. The incumbent is better off upgradingearlier when anticipating the late arrival of its competitor.This result explains the speed with which Apple Inc.replaced the iPod Mini with the iPod Nano. As a dominantplayer that is technologically advanced, Apple did notanticipate any significant rivalry any time soon and, hence,preferred to enjoy the (near) monopoly profit marginsgenerated by the iPod Nano earlier rather than later.
Although the incumbent is not automatically the firstone to launch the next generation product, the incumbentis better positioned in the race of market leadership. Havingalready invested in the current generation product allowsthe incumbent to materialize the learning effects. Mean-while, sunk costs need not be factored into future planning.Thus, the incumbent has a cost advantage over the entrantas far as market leadership is concerned. Nevertheless, apowerful entrant with superior market power can over-come the incumbent’s cost advantage.
In the interest of tractability, we assume in this paperthat a firm’s profit margins are determined by thegeneration of the product the firm offers and the competi-tion status of the market, namely monopoly or duopoly.Doing so, however, implies that the two firms areessentially offering their products to separate marketsegments (decided by brand name). One interestingextension for future research is to consider a more generalsetting. For example, the firms’ profit margins can bemodeled as a result of a pricing and market share game likethe one in Filippini (1999). In such a general setting, morestrategic decisions are to be expected. In our currentresearch, the incumbent reacts strategically to the entrant’smarket entry timing, but the entrant’s optimal timings arenot dependent upon the specific generation of the productoffered by the incumbent. When a pricing and market sharegame is directly modeled, both the incumbent and theentrant will have to respond strategically to each other’sactions.
This research can be further extended by relaxing theassumption that each firm has at most a single generationof the product on the market. Moorthy and Png (1992)found examples where firms offered both high-end andlow-end versions of a product in order to capture differentmarket segments. Given our current profit margin setting, afirm would not have any incentive to replace a high-endversion with a low-end one because the low-end versionhas an inferior profit margin. If the firms do not have to
discontinue existing generations/versions of a productwhen introducing other generations/versions, and if thefirms’ profit margins are determined by a pricing andmarket share game, based on what we have learned fromthis research we envision that various introduction timingscould become an equilibrium outcome. Ultimately, thefirms must balance the fixed costs of offering a product anda stream of profit margins reaped from the product.Different combinations of cost and market conditionparameters will generate different points of balance.
Appendix
Proof of Lemma 1. Note that cc2ðtÞ ¼ rFc
2ðtÞ � ðd=dtÞFc2ðtÞ is
continuous and strictly decreasing because Fc2 is continu-
ously differentiable, strictly decreasing, and strictly convex(see Assumption 1a).
If limt!1cc2ðtÞXDc
2 then for any tc2 2 ½0;1Þ we have
�Dc2 þ cc
2ðtc2Þ4� Dc
2 þ limt!1cc2ðtÞX0. It follows that
ðd=dtc2ÞP2ðt
c2; _Þ in Eq. (2) is always positive. That is,
P2ðtc2; _Þ is strictly increasing in tc
2. For any tc2 2 ½0;1Þ we
have P2ðtc2; _Þolimt!1P2ðt; _Þ ¼ 0. Meanwhile, P2ð_; _Þ ¼ 0.
Therefore, tc2 ¼ _ maximizes the entrant’s profit when
limt!1cc2ðtÞXDc
2.
If limt!1cc2ðtÞoDc
2 then the continuity and monotonicity
of cc2 imply that either cc
2ðtc2ÞoDc
2 for any tc2 2 ½0;1Þ or
there exists a unique tc2 2 ½0;1Þ such that cc
2ðtc2Þ ¼ Dc
2.
When cc2ðt
c2ÞoDc
2 for any tc2 2 ½0;1Þ we have
ðd=dtc2ÞP2ðt
c2; _Þo0. That is, P2 is strictly decreasing. Hence,
P2 is maximized at tc2 ¼ 0 ¼min ftjcc
2ðtÞpDc2g ¼ tc_
2 . So, tc_2
is the maximizer of P2 when cc2ðt
c2ÞoDc
2 for any tc2 2 ½0;1Þ.
On the other hand, when there exists a unique tc2 2 ½0;1Þ
such that cc2ðt
c2Þ ¼ Dc
2, we know that cc2ðtc_
2 Þ ¼ Dc2
from the definition of tc_2 . Due to the monotonicity of cc
2,
we have
cc2ðt
c2Þ
4Dc2 if tc
2otc_2 ;
¼ Dc2 if tc
2 ¼ tc_2 ;
oDc2 if tc
24tc_2 ;
8>><>>: and, thus,
d
dtc2
P2ðtc2; _Þ ¼ �Dc
2 þ cc2ðt
c_2 Þ
40 if tc2otc_
2 ;
¼ 0 if tc2 ¼ tc_
2 ;
o0 if tc24tc_
2 :
8>><>>:
Hence, tc_2 is the maximizer of the entrant’s profit when
there exists a unique tc2 2 ½0;1Þ such that cc
2ðtc2Þ ¼ Dc
2. Thiscompletes the proof of the lemma. &
Proof of Lemma 2. The derivations are the same as thosein the proof of Lemma 1 except that the current generationis replaced by the next generation. That is, Fc
2 in the proof ofLemma 1 is replaced by Fn
2, Dc2 is replaced by Dn
2, and tc_2 is
replaced by t_n2 . &
Proof of Lemma 3. Note that cd2ðDÞ ¼ rFd
2ðDÞ � ðd=dDÞFd2ðDÞ
is continuous and strictly decreasing in D because Fd2 is
continuously differentiable, strictly decreasing, and strictlyconvex in D (see Assumption 1b). Hence, Dc
2 � Dn2 þcd
2ðDÞ isalso continuous and strictly decreasing in D. The rest of the
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 295
derivations are the same as those in the proof of Lemma 1except that Fc
2ðtc2Þ is replaced by Fd
2ðDÞ, Dc2 by Dn
2 � Dc2, and
cc2ðt
c2Þ by cd
2ðDÞ. &
Proof of Lemma 4. Lemma 3shows that d� is well definedbecause limD!1c
d2ðDÞoDn
2 � Dc2.
�
When d� ¼ 0, based on Lemma 3 we note thatP2ðtc2; t
c2Þ4P2ðt
c2; t
c2 þDÞ for any D 2 ð0;1Þ. Meanwhile,
P2ðtc2; t
c2Þ ¼ limg!0P2ðt
c2; t
c2 þ gÞ
¼ limg!0
Z tc2þg
tc2
Dc2 e�rs dsþ
Z 1tc
2þgDn
2 e�rtn2 � Fc
2ðtc2Þ e�rtc
2 � Fd2ðgÞ e
�rðtc2þgÞ
( )
¼
Z 1tc
2
Dn2 e�rtn
2 � ðFc2ðt
c2Þ þ Fd
2ð0ÞÞ e�rtc
2
¼
Z 1tc
2
Dn2 e�rs ds� Fn
2ðtc2Þ e�rtc
2 (based on Assumption 2)
¼ P2ð_; tc2Þ. (10)
We have P2ð_; tc2Þ ¼ P2ðt
c2; t
c2Þ4P2ðt
c2; t
n2Þ for any tc
2 2
½0;1Þ and any tn2 2 ðt
c2;1Þ. Therefore, any decision
to launch both generations of the product (i.e.,ðtc
2; tn2Þ 2 ½0;1Þ � ½t
c2;1Þ) is dominated by the maximi-
zer of P2ð_; tn2Þ over the region: tn
2 2 ½0;1Þ [ f_g. Thiscompletes the proof of part (i) of the lemma.
� When d�40 and limt!1cc2ðtÞXDc
2 � e�rd� ðd=dDÞFd2ðd�Þ,
for any tc2 2 ½0;1Þ we have e�rtc
2 ð�Dc2 þ cc
2ðtc2Þ þ
e�rd� ðd=dDÞFd2ðd�ÞÞ40 because cc
2ðtÞ is strictly decreas-ing in t for any t 2 ½0;1Þ. Consequently, for anytc
2 2 ½0;1Þ, ðd=dtc2ÞP2ðt
c2; t
c2 þ d�Þ40 and, hence,
P2ðtc2; t
c2 þ d�Þ is strictly increasing in tc
2. The continuityof P2ðt
c2; t
c2 þ d�Þ further implies that limt!1P2ðt; t þ
d�Þ4P2ðtc2; t
c2 þ d�Þ for any tc
2 2 ½0;1Þ.Lemma 3 showsthat P2ðt
c2; t
c2 þ d�Þ4P2ðt
c2; t
n2Þ for ðtc
2; tn2Þ 2 ½0;1Þ �
½tc2;1Þ as long as tn
2atc2 þ d�. Meanwhile,
P2ð_; _Þ ¼ 0 ¼ limt!1P2ðt; t þ d�Þ. Hence, any decisionto launch both generations (i.e., ðtc
2; tn2Þ 2 ½0;1Þ� ½t
c2;1Þ)
is dominated by ðtc2; t
n2Þ ¼ ð_; _Þ. This completes the proof
of part (ii) of the lemma.�
�
When d�40 and limt!1cc2ðtÞoDc2 � e�rd ðd=dDÞFd2ðd�Þ,
we further examine two possibilities: cc2ð0ÞoDc
2 � e�rd�
ðd=dDÞFd2ðd�Þ, or cc
2ð0ÞX Dc2 � e�rd� ðd=dDÞFd
2ðd�Þ.
If cc2ð0ÞoDc
2 � e�rd� ðd=dDÞFd2ðd�Þ, then �Dc
2 þ cc2ðt
c2Þþ
e�rd� ðd=dDÞFd2ðd�Þo0 for any tc
2 2 ½0;1Þ because cc2ðtÞ
is strictly decreasing in t. Hence, we have ðd=dtc2Þ
P2ðtc2; t
c2þd
�Þo0. For any tc
2 2 ð0;1Þ we have P2ð0; d�Þ4
P2ðtc2; t
c2 þ d�Þ4limt!1P2ðt; t þ d�Þ ¼ P2ð_; _Þ. Mean-
while, tcd2 ¼ 0 when cc
2ð0ÞoDc2 � e�rd� ðd=dDÞFd
2ðd�Þ. So,
P2ðtcd2 ; tcd
2 þ d�Þ ¼ P2ð0; d�Þ4P2ðt
c2; t
c2 þ d�Þ for any
tc2 2 ð0;1Þ [ f_g.
If cc2ð0ÞXDc
2 � e�rd� ðd=dDÞFd2ðd�Þ, then, by the definition
of tcd2 in the lemma and the monotonicity of cc
2ðtÞ, we have
cc2ðt
cd2 Þ
4Dc2 � e�rd� d
dD Fd2ðd�Þ if tc
2otcd2 ;
¼ Dc2 � e�rd� d
dD Fd2ðd�Þ if tc
2 ¼ tcd2 ;
oDc2 � e�rd� d
dD Fd2ðd�Þ if tc
24tcd2 :
8>>><>>>:
Applying Eq. (5) we have ðd=dtc2ÞP2ðt
c2; t
c2 þ d�Þ is
positive if tc2otcd
2 ; negative if tc24tcd
2 ; and zero otherwise.
Note that P2ð_; _Þ ¼ limt!1P2ðt; t þ d�Þ. Therefore,for any ðtc
2; tn2Þ 2 f½0;1Þ� ½t
c2;1Þg [ fð_; _Þg, we have
P2ðtcd2 ; tcd
2 þ d�Þ4P2ðtc2; t
n2Þ as long as ðtc
2; tn2Þa
ðtcd2 ; tcd
2 þ d�Þ.
This completes the proof of the lemma. &
Proof of Proposition 1. We examine three cases based onn c
the value of D2 � D2.�
When Dn2 � Dc2plimD!1cd2ðDÞ, P2ðt
c2; t
n2ÞoP2ðt
c2; _Þ for
any ðtc2; t
n2Þ 2 ½0;1Þ � ½t
c2;1Þ from Lemma 3. In particular,
P2ðtc2; _Þ4P2ðt
c2; t
c2Þ ¼ P2ð_; t
c2Þ with the last equality
following from Eq. (10). Now the entrant’s profitmaximization is reduced to maximizing P2ðt
c2; _Þ over
the region ½0;1Þ [ f_g for tc2. From Lemma 1 we know
that P2ðtc2; _Þ is maximized by tc
2 ¼ tc_2 if limt!1c
c2ðtÞo
Dc2, and by tc
2 ¼ _ otherwise.
� When cd2ð0ÞpDn2 � Dc
2, we have d� ¼ 0 from Lemma 3.Applying Lemma 4, we know that any ðtc
2; tn2Þ 2 ½0;1Þ �
½tc2;1Þ is dominated by the maximizer of P2ð_; t
n2Þ over
the region ½0;1Þ [ f_g for tn2. The equality d� ¼ 0 also
implies that P2ðtc2; t
c2Þ4P2ðt
c2; _Þ (from Lemma 3).
Now, the entrant’s profit maximization is reducedto maximizing P2ð_; t
n2Þ over the region ½0;1Þ [ f_g
for tn2. From Lemma 2 we know that P2ð_; t
n2Þ is
maximized by tn2 ¼ t_n
2 if limt!1cn2ðtÞoDn
2, andbytn
2 ¼ _ otherwise.
� When limD!1cd2ðDÞoDn
2 � Dc2ocd
2ð0Þ, we have d�40from Lemma 3. That is, P2ðt
c2; t
c2 þ d�Þ4P2ðt
c2; t
c2Þ ¼
P2ð_; tc2Þ and P2ðt
c2; t
c2 þ d�Þ4P2ðt
c2; _Þ for any tc
2 2
½0;1Þ. So, the entrant’s profit is maximized by eitherðtc
2; tn2Þ ¼ ð_; _Þ or the maximizer of P2ðt
c2; t
c2 þ d�Þ over
the region: tc2 2 ½0;1Þ. Applying Lemma 4 we conclude
that the entrant’s profit is maximized by ðtc2; t
n2Þ ¼
ðtcd2 ; tcd
2 þ d�Þ if limt!1cc2ðtÞoDc
2 � e�rd� ðd=dDÞFd2ðd�Þ,
and by ðtc2; t
n2Þ ¼ ð_; _Þ otherwise.
Combining the above three cases, we obtain the neces-
sary and sufficient conditions for the entrant’s optimal
timing decisions.
Below we show that tcd2 ptc_
2 is satisfied. The condition
that limt!1cc2ðtÞoDc
2 implies that tc_2 is well defined (from
Lemma 1). Note that �e�rd� ðd=dDÞFd2ðd�Þ is positive because
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297296
Fd2 is strictly decreasing in D (Assumption 1b). If tc_
2 40, from
the definition of tc_2 in Lemma 1, we have cc
2ðtc_2 Þ ¼
Dc2oDc
2 � e�rd� ðd=dDÞFd2ðd�Þ. Since cc
2 is continuous, there
exists an �40 such that cc2ðtc_
2 � �ÞoDc2 � e�rd� ðd=dDÞ
Fd2ðd�Þ.We also have cc
2ðtcd2 Þ ¼ minftjcc
2ðtÞpDc2 � e�rd� ðd=dDÞ
Fd2ðd�Þg from the definition of tcd
2 in Lemma 4. Since cc2 is
strictly decreasing, we know that tcd2 otc_
2 if tc_2 40. Similarly,
we can show that tcd2 ¼ 0 if tc_
2 ¼ 0. Hence, we have tcd2 ptc_
2 .
This completes the proof of the proposition. &
Proof of Corollary 1. There are three possible entry times:tcd
2 , tc_2 and t_n
2 . From Lemmas 1, 2 and 4, we know thatdetermining these entry times is essentially comparing thefollowing pairs: cc
2ðtÞ vs Dc2, cn
2ðtÞ vs Dn2, and cc
2ðtÞ vsDc
2 � e�rd� ðd=dDÞFd2ðd�Þ. Meanwhile, cn
2ðtÞ ¼ cc2ðtÞ þ rFd
2ð0Þbecause Fn
2ðtÞ ¼ Fd2ð0Þ þ Fc
2ðtÞ from Assumption 2, cn2ðtÞ ¼
rFn2ðtÞ � ðd=dtÞFn
2ðtÞ from Lemma 2, and cc2ðtÞ ¼ rFc
2ðtÞ�
ðd=dtÞFc2ðtÞ from Lemma 1. The comparisons break down
to cc2ðtÞ vs Dc
2, Dn2 � rFd
2ð0Þ, and Dc2 � e�rd� ðd=dDÞFd
2ðd�Þ. By
replacing Fc2ðtÞ with Fc
2ðtÞ � �, we are replacing cc2ðtÞ with
cc2ðtÞ � r�. Given the decreasing trend of cc
2ðtÞ, we knowthat replacing cc
t ðtÞ with cc2ðtÞ � r� results in a smaller tc_
2 ,t_n
2 and tcd2 if these times are well defined.
From Proposition 1, we know that as long as
limt!1cc2ðtÞoDc
2, limt!1cn2ðtÞoDn
2, and limt!1cc2ðtÞoDc
2 �
e�rd� ðd=dDÞFd2ðd�Þ all hold, the entrant will enter the market
regardless of the value of Dn2 � Dc
2. Having the three
inequalities hold is equivalent to having the inequality
limt!1cc2ðtÞominfDc
2;Dn2 � rFd
2ð0Þg hold. Replacing Fct ðtÞ
with Fc2ðtÞ � � also means replacing limt!1c
c2ðtÞ with
limt!1cc2ðtÞ � r�. It is clear that if a pair of Dc
2 and Dn2
satisfies limt!1cc2ðtÞominfDc
2;Dn2 � rFd
2ð0Þg, they must also
satisfy limt!1cc2ðtÞ � r�ominfDc
2;Dn2 � rFd
2ð0Þg, but not vice
versa. This completes the proof of the corollary. &
Proof of Proposition 2. Since cn1ðtÞ ¼ rFn
1ðtÞ � ðd=dtÞFn1ðtÞ is
continuous and strictly decreasing (from Assumption 1a),we have limt!1c
n1ðtÞocn
1ðtn1Þocn
1ð0Þ for any tn1 2 ð0;1Þ.
When limt!1cn1ðtÞXMn
�Mc , cn1ðt
n1Þ4Mn
�Mc for any
tn1 2 ½0;1Þ. That is, the first order derivative in Eq. (7) is
positive for any tn1 2 ½0;1Þ. Therefore, we have P1ð_; _; _Þ ¼
limt!1P1ðt; _; _Þ4P1ðtn1; _; _Þ for any tn
1 2 ½0;1Þ. Hence, the
incumbent is better off never upgrading. This completes the
proof of part (i) of the proposition.
When cn1ð0ÞpMn
�Mc , we know from the first order
derivative in Eq. (7) that P1ð0; _; _Þ4P1ðtn1; _; _Þ4
limt!1P1ðt; _; _Þ ¼ P1ð_; _; _Þ for any tn1 2 ð0;1Þ. That is,
tn1 ¼ 0 maximizes the incumbent’s profit. Since tm
1 ¼ 0 here,
we conclude that tn1 ¼ tm
1 is the maximizer of the incum-
bent’s profit.
When limt!1cn1ðtÞoMn
�Mcocn1ð0Þ, we know from the
definition of tm1 that tm
1 40. Furthermore, we know
cn1ðt
n1Þ
4Mn�Mc if tn
1otm1 ;
¼ Mn�Mc if tn
1 ¼ tm1 ;
oMn�Mc if tn
14tm1 ;
8>><>>: and, thus,
d
dtn1
P1ðtn1; _; _Þ
40 if tn1otm
1 ;
¼ 0 if tn1 ¼ tm
1 ;
o0 if tn14tm
1 :
8>><>>:
We, therefore, conclude that tn1 ¼ tm
1 is the uniquemaximizer of the incumbent’s profit when limt!1c
n1ðtÞo
Mn�Mc . This completes the proof of part (ii) of the
proposition. &
Proof of Lemma 5. When limt!1cn1ðtÞoDn
1 � Dc1o
Mn�Mc , td
1 and tm1 are well defined. If td
1 ¼ 0, thencn
1ð0ÞpDn1 � Dc
1oMn�Mc . In this case, tm
1 ¼ 0 ¼ td1. On
the other hand, if td140, then cn
1ðtd1Þ ¼ Dn
1 � Dc1oMn
�Mc.Given the continuity and monotonicity of cn
1, we know thatthere exists an �40 such that cn
1ðtd1 � �ÞpMn
�Mc . There-fore, tm
1 ¼ minftjcn1ðtÞpMn
�Mcgptd
1 � �otd1. This com-
pletes the proof of the lemma. &
Proof of Proposition 3. Consider scenarios where ðtc2; t
n2Þa
ð_; _Þ. Namely ~t2 is well defined. From Dn1 � Dc
1oMn�Mc we
have tm1 ptd
1 (see Lemma 5).
�
Consider the case where Mn�Mcplimt!1cn1ðtÞ.
d
dtn1
P1ðtn1; t
c2; t
n2Þ ¼
e�rtn1 ðMc
�Mnþ cn
1ðtn1ÞÞ40 if tn
1 2 ½0; ~t2Þ;
e�rtn1 ðDc
1 � Dn1 þcn
1ðtn1ÞÞ40 if tn
1 2 ð~t2;1Þ:
(
The above inequalities imply that tn1 ¼
~t2 maximizesP1 on ½0; ~t2�, and P1ð_; t
c2; t
n2Þ ¼ limt!1P1ðt; t
c2; t
n2Þ4
P1ðtn1; t
c2; t
n2Þ for tn
1 2 ½~t2;1Þ. We can then conclude that
tn1 ¼ _ is the unique maximizer of the incumbent’s profit
when Mn�Mcplimt!1c
n1ðtÞ. This completes the proof
of part (i) of the proposition.
� Consider the case where Dn1 � Dc1plimt!1c
n1ðtÞo
Mn�Mc . We use the notation ^ to denote the operation
of choosing a smaller number, and the notation _ todenote the operation of choosing a larger number.
d
dtn1
P1ðtn1; t
c2; t
n2Þ ¼
e�rtn1 ðMc
�Mnþcn
1ðtn1ÞÞ40 if tn
1 2 ½0; tm1 ^
~t2Þ;
e�rtn1 ðMc
�Mnþcn
1ðtn1ÞÞ ¼ 0 if tn
1 ¼ tm1 o~t2;
e�rtn1 ðMc
�Mnþcn
1ðtn1ÞÞo0 if tn
1 2 ðtm1 ^
~t2; ~t2Þ;
e�rtn1 ðDc
1 � Dn1 þcn
1ðtn1ÞÞ40 if tn
1 2 ð~t2;1Þ:
8>>>>><>>>>>:
Therefore, P1ð_; tc2; t
n2Þ ¼ limt!1P1ðt; t
c2; t
n2Þ4P1
ðtn1; t
c2; t
n2Þ for tn
1 2 ½~t2;1Þ. Furthermore, if ~t2ptm
1 then tn1 ¼
~t2 maximizes P1 on ½0; ~t2�. In this case, tn1 ¼ _ is the
overall maximizer of the incumbent’s profit.If tm
1 o~t2 then tn1 ¼ tm
1 o~t2 maximizes P1 on ½0; ~t2�. In thelatter case, the incumbent’s profit is maximized by eithertn
1 ¼ tm1 or tn
1 ¼ _.This completes the proof of part (ii) ofthe proposition.
� Consider the case where limt!1cn1ðtÞoDn
1 � Dc1.
d
dtn1
P1ðtn1; t
c2; t
n2Þ ¼
e�rtn1 ðMc
�Mnþcn
1ðtn1ÞÞ40 if tn
1 2 ½0; tm1 ^
~t2Þ;
e�rtn1 ðMc
�Mnþcn
1ðtn1ÞÞ ¼ 0 if tn
1 ¼ tm1 o~t2;
e�rtn1 ðMc
�Mnþcn
1ðtn1ÞÞo0 if tn
1 2 ðtm1 ^
~t2; ~t2Þ;
e�rtn1 ðDc
1 � Dn1 þ cn
1ðtn1ÞÞ40 if tn
1 2 ð~t2; ~t2 _ td
1Þ;
e�rtn1 ðDc
1 � Dn1 þ cn
1ðtn1ÞÞ ¼ 0 if tn
1 ¼ td14~t2;
e�rtn1 ðDc
1 � Dn1 þ cn
1ðtn1ÞÞo0 if tn
1 2 ð~t2 _ td
1;1Þ:
8>>>>>>>>>><>>>>>>>>>>:
ARTICLE IN PRESS
Y. Li, Y.H. Jin / Int. J. Production Economics 119 (2009) 284–297 297
Recall that tm1 ptd
1. We examine three subcases below.If ~t2ptm
1 , tn1 ¼
~t2 maximizes P1 on ½0; ~t2�, tn1 ¼ td
1
maximizes ½~t2;1Þ on ½~t2;1Þ, and P1ðtd1; t
c2; t
n2Þ4
P1ðtn1; t
c2; t
n2Þ4limt!1P1ðt; t
c2; t
n2Þ ¼ P1ð_; t
c2; t
n2Þ for
tn14td
1. Firm 1’s profit is then maximized by tn1 ¼ td
1.If td
1p~t2,tn1 ¼ tm
1 maximizes P1 on ½0; ~t2�, and P2ð~t2;
tc2; t
n2Þ4P2ðt
n1; t
c2; t
n2Þ4limt!1P1ðt; t
c2; t
n2Þ ¼P1ð_; t
c2; t
n2Þ for
any tn14~t2. Firm 1’s profit is then maximized by tn
1 ¼ tm1 .
If tm1 o~t2otd
1, tn1 ¼ tm
1 maximizes P1 on ½0; ~t2�, tn1 ¼ td
1
maximizes ½~t2;1Þ on ½~t2;1Þ,and P1ðtd1; t
c2; t
n2Þ4P1ðt
n1;
tc2; t
n2Þ4limt!1P1ðt; t
c2; t
n2Þ ¼ P1ð_; t
c2; t
n2Þ for tn
14td1. In
this case, either tn1 ¼ tm
1 or tn1 ¼ td
1 maximizes theincumbent’s profit.
This completes the proof of the proposition. &
Proof of Corollary 2. When the entrant’s arrival is certain(i.e., ~t2 is well defined), the entrant’s decisions affect theincumbent’s profit through ~t2. We, therefore, express theincumbent’s profit as a function of tn
1 and ~t2.
On Region (II), ðtn1Þ
opt is either tm1 or td
1. We need to show
that the difference, P1ðtm1 ;~t2Þ �P1ðtd
1;~t2Þ, increases with
~t2. We have tm1 o~t2otd
1 on Region (II). So,
P1ðtm1 ; ~t2Þ ¼
Z tm1
0Mc e�rs dsþ
Z ~t2
tm1
Mn e�rs ds
þ
Z 1~t2
Dn1 e�rs ds� Fn
1ðtm1 Þ e
�rtm1
and
P1ðtd1; ~t2Þ ¼
Z ~t2
0Mc e�rs dsþ
Z td1
~t2
Dc1 e�rs ds
þ
Z 1td
1
Dn1 e�rs ds� Fn
1ðtd1Þ e�rtd
1 .
Consequently,
P1ðtm1 ; ~t2Þ �P1ðtd
1; ~t2Þ ¼
Z ~t2
tm1
ðMn�Mc
Þ e�rs ds
þ
Z td1
~t2
ðDn1 � Dc
1Þ e�rs dsþ xðtm
1 ; td1Þ
¼1
re�r~t2 ½Dn
1 � Dc1 � ðM
n�Mc
Þ� þ yðtm1 ; t
d1Þ,
where both xðtm1 ; td
1Þ and yðtm1 ; td
1Þ are functions indepen-
dent of ~t2. Since Dn1 � Dc
1oMn�Mc , we know that the larger
~t2 is, the larger P1ðtm1 ;~t2Þ �P1ðtd
1;~t2Þ is, the more the
incumbent prefers the upgrade time tm1 .
Similarly, on Region (III) we can show that P1ðtm1 ;~t2Þ �
P1ð_; ~t2Þ increases with ~t2. Since ðtn1Þ
opt is either tm1 or _ on
Region (III), the proof of the corollary is completed. &
Proof of Proposition 4. There are two cases in which theentrant launches the next generation product: ðtc
2; tn2Þ
opt¼
ð_; t_n2 Þ or ðtc
2; tn2Þ
opt¼ ðtcd
2 ; tcd2 þ d�Þ.
If ðtc2; t
n2Þ
opt¼ ð_; t_n
2 Þ then cd2ð0ÞpDn
2 � Dc2 and limt!1
cn2ðtÞoDn
2 from Proposition 1. So, we have cn1ð0Þpcd
2ð0ÞpDn
2 � Dc2 ¼ Dn
1 � Dc1oMn
�Mc .
By the definition of tm1 and td
1, we know that tm1 ¼ td
1 ¼ 0
and ðtn1Þ
opt¼ 0 in this case. Therefore, the entrant’s launch
time of the next generation product t_n2 cannot be earlier
than the incumbent’s launch time of the same generation
product.
Ifðtc2; t
n2Þ
opt¼ ðtcd
2 ; tcd2 þ d�Þ then limD!1c
d2ðDÞoDn
2 � Dc2o
cd2ð0Þ and limt!1c
c2ðtÞoDc
2 � e�rd� ðd=dDÞFd2ðd�Þ from Propo-
sition 1. So, we have cn1ðd�Þpcd
2ðd�Þ ¼ Dn
2 � Dc2 ¼
Dn1 � Dc
1oMn�Mc.
By the definition of tm1 and td
1, we know that tm1 ptd
1 ¼ d�
and the incumbent’s optimal launch time of the next
generation product is either tm1 or td
1. So, the entrant’s
launch time of the next generation product: tcd2 þ d� cannot
be earlier than the incumbent’s launch time of the same
generation product. This completes the proof of the
proposition. &
References
Bohlmann, J., Golder, P., Mitra, D., 2002. Deconstructing the pioneer’s
advantage: examining vintage effects and consumer valuations of
quality and variety. Management Science 48, 1175–1195.Boulding, W., Christen, M., 2003. Sustainable pioneering advantage?
Profit implications of market entry order. Marketing Science 22,
371–392.Chung, K., Tsou, C., 1998. Analysis and algorithm for the optimal
investment times of new manufacturing technologies in a duopoly.
European Journal of Operational Research 109, 632–645.Doraszelski, U., 2004. Innovations, improvements, and the optimal
adoption of new technologies. Journal of Economic Dynamics &
Control 28, 1461–1480.Farzin, Y., Huisman, K., Kort, P., 1998. Optimal timing of technology
adoption. Journal of Economic Dynamics & Control 22, 779–799.Filippini, L., 1999. Leapfrogging in a vertical product differentiation model.
International Journal of the Economics of Business 6, 245–256.Fudenberg, D., Tirole, J., 1985. Preemption and rent equalization in the
adoption of new technology. Review of Economic Studies LII, 383–401.Hammonds, K., 1990. How a $4 razor ends up costing $300 million.
Business Week January 29, 62–63.Hoppe, H., 2002. The timing of new technology adoption: theoretical
models and empirical evidence. The Manchester School 70, 56–76.Huisman, K., Kort, P., 2003. Strategic investment in technological
innovations. European Journal of Operational Research 144, 209–223.Jensen, R., 1982. Adoption and diffusion of an innovation of uncertain
profitability. Journal of Economic Theory 27, 182–193.Jensen, R., 1988. Information cost and innovation adoption policies.
Management Science 34, 230–239.Kyle, M., 2006. The role of firm characteristics in pharmaceutical product
launches. The Rand Journal of Economics 37, 602–618.Lieberman, M., Montgomery, D., 1998. First-mover (dis)advantages: retro-
spective and link with the resource-based view. Strategic Management
Journal 19, 1111–1125.Lin, P., Saggi, K., 2002. Timing of entry under externalities. Journal of
Economics 75, 211–225.Mamer, J., McCardle, K., 1987. Uncertainty, competition, and the adoption
of new technology. Management Science 33, 161–177.McCardle, K., 1985. Information acquisition and the adoption of new
technology. Management Science 31, 1372–1389.Moorthy, K., Png, I., 1992. Market segmentation, cannibalization, and the
timing of product introductions. Management Science 38, 345–359.Narasimhan, C., Zhang, Z., 2000. Market entry strategy under firm
heterogeneity and asymmetric payoffs. Marketing Science 19,
313–327.Ofek, E., Sarvary, M., 2003. R&D, marketing, and the success of next-
generation products. Marketing Science 22, 355–370.Rahman, A., Loulou, R., 2001. Technology acquisition with technological
progress: effects of expectations, rivalry and uncertainty. European
Journal of Operational Research 129, 159–185.Reinganum, J., 1981. On the diffusion of new technology: a game theoretic
approach. Review of Economic Studies 48, 395–405.Souza, G., 2004. Product introduction decisions in a duopoly. European
Journal of Operational Research 152, 745–757.
