Innovation Threats and Strategic Responses in Oligopoly ...

Innovation Threats and Strategic Responses in

Oligopoly Markets∗

Herbert Dawid†, Michael Kopel‡, Peter M. Kort§

Abstract

This paper deals with the strategic reaction of firms to competitivethreats stemming from newly developed products of current competitors.Due to the fact that product innovation projects go through multiple timeconsuming stages with multiple continuation/termination decisions, com-petitors can react to the threat before the new product is introduced andthereby may prevent or facilitate the product introduction. We consider aquantity-setting duopoly model where Firm 1 can start a two-stage prod-uct innovation project for obtaining a horizontally and vertically differen-tiated product. In-between the two stages Firm 2 can react by investingin cost-reducing process innovation. We find that under weak verticaldifferentiation Firm 2 wants Firm 1 to innovate. Horizontal differenti-ation softens competition and Firm 2 over-invests in process innovationto induce Firm 1 to launch the new product. Second, under strong verti-cal differentiation Firm 1 starts the product innovation project –triggeringunder-investment by Firm 2 – but never finishes it. The under-investmentleads to higher production costs for Firm 2, which induces Firm 1 not toinnovate. Third, under very strong vertical differentiation Firm 2 pre-vents a launch of the new product by over-investing in process innovation.Due to the strong decrease in production costs Firm 2 captures such a bigmarket share that Firm 1 will not introduce the new product. In such ascenario the situation of Firm 1 in the old market also has become worse,and therefore the option to complete the innovation, which is created byinitiating the first stage of the innovation project, has a negative value forFirm 1.

Keywords: product innovation, strategic response, multi-stage innovation, op-tion value

JEL Classification: L13, O31

∗We thank Thomas Ehrmann for helpful comments on an earlier draft of this paper. Fi-nancial support from the German Science Foundation (DFG) under grant GRK1134/1: Inter-national Research and Training Group ’Economic Behavior and Interaction Models (EBIM)’is gratefully acknowledged.

†Department of Business Administration and Economics and Institute of MathematicalEconomics, Bielefeld University, Germany, email: [email protected]

‡Institute of Management Science, Vienna University of Technology, Austria§Department of Econometrics and Operations Research and CentER, Tilburg University,

The Netherlands

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1 Introduction

Managers are frequently confronted with competitive threats stemming fromnewly developed products of competitors.1 Empirical studies demonstrate thatthe introduction of new products has a negative wealth effect on industry rivals(see e.g. Chen et al. (2005)), and as a consequence, competitive reactions to newproduct entry can be anticipated (Hultink and Langerak (2002), Debruyne et al.(2002)). Kuester et al. (1999) write ”Threatening moves, such as new productintroductions [...], have a potentially negative impact on profitability of otherplayers in the industry...”, and from this they conclude that ”...countermovesmust be expected.” Finding appropriate reactions to such threats is on the onehand key to defending market shares and sustaining profits, and on the otherhand will determine the success of newly developed products.

A significant amount of research has been devoted to study the relationbetween new product development strategies and the competitors’ defense moves(see the survey by Gatignon and Soberman (2002)). The game-theoretic modelsof IO and marketing typically predict aggressive behavior of the incumbentfirms. In order to avoid the entry of new firms or the introduction of a newlydeveloped rival product, incumbents e.g. invest in excess capacity or engagein aggressive price reductions for their existing products (see for example thesurvey by Neven (1989)). In the terminology of Fudenberg and Tirole (1984)these results suggest that the best way to discourage (or lessen the effects ofa) new product entry is to choose a top-dog strategy. However, there is littleempirical support for limit pricing strategies2 and excess capacity investments;instead many firms adjust their efforts in R&D and advertising (see Thomas(1999), Liebermann (1987a, b), Smiley (1988), Singh et al. (1998), Geroski(1995), and Chang and Tang (2001)). Once their business is threatened by newproduct entry, incumbents are often encouraged to increase efficiency and cutcosts by introducing new processes and technologies (Geroski (1995), Singh et al(1998)). Empirical studies on the determinants and the intensity, breadth, anddirection of various defense strategies also show that aggressive reactions to newproduct entry are less often observed in practice than is predicted by existinggame-theoretic models, and that passive or accommodating moves are quitecommon (Thomas (1999), Kuester et al. (1999), Robinson (1988), Gatignonand Soberman (2002), Gatignon et al. (1989)).

Furthermore, the focus of the entry-deterrence literature on entry by firmswhich are new to the industry is in contrast with the observation that incum-bent firms believe that existing rivals are the main source of new product entry(Smiley (1988)). New product introduction by existing rivals has been studiedin numerous papers dealing with innovation incentives in oligopolistic markets,

1A recent study by the product development and management association (PDMA) showsthat the best performing firms in the US generate almost 50% of sales and profits from newproducts and that new product sales are 28% of total company sales (Adams and Boike(2004)).

2In a recent contribution, Simon (2005) examines pricing responses in more detail andprovides conditions under which price cutting can be expected in the US consumer magazinemarket.

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but this research assumes that any innovation project that is started is com-pleted and therefore deals with strategic reactions to actual introduction of newproducts rather than with the reaction to the threat of a product innovation ofa competitor (e.g. Athey and Schmutzler (1995), Lin and Saggi (2002), Boone(2000), Symeonidis (2003), Rosenkranz (2003)). Taken together, these argu-ments show that there is a gap between theoretical predictions and empiricalevidence and that existing game-theoretic models dealing with new product en-try and competitive reactions are not sufficiently rich to capture some of themain aspects of these reactions in real world industries.

In this paper we make a contribution to close this gap. Our aim is to char-acterize under which circumstances aggressive or accommodating measures areoptimal reactions to threats of new product introduction in markets with exist-ing competition. Furthermore, we explore how the anticipation of such reactionsinfluences the incentives to initiate a product innovation project. We consider aduopoly market where both competitors currently offer a homogeneous productand one of the firms (Firm 1) might try to gain an advantage by developing anew horizontally and vertically differentiated product. The competitor (Firm2) reacts by adjusting its cost-reducing process innovation efforts for the oldproduct. We explicitly take into account that product innovation projects aretime consuming and have multiple stages which allows the competitor to re-act to the threat before the decision about the market introduction of the newproduct has been taken. Although the multi-stage nature of product innovationprojects is well documented (see e.g. the Stage-Gate model in Cooper (2001))and in many industries only a small percentage of product innovation projectsis carried through until market introduction3, the industrial organization liter-ature so far has largely neglected this sequential nature (see our brief literaturereview below). Given our agenda to examine the strategic reaction to threatsof new product introduction in markets with existing oligopolistic competition,the explicit consideration of the distinction between the stage where the optionto introduce a new product is created and the stage where the product is ac-tually introduced to the market seems essential. For this reason we examine aframework where Firm 1 can carry out a two-stage product innovation project,while in-between the two stages Firm 2 can react by investing in cost-reducingprocess innovation.

It turns out that by taking the sequential decision making process into ac-count, new strategic effects are revealed. Firm 2’s investment in process inno-vation clearly influences the incentive of Firm 1 to complete the product inno-

3Empirical data reveal the importance of multi-stage decision-making procedures in R&D-intensive firms: the share of innovative products which firms launch after development istypically rather small. Mansfield and Wagner (1975) and Mansfield et al. (1977) estimate theprobability of commercialization of R&D projects given technical completion in the chemical,drug, petroleum and electronics industries. The average probability in the sample is 65%, butvalues differ significantly between firms ranging from 12% to more than 90%. Cooper (2001,p. 11) shows that the share of new product ideas which actually lead to a market launch issomewhere between 10% and 22%. Astebro (2003) and Astebro and Simons (2003) employdata from the Canadian Innovation Centre to show that only 7% of the inventions recordedfrom independent inventors lead to a successful commercialization.

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vation process. There is a trade-off between the following two effects. First,higher process innovation investments make Firm 2 a stronger competitor inthe market for the existing product, which increases the incentive for Firm 1 tocomplete the product innovation project and to launch the new product. Werefer to this effect as the competition effect. Second, the fact that higher processinnovation investments make Firm 2 a stronger competitor also implies that itis less tempting for Firm 1 to incur any investment costs in this market, sinceeven after Firm 1 has launched a new product, the firms still operate in thesame market. The only change occurs with regard to the degree of differentia-tion of the products. Hence, Firm 1’s incentives to invest in completion of theproduct innovation project is reduced by higher process innovation investments.We denote this second effect as the size effect. Firm 2 will thus try to influenceFirm 1’s decision to complete the product innovation project by (over- or under-)investing in process R&D4. Observe that it is a priori not clear in what wayFirm 2 might want to influence Firm 1. Due to the horizontal differentiationeffect, Firm 2 wants Firm 1 to launch the new product. On the other hand, dueto the vertical differentiation effect Firm 2 would prefer the opposite. There-fore, in this way the characteristics of the new product determine if aggressiveor defensive moves are observed in the industry.

To provide more details on the strategic effects occurring in our setup, wesummarize our main results. In the case of weak vertical differentiation thehorizontal differentiation effect dominates. This implies that Firm 2 wants Firm1 to complete its product innovation project. Firm 2 accomplishes this byover-investing in process innovation, thus making the market for the existing(homogeneous) product less profitable for Firm 1. In terms of the Fudenberg-Tirole taxonomy of business strategies (Fudenberg and Tirole (1984)), Firm 2selects a top dog strategy. Under strong vertical differentiation Firm 1 starts theproject but never finishes it. Although it has to incur the start-up investmentswithout ever having the possibility to take advantage by selling a new product,it is still optimal to make this investment for the following strategic reason. Dueto the strong vertical differentiation of the newly developed product, Firm 2 doesnot want Firm 1 to innovate. To achieve this, Firm 2 makes the market for theexisting product more attractive to Firm 1 by strategic under-investment whichleads to a smaller reduction of Firm 2’s production costs. As a consequence,this makes Firm 2 a weaker competitor for Firm 1. Therefore, for Firm 1 theaim of starting up the innovation process is not to innovate, but to trigger adesired reaction by its competitor. In terms of the Fudenberg-Tirole taxonomywe can say that Firm 2 under-invests to be soft, which is a puppy dog strategy.Under very strong vertical differentiation Firm 2 also does not want Firm 1 tocomplete the product innovation project, but this time it achieves its goal bystrategically over-investing in process R&D which results in very low productioncosts. This makes Firm 2 a very strong competitor and significantly reduces thepotential market share of Firm 1. Firm 2 exploits the size effect, so that the

4By over-investment (under-investment) we refer to an investment level which is higher(lower) than the corresponding investment level which is optimal ex post, i.e. once Firm 1has decided whether to introduce the new product.

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market as a whole, thus including the heterogeneous product market that wouldarise after Firm 1 launches the new product, is not sufficiently profitable forFirm 1 to undertake the necessary R&D investment for completing the productinnovation project. Note that Firm 1 never intends to trigger over-investmentby Firm 2, since this makes Firm 2 a stronger competitor. For this reason itis never optimal for Firm 1 to start the product innovation project in the firstplace, given that the degree of vertical product differentiation is very strong.

It seems worth considering this last result in more detail. Initiating the prod-uct innovation project creates the option to innovate and thus Firm 1 acquiresthe possibility to produce the new product. However, since in the case of verystrong vertical differentiation this triggers an undesirable reaction, namely over-investment by Firm 2, the value of the option to innovate is negative for Firm 1.This implies the following interesting and counterintuitive result. Assume Firm1 can start the product innovation project for free, i.e. without incurring anyinvestment costs. Then even in this situation Firm 1 would have been better offnot having this opportunity since the sheer existence of this flexibility resultsin very low production costs of Firm 2. Observe that this conclusion is com-pletely opposite to the standard insights in the real option theory, where thepositive value of flexibility, in our case having options to invest, is stressed (see,e.g., Dixit and Pindyck (1994)). On the other hand, these arguments highlightthe value of being committed to abstain from introducing new products, whichreduces the competitor’s incentive to make strategic moves.

The paper is organized as follows. In the next section we briefly summarizethe literature which is related to our work and indicate how the present paperdiffers. We introduce our model in Section 3 and the stages of the game aresubsequently analyzed in Section 4. The results of Section 4 are transformedin sub-game perfect equilibria in Section 5 where we also give an economicinterpretation of our results. Concluding remarks are provided in Section 6.Whereas in the main body of the paper it is assumed that Firm 1 stops producingthe old product after introducing the new one, in Appendix A this assumptionis relaxed and it is shown that the obtained qualitative results carry over tosuch a case. The formal equilibrium analysis and all proofs are relegated toAppendix B.

2 Related Literature

A first step in exploring the effects of the multi-stage nature of innovation pro-cesses has been done in several recent papers on the value of R&D projectsunder uncertainty. The focus of work in this research area is either on deter-mining the value of flexibility in the R&D process using a real options approach(e.g. Huchzermeier and Loch (2001), Jagle (1999), Lint and Pennings (1998)) oron the trade-off between flexibility and commitment under oligopolistic competi-tion (see, in particular, Smit and Trigeorgis (2004)). Another interesting streamof literature consists of work on patent-races and innovation timing. Althoughthese contributions take into account the dynamic nature of R&D projects and

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provide insights into the resulting strategic effects, the focus is on the adoptionof new technology, technological competition, and the optimal timing of bringinga new product or process to market (see e.g. Hoppe and Lehmann-Grube (2005),Doraszelski (2003), Reinganum (1989)). The difference of these contributionswith respect to the present paper is that the interplay between the incentives toinvest in process and product innovation in the presence of strategic interactionis largely neglected there, whereas it is one of our concerns here. The papers byLukach et al. (2007) and Dawid et al. (2006) are most closely related to ours.In Lukach et al. (2007) the role of sequential investment decisions in processinnovation in a market setting with potential competition is studied. In thispaper technical uncertainty results in a random outcome of the costs of R&D.Then the project may be stopped midstream in case completion turns out tobe more expensive than expected. However, our emphasis here is not on therandomness of innovations but on strategic behavior, which can be another rea-son to stop a project midstream. For instance, less process innovation by thecompetitor may reduce the incentives to complete the product innovation, sothat it can be optimal to stop the product innovation project midstream. Dawidet al. (2006) consider a quantity-setting duopoly where both firms can invest incost-reducing process innovation for an existing product. In addition, one firmalso develops a horizontally differentiated new product. The authors study theincentive to launch this newly developed product and find an equilibrium wherethe competitor strategically over-invests in process innovation in order to in-duce the introduction of this new (horizontally differentiated) product, therebyreducing the competition for the existing product. The present paper differson two accounts. First, product innovation in our case leads to horizontal andvertical differentiation. Second, in our setup we explicitly take into accountthe sequential nature of R&D processes by incorporating decisions to start thenew-product project and to continue or terminate it at a subsequent stage (andmake the necessary investments), whereas in Dawid et al. (2006) only the launchdecision is considered.

3 The Model

Consider a duopoly where two firms denoted as Firm 1 and Firm 2 engage inCournot (quantity) competition. Both firms have the possibility to producea homogeneous product at unit costs c. The inverse demand function for thisproduct is

p = 1 − (q1 + q2), (1)

where p denotes the market price and qi is the quantity of Firm i (i = 1, 2).Firm 1 has the option to initiate a product innovation project, whereas Firm2 can reduce its unit costs by investing in process innovation5. To incorporatethe multi-stage structure of R&D projects, we assume that Firm 1’s product

5This assumption is made in order to focus on the main topic of this paper, the strategiceffect of a firm’s option to introduce a product innovation. Allowing both firms to invest in

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innovation project develops in two stages and that the project can be terminatedafter the first stage. We impose that once Firm 1 decides to introduce the newproduct it stops producing the old product6. The details of the four stages ofour game are as follows.

Product Innovation Stage 1 (Firm 1):

Firm 1 makes the decision to either start a two-stage product innovation projector not, where the necessary investment to start the project is I1. We assumethat the outcome of the innovation project can be perfectly predicted.7 The newproduct would be horizontally and vertically differentiated from the existingproduct, but further investment at a later stage is needed for completion. Inwhat follows we denote the binary decision variable of Firm 1 in the ProductInnovation Stage 1 by P1. We write P1 = S if the firm decides to start theproduct innovation and P1 = D if it declines to start the project.

Process Innovation Stage (Firm 2):

After observing Firm 1’s decision, at the second stage Firm 2 can select thelevel of investments in process innovation x2 ∈ [0, c], which reduces productioncosts for its product to c2 = c−x2. The associated costs for this investment arek(x2) = βx2

2.

Product Innovation Stage 2 (Firm 1):

Firm 1 observes the level of investments in process innovation selected by Firm2 and decides whether to continue with the product innovation project, whereinvestments of I2 are necessary for completion. We denote the binary decisionvariable of Firm 1 in this stage by C1 and write C1 = C if the firm decidesto continue with the product innovation project and C1 = T if the project isterminated.

Quantity Competition Stage (Both Firms):

All innovation projects are finished and firms engage in quantity competition.If Firm 1 has decided to complete the new-product project, the new product

product and process innovation would make the model more complex and hardly tractable.Furthermore, in Dawid et al. (2006) it has been demonstrated that giving the productinnovator the option to invest in process innovation does not lead to qualitative changes.

6In Appendix A we show that the qualitative results of our analysis carry over to anextended model where Firm 1 has the option to keep its old product on the market and tooffer positive quantities of both products simultaneously.

7Although innovation projects are typically associated with (technical and/or market) un-certainty, we abstract from this issue to highlight the effects of strategic uncertainty.

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with differentiation parameters η, γ is taken into production and Firm 1 offersonly this new product on the market. The inverse demand functions in this caseare given by8

p1 = 1 − γq1 − ηq2,p2 = 1 − q2 − ηq1,

(2)

where 0 < η ≤ γ ≤ 1. The new product is thus vertically and horizontallydifferentiated from the product of Firm 2, where vertical differentiation is re-flected by γ being smaller than one and horizontal differentiation is captured byη being also less than one (cf. Qiu (1997), Lin and Saggi (2002)). Furthermore,as usual it also holds that the price of one product always depends more on thequantity of that same product than on the quantity of the other product, i.e.η ≤ γ ≤ 1. If Firm 1 has decided to terminate the new-product project or didnot even start the project, it competes with its homogeneous product on themarket. Marginal costs for producing the new product are c1 = c. Values ofci, η, γ are common knowledge.

Observe that in order to focus exclusively on the interplay between the ter-mination of the product innovation project midstream and the competitor be-havior we abstract from any random influences. Since the investment in processinnovation x2 by Firm 2 takes place in between the two product innovationstages of Firm 1, Firm 2 can strategically over- or under-invest to influence thecontinuation decision of Firm 1. Over-investing implies that Firm 2 can produceat lower costs, which makes this firm a stronger competitor for Firm 1. Due tothe competition effect of process innovation, this can give Firm 1 the incentiveto ”move away from Firm 2’s product” by launching the new product. On theother hand, if Firm 2 conquers such a high market share that even innovatingleads to low profits, then Firm 1 refrains from undertaking any investments indeveloping the new product (size effect of process innovation). Finally note thatit is also possible for Firm 1 to influence Firm 2’s innovation behavior by justinitiating the product innovation project. From a strategic perspective this isvaluable if creating the option to innovate triggers under-investment by Firm 2.However, the value of creating the option to innovate is negatively affected ifover-investment is triggered.

Since we want to focus on the economically most relevant cases, we imposesome additional assumptions. First, we assume that

c ≤ 0.5

in order to guarantee positive quantities of both firms. This indicates non-drastic innovation.

Second, for the process innovation cost parameter β it holds that

β ≥2(1 + c)

9c, (3)

8This is a variant of the standard linear demand model as in Singh and Vives (1984),and has been used e.g. by Symeonidis (1999, 2003). Inverse demand functions of this typecan be derived from a quality-augmented version of a standard quadratic utility function, seeSymeonidis (2003) and the references given therein.

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which implies that optimal process innovation effort x2 is in (0, c).Third, we impose that

I2 <4γ(1 − η)2(1 − c)2

(4γ − η2)2 − 9γη2. (4)

It turns out that if (4) is violated the costs of completing the product inno-vation project are so high that Firm 1 would never complete it regardless of theprocess innovation efforts of Firm 2. Obviously in such a case there would beno incentives for Firm 1 to start a product innovation project in the first place.

4 Analysis of the Stages of the Game

The model described above can be analyzed as an extensive form game with fourstages. A strategy for Firm 1 is given by a triple {P1, C1(P1, x2), q1(P1, C1, x2)},where P1 ∈ {S,D}, C1 : {S,D} × [0, c] 7→ {C, T} and q1 : {S,D} × {C, T} ×[0, c] 7→ [0,∞). A strategy for Firm 2 is given by a pair {x2(P1), q2(P1, C1, x2)},where x2 : {S,D} 7→ [0, c] and q2 : {S,D}×{C, T}× [0, c] 7→ [0,∞). Employinga backward induction approach, we analyze the four stages in reversed order.In Section 4.1 we start with the final stage, which is the quantity competitionstage. In Section 4.2 we study the product innovation stage 2, followed bythe process innovation stage in Section 4.3. Finally, we analyze the productinnovation stage 1 in Section 4.4.

4.1 Quantity Competition Stage

At the final stage Firm 1 has already made the start-up and completion deci-sions of its product innovation project and it is thus known whether the marketdemand is described by (1) or (2). Also, Firm 2 has already determined the op-timal process innovation effort x2. The next proposition presents the quantitiesand profits for all possible cases.

Proposition 1 For any subgame at the quantity competition stage there existsa unique Nash equilibrium. If Firm 1 has introduced the new product (P1 =S,C1 = C), quantities and profits in equilibrium read

q∗SC1 (x2) =

2 − η + η(c − x2) − 2c

4γ − η2, (5)

q∗SC2 (x2) =

2γ − η + ηc − 2γ(c − x2)

4γ − η2> 0, (6)

π∗SC1 (x2) = γ

[

2 − η + η(c − x2) − 2c

4γ − η2

]2

− I1 − I2, (7)

π∗SC2 (x2) =

[

2γ − η + ηc − 2γ(c − x2)

4γ − η2

]2

− βx22. (8)

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If Firm 1 starts the project but does not complete it (P1 = S,C1 = T ) quantitiesand profits in equilibrium read

q∗ST1 (x2) =

1 − c − x2

3, (9)

q∗ST2 (x2) =

1 − c + 2x2

3> 0, (10)

π∗ST1 (x2) =

[

1 − c − x2

3

]2

− I1, (11)

π∗ST2 (x2) =

[

1 − c + 2x2

3

]2

− βx22. (12)

In all subgames where Firm 1 did not start the project, equilibrium quantities aregiven by q∗O

i (x2) = q∗STi (x2), i = 1, 2 and profits read π∗D

1 (x2) = π∗ST1 (x2) + I1

for Firm 1 and π∗D2 (x2) = π∗ST

2 (x2) for Firm 2.

4.2 Product Innovation Stage 2

At the time Firm 1 has to decide about continuing the product innovationproject, Firm 2 has already determined the optimal process innovation effort,x2, and Firm 1 has decided about starting up the project or not. Clearly, itonly makes sense to analyze this continuation decision if in stage 1 of the gameFirm 1 has started the product innovation project.

To determine whether it is optimal or not to continue the product innovationproject, we have to compare the ”continuation profit” and the ”terminationprofit”, i.e. the profits which are earned if Firm 1 continues with or terminatesthe product innovation project respectively. To do so, we define the functionF1(x2) to be the difference between continuation and termination profit, i.e.

F1(x2) = π∗SC1 (x2) − π∗ST

1 (x2)

Employing (7) and (11) it is easy to check that F1 is a concave function9.Furthermore, straightforward calculations show that if condition (4) holds, thereexist real thresholds xT1

2 (I2) < xT22 (I2) such that F1(x2) ≥ 0 iff x2 ∈ [xT1

2 , xT22 ].

This implies that it is optimal for Firm 1 to complete the innovation project ifand only if Firm 2’s process innovation effort is larger or equal than xT1

2 butsmaller or equal than xT2

2 . Note that these thresholds might lie outside theinterval [0, c]. As I2 increases, xT1

2 becomes larger and xT22 becomes smaller.

Figure 1 illustrates the shape of F1(x2) for the case in which both thresholdsare situated in the interval [0, c].

All this leads to the following characterization of the equilibrium strategy atthe product innovation stage 2.

9The condition F ′′1 (x2) = 2γη2

(4γ−η2)2− 2

9< 0 is equivalent to (−16γ + η2)(γ − η2) < 0,

which holds due to η ≤ γ ≤ 1.

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1

2

Tx 2

2

Tx 2x

)( 21 xF

0c

Termination

domain

Completion

domain

Termination

domain

Figure 1: The function F1 and the thresholds xT12 , xT2

2 .

Proposition 2 In any SPE the choice at the product innovation stage 2 has tosatisfy

C∗1 (S, x2) = C for x2 ∈ (xT1

2 , xT22 )

C∗1 (S, x2) = T for x2 ∈ [0, xT1

2 ] ∪ [xT22 , c]

C∗1 (S, x2) ∈ {C, T} for x2 ∈ {xT1

2 , xT22 }

.

We call the set [xT12 , xT2

2 ], which consists of all x2 levels for which firm1 completes the project, the completion domain and by [0, xT1

2 ] ∪ [xT22 , c] the

termination domain. Clearly, Proposition 2 reflects that by choosing its processinnovation effort x2, Firm 2 can directly influence Firm 1’s decision to continuethe product innovation or not. When determining the optimal level of x2, whichis done in the next subsection, this of course has to be taken into account.

4.3 Process Innovation Stage

When Firm 2 decides about the process innovation level, it already knows Firm1’s decision about starting up its product innovation process or not. First,

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consider the case where Firm 1 did not start the product innovation. ThenFirm 2’s optimal process innovation level is given in Proposition 3.

Proposition 3 Consider the subgame where P1 = D. In a subgame perfectequilibrium the amount of process innovation is given by x∗D

2 = xO2 , where

xO2 :=

2 [1 − c]

9β − 4. (13)

Next, consider the case where Firm 1 has started the product innovationproject, thus where P1 = S. The characterization of optimal behavior in theproduct innovation stage 2 shows that by selecting the appropriate level of pro-cess innovation, Firm 2 can influence Firm 1’s decision whether to continue theinnovation project or not. There are two possibilities. First, it can be optimalfor Firm 2 that Firm 1 does not complete the product innovation project. Thismay happen when the new product is of such a high quality that hardly anyconsumer will be interested anymore in buying the old product. Second, it canalso be the case that Firm 2 wants Firm 1 to introduce the new product. Thisoccurs when the horizontal differentiation aspect dominates. Competition onthe output market is reduced if the new product is introduced.

Let us first consider in detail the situation where Firm 2 does not want Firm1 to complete the innovation project. The resulting process innovation level x2

must be located in the termination domain, so that Firm 2 has to solve thefollowing constrained maximization problem:

maxx2∈[0,xT1

2]∪[xT2

2,c]

π∗ST2 (x2). (14)

We denote the maximum by πST2 . An obvious candidate for the optimal process

innovation level of Firm 2 is xO2 , the optimal process innovation level for the case

where Firm 1 did not start the product innovation, because also in this case Firm1 does not innovate. Clearly, if setting x2 = xO

2 induces Firm 1 to terminatethe project at the product innovation stage 2 (i.e. xO

2 ∈ [0, xT12 ]∪ [xT2

2 , c]), thenx2 = xO

2 maximizes Firm 2’s profits over the termination domain. However,if xO

2 is located in the completion domain, i.e. xO2 ∈ (xT1

2 , xT22 ), then Firm 2

has to deviate from xO2 in order to make Firm 1 terminate the project. Taking

into account that π∗ST2 (x2) is a strictly concave function of x2, the maximum of

Firm 2’s profit over the termination domain is attained at one of the boundariesof the completion domain. Hence, if the completion domain is in the interiorof [0, c], then choosing either xT1

2 or xT22 maximizes Firm 2’s profits over the

termination domain10.Next, consider the other case where Firm 2 wants Firm 1 to complete its

product innovation project. Accordingly, we have to determine the maximum

10If the completion domain is not in the interior of [0, c] then one or both of xT12 and xT2

2might lie outside the feasible range of x2 values. Explicit conditions on the range of values ofγ that guarantee that the completion domain is in the interior of [0, c] can be derived. Sincethese expressions are rather involved and do not provide additional qualitative insights we donot present them here.

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of Firm 2’s profits over the completion domain. Analogous to (14) we define

πSC2 = max

x2∈[xT1

2,xT2

2]π∗SC

2 (x2). (15)

On obvious candidate for the optimal solution of this constrained maximizationproblem is the process innovation level which Firm 2 would have chosen incase Firm 1 is committed to innovate. This is the solution to the problem ofmaximizing π∗SC

2 (x2) over the domain [0, c], which is given by

xN2 :=

2γ [2γ − η] [1 − c][

β [4γ − η2]2− 4γ2

] . (16)

If xN2 lies in the completion domain then πSC

2 can be attained by setting x2 =xN

2 . Otherwise, similar considerations as above lead to the conclusion that Firm2 should choose either x2 = xT1

2 (if xN2 ∈ [0, xT1

2 )) or x2 = xT22 (if xN

2 ∈ (xT22 , c]).

To sum up, there are four candidates for the optimal choice of x2 in thesubgame where Firm 1 has started the product innovation project. These fourcandidates are xN

2 , xO2 , xT1

2 and xT22 . It should be noted that xN

2 and xO2 can be

interpreted as the ex-post optimal investment levels of Firm 2 once the contin-uation decision of Firm 1 has been determined.

In order to avoid a cumbersome analysis of many different cases, we willconcentrate on scenarios where xO

2 > xN2 holds, i.e. where the ex-post optimal

investment level for Firm 2 is larger if Firm 1 offers the homogeneous prod-uct compared to the case where it introduces the new product. This wouldimply that the investment in process innovation is higher when there is morecompetition on the product market. Straightforward calculations show that

γ ≤η

2(9 − 8η) (17)

is a sufficient condition for this inequality. In particular, if η ∈[

9−√

1716 , 9+

√17

16

]

≈

[0.3, 0, 82] we have xO2 > xN

2 for any values of c0, β and γ ∈ [η, 1]. In what followswe only consider values of η in this interval.

An important factor for determining which of the four candidates should bechosen by Firm 2 is the order of the candidates. Taking into account xT1

2 < xT22

and xN2 < xO

2 we have to consider six scenarios:

(a) xN2 < xO

2 < xT12 < xT2

2

(b) xN2 < xT1

2 < xO2 < xT2

2

(c) xN2 < xT1

2 < xT22 < xO

2

(d) xT12 < xN

2 < xO2 < xT2

2

(e) xT12 < xN

2 < xT22 < xO

2

(f) xT12 < xT2

2 < xN2 < xO

2

13

In the following proposition we cover all scenarios where πST2 6= πSC

2 . It isstraightforward (but cumbersome) to extend our characterization to the non-generic case where πST

2 = πSC2 .

Proposition 4 Consider the subgame where firm 1 has chosen P1 = S. Theequilibrium level of x2 in a subgame-perfect equilibrium can be characterized asfollows:

i) If scenario (a) or scenario (c) applies then

x∗S2

{

= xT12 πSC

2 > πST2

= xO2 πST

2 > πSC2

ii) If scenario (b) applies then

x∗S2

= xT12 πSC

2 > πST2 or

πST2 > πSC

2 and xO2 − xT1

2 < xT22 − xO

2

= xT22 πST

2 > πSC2 and xO

2 − xT12 > xT2

2 − xO2

∈ {xT12 , xT2

2 } πST2 > πSC

2 and xO2 − xT1

2 = xT22 − xO

2

iii) If scenario (d) applies then

x∗S2

= xN2 πSC

2 > πST2

= xT12 πST

2 > πSC2 and xO

2 − xT12 < xT2

2 − xO2

= xT22 πST

2 > πSC2 and xO

2 − xT12 > xT2

2 − xO2

∈ {xT12 , xT2

2 } πST2 > πSC

2 and xO2 − xT1

2 = xT22 − xO

2

iv) If scenario (e) applies then

x∗S2

{

= xN2 πSC

2 > πST2

= xO2 πST

2 > πSC2

v) If scenario (f) applies then

x∗S2

{

= xT22 πSC

2 > πST2

= xO2 πST

2 > πSC2

4.4 Product Innovation Stage 1

At the first stage of the game, Firm 1 has to decide to start the product innova-tion project or not. The relevant data are the value of starting the project, V1S ,the value of declining to start the project, V1D, and the sunk cost investmentrequired to start the project, I1.

14

V1S is the continuation value for Firm 1 in the subgame generated by theaction P1 = S, if it is assumed that both firms follow a subgame perfect equi-librium in all stages of this subgame. In case more than one subgame perfectequilibrium exists for this subgame, V1S gives the maximum of the continuationvalues for Firm 1 in all subgame perfect equilibria. In determining V1S we haveto account for the process innovation decision of Firm 2, since this decision caneither lead to continuation or termination of the innovation project in stage 3of the game. Consequently, it holds that

V1S =

{

π∗SC1 (x∗S

2 ) + I1 x∗S2 ∈ [xT1

2 , xT22 ]

π∗ST1 (x∗S

2 ) + I1 x∗S2 ∈ [0, xT1

2 ) ∪ (xT22 , c],

where it should be noted that π∗SC1 (x∗S

2 ) = π∗ST1 (x∗S

2 ) for x∗S2 ∈ {xT1

2 , xT22 }.

V1D is the continuation value for Firm 1 in the subgame where P1 = D. Forthis subgame there is always a unique subgame perfect equilibrium and we have

V1D = π∗D1 (x∗D

2 ) with x∗D2 = xO

2 .

The (equilibrium) value for Firm 1 of starting the innovation project is givenby

V1 = V1S − V1D,

and we directly obtain a characterization of the actions to be chosen at thisstage in a subgame perfect equilibrium.

Proposition 5 There exists a SPE with P ∗1 = S if and only if V1 ≥ I1. Con-

versely there exists a SPE with P ∗1 = D whenever V1 ≤ I1.

5 Economic Analysis

The analysis in the previous section demonstrates that there are four levelsof process innovation investment that might arise in equilibrium. Based onthis observation a reduced representation of the game can be derived, which isdepicted in Figure 2.

All-together there are 9 potential outcomes of this reduced game. We haveattached a label to each outcome where the different choice of labels will becomeapparent as we continue the discussion. Three of the outcomes, no innovator,standard innovator and non-strategic termination are non-strategic outcomesin the sense that the process innovation level of Firm 2 is ex-post optimal (noover- or under-investment is necessary to induce the preferred action from Firm1). The outcomes pushed innovator, threatening innovator, lucky innovator andcursed innovator are strategic outcomes. As we will demonstrate, in a pushedinnovator outcome Firm 2 over-invests in order to induce Firm 1 to launch thenew product, whereas in a lucky innovator outcome Firm 2 under-invests. Onthe other hand, in a threatening innovator equilibrium Firm 2 under-invests inorder to induce Firm 1 to terminate the new-product project, whereas in a cursed

15

DS

C

T

C

T

C

T

C

0

2x

Nx

2

1

2

Tx

2

2

Tx

0

2x

No Innov.

Lucky

Innov.

Threatening

Innov.Pushed

Innov.

Irrational

Standard

Innov.Non-strategic

termination

Irrational

F1F1

F1F1

T

F1

F2F2

Cursed

Innov.

Figure 2: Reduced game tree where only potential candidates for equilibriumactions are depicted.

innovator outcome Firm 2 over-invests. In all these cases Firm 2’s investmentin process innovation is given either by xT1

2 or by xT22 , which means that it is

different from the ex-post optimal level. In addition to these outcomes thereare two nodes labelled as irrational. To realize that these outcomes correspondto irrational behavior of Firm 2 one just has to observe that the only strategicchoice of x2 that might be rational are xT1

2 and xT22 . In the two irrational

outcomes Firm 2 chooses an x2 different from these two levels that would only beex-post optimal if Firm 1 would have taken the opposite continuation decision.In what follows we will explore the question whether all of the outcomes thatare not labeled as ’irrational’ can be attained in a subgame-perfect equilibriumof the game and characterize the parameter constellations that give rise to eachof them. Furthermore, we will discuss the economic intuition and implicationsof each outcome.

We start with the non-strategic outcomes. If an outcome with a certainlabel is reached in a subgame perfect equilibrium we give the correspondingequilibrium profile the same label.

Proposition 6

16

i) For sufficiently small I1 and I2, there exists a standard innovator equilib-rium where the set of actions chosen in equilibrium reads P1 = S, x2 =xN

2 , C1 = C.

(ii) For sufficiently large I1 or I2 there exists a no innovator equilibrium whereFirm 1 does not start the project.

(iii) There exists no non-strategic termination equilibrium where the set of ac-tions chosen in equilibrium reads P1 = S, x2 = xO

2 , C1 = T .

Let us now turn to scenarios where due to strategic reasons Firm 2 choosesinvestments different from the ex-post optimal level. We treat each of the fourcases in different subsections.

5.1 Pushed Innovator Outcome

Firm 2 over-invests (x2 = xT12 > xN

2 ) and, therefore, Firm 1 completes theinnovation project (P1 = S,C1 = C), see Figure 311. In other words, by over-investing Firm 2 ”pushes” Firm 1 to complete the product innovation project.The following proposition shows that such an outcome can indeed arise fromequilibrium behavior.

Proposition 7 Assume that

η <2(1 − 2c)

1 + c.

Then, there exists a γ1 < 1 such that for all γ ∈ [γ1, 1] there exist β, I1, I2 suchthat the unique subgame-perfect equilibrium of the game is a pushed innovatorequilibrium where the set of actions chosen in equilibrium reads P1 = S, x2 =xT1

2 , C1 = C.

This equilibrium thus emerges if γ is large, i.e if the degree of vertical differ-entiation is low, and η is small, which indicates a high horizontal differentiation.In such a situation the quality of the new product is not much higher than thequality of the old product. On the other hand high horizontal differentiation im-plies that after the new product is launched competition on the output marketdeclines considerably, which leads to higher output prices. Both implicationsin fact make it attractive for Firm 2 to induce Firm 1 to complete the prod-uct innovation project. Figure 3 shows that xN

2 , the ex-post optimal processinnovation level of Firm 2 in the heterogeneous product market that arises after

11In principle the outcome x2 = xT12 , P1 = S, C1 = C could also arise in our reduced game if

xT12 < xN

2 , but it is obvious from our arguments in Section 3 that this would involve irrationalbehavior of Firm 2. In such a case it holds that either xN

2 is already in the completion domain,then no deviation is needed to induce completion. Or xN

2 lies above the completion domain,then xT2

2 is the optimal level to induce completion.

17

1

2

Tx

2

2

Tx 2x

)( 21 xF

0cN

x2

Termination

domain

Termination

domain

Completion

domain

Figure 3: Constellation of the function F1 and the ex-post optimal investmentlevels xN

2 , xO2 leading to a pushed innovator outcome.

Firm 1 has innovated, lies in the termination domain. Hence, if Firm 2 choosesxN

2 , Firm 1 will not innovate. For this reason Firm 2 has to select a level in thecompletion domain which is as close as possible to xN

2 , and Figure 3 reveals thatthis must be xT1

2 . Since xT12 > xN

2 , Firm 2 over-invests to induce the comple-tion of Firm 1’s product innovation project. In terms of the Fudenberg-Tiroletaxonomy (Fudenberg and Tirole (1984)) this is a top dog policy.

The result that Firm 2’s over-investment induces Firm 1 to launch the newproduct is driven by the competition effect. Over-investment by Firm 2 resultsin low unit production costs and this makes Firm 2 a strong competitor in themarket for the old product. For this reason Firm 1 decides to launch the newproduct by completing the product innovation process.

5.2 Threatening Innovator Outcome

Here Firm 2 under-invests (x2 = xT12 < xO

2 ) and Firm 1 terminates the innova-tion project (P1 = S, C1 = T ), see Figure 4. Hence, by starting up the project,Firm 1 ”threatens” to innovate, which induces Firm 2 to under-invest. Thefollowing proposition shows that a parameter scenario exists under which this

18

1

2

Tx

2

2

Tx 2x

)( 21 xF

0cO

x2

Termination

domain

Termination

domain

Completion

domain

Figure 4: Constellation of the function F1 and the ex-post optimal investmentlevel xN

2 leading to a threatening innovator outcome.

outcome is a unique subgame-perfect equilibrium.

Proposition 8 For all η ∈ [0, 1) there exist values

γ2 := η18 − η +

√

(18 − η)2 − 64η2

32

γ3 := η3 − η

2

with η < γ2 < γ3 ≤ 1 such that for all γ ∈ [γ2, γ3] there exist β, I1, I2 such thatthe unique subgame-perfect equilibrium of the game is a threatening innovatorequilibrium with x∗S

2 = xT12 < xO

2 and Firm 1 terminates the product innovationproject.

In this equilibrium we have a small γ, thus strong vertical differentiation.This implies that the quality of the new product is significantly higher thanthe quality of the old product. For this reason Firm 2 does not like Firm 1to complete its product innovation project. If Firm 1 does not innovate, Firm2’s ex-post optimal process innovation level is xO

2 . However, Figure 4 shows

19

that xO2 is situated in the completion domain, implying that if Firm 2 chooses

this process innovation level, it will induce Firm 1 to innovate. For this reasonFirm 2 chooses an appropriate level of process innovation of x2 that is in thetermination domain, while at the same time this level of x2 is as close as possibleto the ex-post optimal level xO

2 . In Figure 4 it is easy to see that this leads tothe choice xT1

2 . Hence, since xT12 < xO

2 , Firm 2 will under-invest in order totempt Firm 1 to compete with the homogeneous product. Firm 2 chooses inthis case a puppy dog strategy.

Under-investment results in Firm 2’s unit production costs being high, whichmakes Firm 2 a weaker competitor. For this reason Firm 1 will not innovateand will thus offer the homogeneous product. So, like in the pushed innovatorequilibrium, it is again the competition effect that generates this outcome. It isimportant to realize that in the threatening innovator equilibrium the value forFirm 1 of starting the project is strictly positive, although in equilibrium theproject is never completed. The reason is that starting the project has strategicvalue for Firm 1. It creates an option to innovate for Firm 1, and this forcesFirm 2 to under-invest in process innovation.

Proposition 8 shows that strategic under-investment occurs if there is a cer-tain degree of horizontal differentiation, i.e. η is sufficiently below one. Ifhorizontal differentiation is limited, i.e. η is close to one, the competition effectof process innovation is negligible, because by innovating Firm 1 now does not”move away from the old product market”. This implies that the size effect al-ways dominates, meaning that increased process innovation by Firm 2 reducesFirm 1’s market share for both the old and the new product, and thus Firm1’s incentive to innovate. Mathematically, this implies that the profit differencefunction F1(x2) is decreasing in x2 on the entire interval [0, c] and, consequently,only over-investment can induce Firm 1 to terminate the innovation project.

5.3 Lucky Innovator Outcome

We now turn to the outcome where x2 = xT22 and Firm 1 completes the product

innovation project, see Figure 5. Figure 5 shows that this outcome occurs inconjunction with under-investment of Firm 2 (xT2

2 < xN2 ). We call this a ”lucky

innovator outcome”, because Firm 1 not only benefits from its new productbut also from the fact that for strategic reasons Firm 2 reduces its processinnovation investment. However, it turns out that such an outcome cannotresult from equilibrium behavior.

Proposition 9 There exists no admissible parameter constellation such thatthere exists a lucky innovator equilibrium where xS∗

2 = xT22 < xN

2 and Firm 1completes the innovation project.

The intuition behind this result is that, as x2 goes up, it becomes less attrac-tive for Firm 2 that Firm 1 completes the project12. This is because increas-ing x2 raises the comparative cost advantage of Firm 2 and this comparative

12Formally, this means that π∗SC2 (x2) − π∗ST

2 (x2) is a strictly decreasing function of x2.This fact is proved in Lemma 15 in the Appendix.

20

1

2

Tx

2

2

Tx 2x

)( 21 xF

0cN

x2

Termination

domain

Completion

domain

Figure 5: Constellation of the function F1 and the ex-post optimal investmentlevel xN

2 leading to a lucky innovator outcome.

advantage is more beneficial if the competitor’s product is not differentiated.Consequently, if x2 is large, in particular for x2 ≥ xT2

2 , Firm 2 prefers that Firm1 terminates the product innovation project and therefore never chooses xT2

2 toinduce completion.

5.4 Cursed Innovator Outcome

In this outcome we have that x2 = xT22 , while Firm 1 terminates the innovation

project, see Figure 6. In this figure we see that, since xO2 < xT2

2 , Firm 2 over-invests to induce termination. Here Firm 1 is a ”cursed innovator”, because Firm1 not only refrains from innovating, but it also suffers from an over-investingcompetitor. The following proposition shows that this outcome can indeed arisein a SPE of the subgame where Firm 1 has already started the project.

Proposition 10 For any η ∈ (0, 1) there exists a γ4 ∈ (η, 1] such that for allγ ∈ [η, γ4] there exist β, I1, I2 such that in the unique subgame-perfect equilib-rium of the subgame following P1 = S Firm 2 chooses x2 = xT2

2 > xO2 and Firm

1 terminates the project at the product innovation stage 2.

21

1

2

Tx

2

2

Tx 2x

)( 21 xF

0cO

x2

Completion

domain

Termination

domain

Termination

domain

Figure 6: Constellation of the function F1 and the ex-post optimal investmentlevel xO

2 leading to a cursed innovator outcome.

This outcome occurs if γ is very small, thus if there is a very strong verticaldifferentiation between the old product and the newly developed product. Sincethe quality of the new product is so much better than the quality of the oldproduct, Firm 2 does not want Firm 1 to complete the product innovationproject. To achieve this, Firm 2 employs the size effect of process innovation:it over-invests to reduce Firm 1’s market share. Then the ”size of the market”becomes too small for Firm 1 to make any further R&D investments optimal.

Note that in scenarios where the cursed innovator outcome results from equi-librium behavior in the subgame P1 = S, Firm 1 offers the homogeneous producteven if it starts the product innovation project. Hence, the only effect of startingthe project is that Firm 2 increases its process innovation for strategic reasons.Obviously, this hurts Firm 1. So, even if the direct costs of starting the projectwere zero (I1 = 0), starting the project has detrimental effects for Firm 1’sprofits. Put differently, the option to complete the product innovation projectwhich arises only in the case when Firm 1 starts the project has a negativevalue for Firm 1: Firm 1 is better off not having the option to innovate. Thisis quite counterintuitive at first sight and in contrast to the standard insightsfrom the real options literature where the positive value of additional flexibility

22

is stressed. On the other hand, this observation highlights the potential posi-tive value of making commitments, in our case the commitment to abstain fromintroducing a new product. Such a commitment takes away the competitor’sincentive for making any strategic moves.

These considerations are summarized in the following proposition.

Proposition 11 If the unique SPE of the subgame P1 = S leads to a cursedinnovator outcome, the value V1 for Firm 1 of starting the project is strictlynegative. Consequently, in such a setting Firm 1 chooses P1 = D in equilibrium,and the cursed innovator outcome cannot be the result of a SPE of the entiregame.

6 Conclusions

This paper considers the strategic effects that emerge if it is explicitly takeninto account that R&D projects are time consuming and go through differentstages. This implies that competitors can react to the innovation threat beforethe final decision about market introduction has been made. The real optionliterature concentrates on the possibility to stop such a project after a badrealization of some stochastic process that, for instance, implies that the valueof the completed project falls or that the expected cost of completing the projectrises (Pindyck (1993), Schwartz and Moon (2000)). Contrastingly, the presentpaper considers the strategic effects of the possibility to terminate an R&Dproject midstream. After one firm has started up an innovation project, byover- or under-investing the competitor can influence this firm’s decision to stopor continue the project after every separate stage is completed. Two contraryeffects drive our results and have to be taken into account. On the one handthere is a competition effect implying that Firm 1 has a higher incentive to moveto the new product market once Firm 2 becomes a stronger competitor due toover-investment in process innovation. On the other hand, there is a size effectin that over-investment can make Firm 2 such a strong competitor that even inthe new product market Firm 1 has a very low market share. The reduced sizeof its market then implies that it is not optimal to incur the R&D investmentsnecessary to complete the product innovation project.

Based on the interplay of these effects we can characterize the factors thatlead to an aggressive (over-investment), accommodating (non-strategic invest-ment) or defensive (under-investment) optimal reaction of a firm to a productinnovation threat of its competitor. This in turn allows us to predict what kindof product innovation projects are pursued by a firm that takes into accountthis optimal reaction. Innovation projects associated with new products of muchhigher quality are never realized by the existing firms in the market. The rea-son is that in such a case the competitor has a strong incentive to induce thethreatening firm to stick with the homogeneous product. The correspondingstrategic reaction to such a threat is over-investment that is so massive that itno longer pays off to introduce the new product and the threat in the end harms

23

the threatening firm. In case the degree of vertical differentiation is smaller, theoptimal reaction to the product innovation threat is under-investment to keepthe old product market sufficiently attractive for Firm 1, so that Firm 1 hasno incentive to complete the product innovation. Hence, creating this option toinnovate has value despite of the fact that the innovation project will never becompleted and Firm 1 has to compare this value with the costs of generating thisoption. A defensive reaction with under-investment might also be optimal in re-sponse to the threat of the introduction of a mainly horizontally differentiatedproduct. In such a scenario however, the potential product-innovator shouldnot only threaten to introduce the new product but should also complete theproject and offer the new product on the market. These patterns are the onlyscenarios where strategic reactions to threats that deviate from ex-post optimalbehavior occur. In all other cases equilibrium behavior implies an accommodat-ing reaction to a product-innovation threat where the competitor adjusts theinvestment to the optimal level given that the new product has been introduced.

Appendix A: Option to Produce both Products

In this Appendix we extend the basic model by allowing Firm 1 to simultane-ously offer the old and the new product. Hence, once Firm 1 has decided tocomplete the product innovation at Product Innovation Stage 2, it has still theoption to offer the old product while producing positive quantities of the newproduct. Accordingly, we have the following inverse demand functions:

pN1 = 1 − γqN

1 − η(qO1 + q2)

pO1 = p2 = 1 − (qO

1 + q2) − ηqN1 ,

where qN1 , qO

1 are the quantities of the new and the old product offered by Firm1 and pN

1 , pO1 are the corresponding unit prices. The next proposition shows

that, if the new product is sufficiently vertically differentiated, in equilibriumFirm 1 does not exercise the option to offer both products simultaneously.

Proposition 12 If γ < γ3 with γ3 defined as in Proposition 8 then in everyequilibrium where P ∗

1 = S and C∗1 = C we have qO∗

1 = 0.

Based on this result we can show that all existence results of the previoussection still hold.

Proposition 13 (i) Assume that

η <2(1 − 2c)

1 + c.

Then, there exists a γ1 < γ3 such that for all γ ∈ [γ1, γ3] there existβ, I1, I2 such that the unique subgame-perfect equilibrium of the game is apushed innovator equilibrium.

24

(ii) For all η ∈ [0, 1) and γ ∈ [γ2, γ3], with γ2, γ3 defined as in Proposition 8,there exist β, I1, I2 such that the unique subgame-perfect equilibrium of thegame is a threatening innovator equilibrium.

(iii) For any η ∈ (0, 1) there exists a γ4 ∈ (η, γ3] such that for all γ ∈ [η, γ4]there exist β, I1, I2 such that the unique subgame-perfect equilibrium im-plies the cursed innovator outcome in the subgame following P1 = S.

Finally, we can also replicate the non-existence result for lucky innovatoroutcomes in equilibria of the extended model.

Proposition 14 There exists no admissible parameter constellation such thatthere exists a lucky innovator equilibrium of the extended model.

These results show that the qualitative insights obtained in our analysiscarry over to the more involved framework where Firm 1 can choose to offerboth products at the same time.

Appendix B

Proof of Proposition 1:

The firms engage in quantity competition. The objective function of Firm 1 is

π1 = maxq1

[1 − γq1 − ηq2 − c] q1,

which gives the first order condition

1 − 2γq1 − ηq2 − c = 0.

Similarly, for Firm 2 we have

π2 = maxq2

[1 − q2 − ηq1 − c2] q2,

with first order condition

1 − 2q2 − ηq1 − c2 = 0.

We combine the two first order conditions to establish the optimal quantities:

q∗SC1 =

2 − η + ηc2 − 2c

4γ − η2, (18)

q∗SC2 =

2γ − η + ηc − 2γc2

4γ − η2> 0. (19)

Note that the denominators are always positive, since

0 < η ≤ γ ≤ 1.

25

Hence, q1 is non-negative if it holds that

x2 ≤[2 − η] [1 − c]

η.

Since x2 ≤ c, non-negativity of q1 is guaranteed only when

η ≤ 2 [1 − c] .

This is always satisfied due to our assumption that c ≤ 12 . The quantities q∗ST

1

and q∗ST2 can be directly obtained by setting γ = η = 1 in (18) and (19). Know-

ing the quantities we can calculate the resulting profits for both firms.


Due to the definition of F1 it is optimal for firm 1 to continue the project when-ever F1 ≥ 0.


Firm 2 maximizes π∗D2 (x2) over the domain x2 ∈ [0, c]. Taking into account (12)

gives xO2 . Note that due to our assumptions that c ≤ 1/2 and β ≥ 2(1 + c)/9c

the investment level xO2 > 0.


Taking into account that π∗SC2 (x2) and π∗ST

2 (x2) are quadratic polynomials inx2 we obtain

π∗SC2 (x2) > π∗SC

2 (y2) ⇔ |x2 − xN2 | < |y2 − xN

2 |

π∗ST2 (x2) > π∗ST

2 (y2) ⇔ |x2 − xO2 | < |y2 − xO

2 |

and the proposition follows directly from the considerations above.


i) If I2 = 0 then for η = γ = 1 we have π∗SC1 (x2) = π∗ST

1 (x2) for all x2. Fur-thermore, π∗SC

1 (x2) increases if γ, η are decreased. Therefore, Firm 1 alwayscompletes if I2 = 0 and we have V1S > V1D. Accordingly, Firm 1 starts theproduct innovation project for I1 sufficiently small.

ii) Trivial

iii) If x∗S2 = xO

2 and C1(S, x∗S2 ) = T, then V1S = V1D. Hence, V1D > V1S − I1

and we must have P ∗1 = D.

26

In order to simplify the proof of the following propositions we define

F2(x2) = π∗SC2 (x2) − π∗ST (x2)

as the difference of Firm 2 profits between the cases where Firm 1 completes theinnovation project and where it terminates. We will use the following lemmaconcerning the monotonicity of F2:

Lemma 15 The function F2 is strictly decreasing on the domain [0, c].

Proof. First we show that F2 is concave. Using the definition of F2 we obtain

F ′′2 (x2) ≤ 0

⇔8γ2

(4γ − η2)2−

8

9≤ 0

⇔ (−7γ + η2)(γ − η2) ≤ 0,

where the last inequality holds due the assumption that 0 < η ≤ γ ≤ 1. Due tothe concavity of F2 monotonicity on [0, c] follows from the condition F ′

2(0) < 0.To establish this inequality we observe

F2′(0)

=4γ(2γ − η)(1 − c)

(4γ − η2)2−

4(1 − c)

9

=4(1 − c)

9(4γ − η2)2(2γ2 − γη(9 − 8η) − η4)

≤4(1 − c)

9(4γ − η2)2(2γ2 − 2γ2 − η4)

< 0,

where the inequality in the second last line follows from (17).


In Proposition 4 the following conditions have been shown to be sufficient forthe existence of a SPE with x∗

2 = xT12 and C∗

1 (S, xT12 ) = C in the subgame

where P1 = S:

i) πSC2 > πST

2 ,

ii) xN2 < xT1

2

As a first step, we show that, for γ sufficiently close to 1, for any β there existvalues of I2 such that xN

2 = xT12 . Fix β and denote by x2 the argmax of F1. Since

F1 is a quadratic function we have xT12 < x2 < xT2

2 and xT22 − x2 = x2 − xT1

2 .Taking into account that xT1

2 goes to minus infinity for I2 → 0 (see the proofof Proposition 6) and becomes close to x2 for sufficiently large I2. It follows

27

from the monotonicity of xT12 with respect to I2 that under the condition that

x2 > 0 there exists a I2 such that xN2 = xT1

2 . The condition x2 > 0 holds trueif F ′

1(0) > 0. Calculating F ′1(x2) we get

F1′(x2)

=−2γη((2 − η)(1 − c) − ηx2)

(4γ − η2)2+

2(1 − c − x2)

9

=2

9(4γ − η2)2(((4γ − η2)2 − 9ηγ(2 − η))(1 − c) − x2((4γ − η2)2 − 9γη2))

. (20)

Setting γ = 1 this gives the following sufficient condition for F ′1(0) > 0:

(((4 − η2)2 − 9η(2 − η)) > 0

⇔ (1 − η)(16 − 2η − η2 − η3) > 0.

Obviously the last inequality is true for all η < 1. Therefore, there exists aγ1 < 1 such that for all γ ∈ [γ1, 1] we have x2 > 0 and there exists a I2 suchthat xN

2 = xT12 . Obviously, condition ii) is satisfied for any I2 > I2.

Concerning i), if follows from Lemma 15 that F2(c) > 0 is a sufficient con-dition for π∗SC

2 (x2) > π∗ST2 (x2) for all x2 ∈ [0, c]. Straightforward calculations

show that after setting γ = 1, the condition F2(c) > 0 is equivalent to

(η − 1)((1 + c)η − 2(1 − 2c)) > 0.

Under the condition on η given in the proposition this inequality holds as astrict inequality and due to continuity with respect to changes in γ there mustexist a γ1 < 1 such that π∗SC

2 (xT12 ) > πST

2 (xT12 ) holds for all γ ∈ [γ1, 1]. For

any such γ we have for I2 = I2

πSC2 = π∗SC

2 (xN2 ) > π∗SC

2 (xO2 ) > π∗ST

2 (xO2 ) ≥ πST

2 .

Since πSC2 is a continuous function of I2 we obtain that for all γ ∈ [γ1, 1] there

exist values of I2 > I2 such that condition i) holds. Setting γ1 = max(γ1, γ1)we have proved that under the conditions given in the proposition and suitablychosen I2 there exists a SPE of the subgame P1 = S where x∗S

2 = xT12 and

C∗1 (S, xT1

2 ) = C. To illustrate this part of the proof we show in Figure 7 thefunctions F1, π

∗SC2 , π∗ST

2 for γ ∈ [γ1, 1] and I2 > I2.To finish the proof we still have to show that there exists an I1 such that in

the SPE Firm 1 starts the product innovation project. To see this note that wehave

V1 = V1S − V1D

= πSC1 (xT1

2 ) + I1 − πD1 (xO

2 )

= πST1 (xT1

2 ) + I1 − πD1 (xO

2 )

= πD1 (xT1

2 ) − πD1 (xO

2 )

> 0,

28

1

2

Tx 2

xcO

x2 2

x

Completion

domain

0

S*

2

!

ST*

2

!

SC*

2

!

1F

Termination

domain

Nx

2

2xc

Figure 7: The functions F1, π∗SC2 , π∗ST

2 for γ ∈ [γ1, 1] and I2 > I2.

where the last inequality follows from xT12 < xO

2 . Accordingly, we have V1 > I1

for sufficiently small I1, which implies by Proposition 5 that P ∗1 = S. This

completes the proof.



2 = xT12 and C∗

1 (S, xT12 ) = T in the subgame where

P1 = S has been chosen:

i) πST2 > πSC

2 ,

ii) xT12 < xO

2 < xT22

iii) xO2 − xT1

2 ≤ xT22 − xO

2 .

Note first that xO2 is monotonously decreasing toward zero for increasing β.

Therefore, if the argmax of F1 denoted by x2 is positive there exist values of βsuch that xO

2 < x2. Using similar arguments as given in the proof of Proposition7 we conclude that for each such β there is a value I2 such that iii) is satisfied

29

and xO2 = xT1

2 . The argmax is positive if and only if F ′1(0) > 0. Using (20) we

obtain that F ′1(0) > 0 is equivalent to

(4γ − η2)2 − 9ηγ(2 − η) > 0.

This inequality is violated for γ = η and the unique root of the left handside with γ > η is the expression γ2 given in the proposition. Hence, for anyγ ∈ [γ2, 1] there exist values β and I2 < I2 such that ii) and iii) hold.

Considering condition (i) we note that for all γ ∈ (η, γ3) with γ3 = η 3−η2

we have F2(0) < 0. Due to F ′2 < 0 this implies π∗SC

2 (x2) < π∗ST2 (x2) for all

x2 ∈ [0, c]. In particular, for I2 = I2 we have

πST2 = π∗ST

2 (xT12 ) > π∗ST

2 (xN2 ) ≥ πSC

2 .

Therefore, for all γ ∈ (η, γ3) condition i) is satisfied for values of I2 < I2 thatare sufficiently close to I2. It is easy to check that for all η ∈ (0, 1) it holdsthat γ2 < γ3. Accordingly, there is a range of γ-values with γ2 < γ < γ3, whereconditions i) - iii) are satisfied for suitable values of β and I2.

Considering the product innovation stage 1 we have

V1 = V1S − V1D

= π∗ST1 (xT1

1 ) + I1 − π∗D1 (xO

2 )

= π∗D1 (xT1

1 ) − π∗D1 (xO

2 )

> 0

due to xT12 < xO

2 . Hence for sufficiently small I1 there exists a SPE withP ∗

1 = S, x∗S2 = xT1

2 and C∗1 (S, xT1

2 ) = T .


We have to show that there is no SPE with C∗1 (S, xT2

2 ) = C and x∗S2 = xT2

2

exists. According to part v) of Proposition 4 equilibrium strategies can onlyhave these properties if xT2

2 < xN2 and πSC

2 > πST2 . In the remainder of the

proof we show that xT22 < xN

2 implies πSC2 < πST

2 , which in turn implies thatthe inequalities xT2

2 < xN2 and πSC

2 > πST2 are incompatible with each other

and therefore no lucky innovator equilibrium can exist.Assume that xT2

2 < xN2 . We show that under this assumption π∗SC

2 (xT22 ) <

π∗ST2 (xT2

2 ) must hold, which then directly implies πSC2 < πST

2 . Define x2 as theunique root of F2. Straight-forward calculations give

x2 =2γ + η2 − 3η

2(γ − η2)(1 − c).

Using (20) we obtain

F ′1(x2) ≥ 0

⇔ ((4γ − η2)2 − 9ηγ(2 − η))(1 − c) − ((4γ − η2)2 − 9γη2)x2 ≥ 0

⇔ 2(γ − η2)((4γ − η2)2 − 9ηγ(2 − η)) − (2γ + η2 − 3η)((4γ − η2)2 − 9γη2) ≥ 0

⇔ 3η(1 − η)(γ − η2)(4γ − η2) ≥ 0.

30

The last inequality holds due to the assumption that 0 ≤ η ≤ γ ≤ 1, so we haveshown that F ′

1(x2) ≥ 0. Furthermore, we have F ′1(x

T22 ) < 0 and it follows from

the concavity of F1 that x2 < xT22 . Since F2(x2) is monotonously decreasing

this implies that F2(xT22 ) < 0 and therefore π∗SC

2 (xT22 ) < π∗ST

2 (xT22 ). This com-

pletes the proof.



2 = xT22 and C∗

1 (xT12 ) = T in the subgame where

P1 = S:

i) πST2 > πSC

2 ,

ii) xT12 < xO

2 < xT22

iii) xO2 − xT1

2 ≥ xT22 − xO

2 .

We deal first with conditions ii) and iii). Denote again by x2 the argmax ofF1. Since F1 is a quadratic function we have xT1

2 < x < xT22 and xT2

2 − x2 =x2 −xT1

2 . If x2 < c we can choose a value of β satisfying (3) such that x2 ≥ xO2 .

This implies that regardless of I2 we have |xO2 − xT1

2 | ≥ |xT22 − xO

2 |. Thereforeconditions iii) holds whenever xT1

2 < xO2 < xT2

2 . Taking into account that xT22

goes to infinity for I2 → 0 (see the proof of Proposition 6) and becomes close tox2 for sufficiently large I2, it follows from the monotonicity of xT2

2 with respectto I2 that there exists a I2 such that xO

2 = xT22 .

Using the same arguments as in the proof of Proposition 8 we obtain thatF2(x2) < 0 for all x2 ∈ [0, c] whenever γ < γ3. In particular, under thatcondition we have π∗ST

2 (xO2 ) > πSC

2 (xN2 ) ≥ πSC

2 . For I2 = I2 this impliesπST

2 = π∗ST2 (xO

2 ) > πSC2 . By continuity this inequality also holds for values of

I2 < I2 if I2 − I2 is sufficiently small. Since xT22 is a decreasing function of I2

for any such value of I2 we also have xT12 < xO

2 < xT22 . Put together, we have

shown that under the conditions γ < γ3 and x2 < c there exist values of β andI2 such that i) - iii) is satisfied. A sufficient condition for x2 < c is F ′

1(c) < 0.Setting γ = η and using (20) we obtain

F ′1(c) < 0

⇔ ((4η − η2)2 − 9η2(2 − η))(1 − c) − c((4η − η2)2 − 9η3) < 0

⇔ η2(−2 + η + η2 − 2c(1 − η)(7 − η)) < 0.

Obviously the last inequality is true for all η < 1. Using continuity of F ′1(c) with

respect to changes in γ this implies that there exists γ4 > η such that F ′1(c) < 0

for all γ ∈ [η, γ4]. Setting γ4 = min(γ3, γ4) completes the proof. .


In a subgame where firm 1 has completed its product innovation project the

31

maximization problems of the two firms read

maxqO

1,qN

1

π1(qO1 , qN

1 , q2) = (1 − γqN1 − η(qO

1 + q2))qN1 + (1 − (qO

1 + q2) − ηqN1 )qO

1 − c(qN1 + qO

1 )

maxq2

π2(qO1 , qN

1 , q2;x2) = (1 − (qO1 + q2) − ηqN

1 )q2 − (c − x2)q2.

Assume that there is an interior solution with qO∗1 (x2) > 0. Exploiting the two

first order conditions for Firm 1 we obtain

qO∗1 (x2) =

(γ − η)(1 − c)

2(γ − η2)−

1

2q∗2(x2). (21)

It is well know that equilibrium quantities in quantity competition are decreasingfunctions of the own marginal costs. Accordingly, we have q∗2(x2) ≥ q∗2(0) forall x2 ≥ 0 and therefore qO∗

1 (x2) ≤ qO∗1 (0). Using the first order condition for

Firm 2 gives

q∗2(0) =(2γ − η)(1 − c) + 2(η2 − γ)qO∗

1 (0)

(4γ − η2.

Inserting this into (21) gives

qO∗1 (0) = (1 − c)

γ((2(γ − η) − η(1 − η)

6γ(γ − η2)

and it is straight forward to see that qO∗1 (0) < 0 if condition stated in the propo-

sition holds. Accordingly, in equilibrium qO∗1 (x2) = 0.


(i): Following the arguments of the proof of Proposition 7 and using Proposition12 we only have to show that there exist values of γ ∈ [η, γ3] such that F ′

1(0) > 0.For γ = γ3 we have

F ′1(0) =

η2(2 − η)(1 − η)(1 − c)

(4γ − η2)2> 0

and we obtain the result by continuity of F ′1(0) with respect to γ.

(ii): By Proposition 12 in equilibrium Firm 1 does not offer both products forall γ ∈ [γ2, γ3] and the proof of Proposition 8 applies.(iii): Follows directly from Propositions 10 and 12 by setting γ4 = min[γ3, γ4].

.Proof of Proposition 14:

If qO∗1 = 0 the identical action profile has to be a lucky innovator equilibrium in

the base model, which contradicts Proposition 9. So, we only need to considerthe case where qO∗

1 > 0, qN∗1 > 0, q∗2 > 0. Analogous to π∗SC

1 (x2) denote byπ∗SC

1 (x2) the profit function of Firm 1 for cases where in equilibrium Firm 1

32

offers both products in the market. Analogous define π∗SC2 (x2). If F1(x2) =

π∗SC1 (x2) − π∗ST

1 (x2) ≥ 0 this implies that it is optimal for Firm 1 to completethe product innovation under the assumption that the new product is added toits product portfolio. Calculating the equilibrium quantities in this setting andinserting into Firm 1’s profit function gives

F1(x2) =(1 − c)2(1 − η)2

4(γ − η2)− I2.

This expression is independent from x2 and therefore if Firm 1 decides to offerpositive quantities of both products for some x2, it would also offer positivequantities of both products for all x2-values in a neighbourhood. Accordingly, inany equilibrium where Firm 1 offers strictly positive quantities of both productsx2 must be given by a local maximum of π∗SC

2 . Because of the strict concavityof π∗SC

2 there exists only one local maximum in [0, c] which coincides with themaximum of π∗SC

2 in [0, c]. Therefore the equilibrium value of x2 corresponds tothe ex-post optimal level of x2 and we cannot have a lucky innovator equilibriumwhere Firm 1 offers positive quantities of both products.

33

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