Technological Change: A Burden or a Chance*

Transcription

1 Technological Change: A Buden o a Chance* Veena Hagspiel 1, Pedo Jesus, Pete M. Kot 3,4, Cláudia Nunes 1 Nowegian Univesity of Science and Technology NTNU, Depatment of ndustial Economics and Technology Management, 7491 Tondheim, Noway Depatment of Mathematics and CEMAT, nstituto Supeio Técnico, Av. Rovisco Pais, Lisboa, Potugal 3 CentER, Depatment of Econometics and Opeations Reseach, Tilbug Univesity, 5000 LE TLBURG, The Nethelands 4 Depatment of Economics, Univesity of Antwep, Belgium Septembe 16, 013 Abstact n this pape we conside a fim that has to deal with technological change facing a declining pofit steam fo its established poduct. The fim has the following options to choose fom: it can eithe exit the industy o invest in a new technology with which it can poduce an innovative poduct. Futhemoe, it has the possibility to tempoaily suspend opeations befoe taking an ievesible decision. n case the fim decides to launch the new poduct we analyze the fim s optimal capacity choice opposite to a scenaio whee the fim has infinity capacity at its disposal. We find that depending on this assumption the investment theshold is monotonic o non-monotonic as a function of uncetainty. t is monotonic unde capacity choice and non-monotonic unde full flexibility. Contay to standad eal options theoy we find that the effect of uncetainty on the exit theshold is non-monotonic when taking into account the capacity choice decision. Futhemoe, we conduct an analysis of the effect of capacity size of the old maket on the investment decision in a new poduct as well as the exit decision. We find that in case the fim invests out of suspension the chosen capacity level stays constant fo diffeent sizes of the old maket s poduction capacity. Regading the fim s suspension option, we conclude that the fim suspends opeation only when uncetainty is high and innovative poduct maket is consideed vey attactive. 1

2 1 ntoduction The photogaphy industy undewent a disuptive change in technology duing the 1990s when the taditional film was eplaced by digital photogaphy see e.g. The Economist Januay 14th 01. n paticula Kodak was lagely affected : by 1976 Kodak accounted fo 90% of film and 85% of camea sales in Ameica. Hence it was a nea-monopoly in Ameica. Kodak s evenues wee nealy 16 billion in 1996 but the pediction is that it will decease to 6. billion in 011. Kodak tied to get squeeze as much money out of the film business as possible and it pepaed fo the switch to digital film. The esult was that Kodak did eventually build a pofitable business out of digital cameas but it lasted only a few yeas befoe camea phones ovetook it. Accoding to M Komoi, the fome CEO of Fujifilm of , Kodak aimed to be a digital company, but that is a small business and not enough to suppot a big company. Fo Kodak it was like seeing a tsunami coming and thee is nothing you can do about it, accoding to M. Chistensen in The Economist Januay 14th 01. This pape focuses on industies that have to deal with technological change. The above example showed that this can be a buden. Howeve thee ae enough industies whee technological change bings fuitful times in tems of pofits. One example is the video game industy, whee innovation plays a big ole. The publishes, Activision, saw thei woldwide sales incease with $650m in the fist five days, when the new video game Call of Duty: Black Ops eplaced its pedecesso, Call of Duty: Moden Wafae, in Novembe 010 The Economist, Decembe 10th 011. Anothe example is the Phone launched by Apple that was descibed by Time Magazine as the invention of the yea 007. n 011 net income was $7.31bn in the thee months to 5 June, 15% highe than a yea ealie and a ecod quately pofit fo the fim. Revenue was $8.6bn, also a quately ecod. We study the poblem of a pice setting fim that poduces with a cuent technology fo which it faces a declining sales volume. The fim has thee options: it can eithe exit this industy, invest in a new technology with which it can poduce an innovative poduct o suspend poduction fo a cetain amount of time. The fim is a monopolist in a maket chaacteized by uncetain demand. Demand is diven by a demand intecept assumed to follow a geometic Bownian motion. We distinguish between two scenaios in the sense that the esulting new maket can be booming o ends up to be smalle than the old maket used to be. We assume that the technology necessay to bing this innovative poduct on the maket is aleady available to the company. This technology was eithe povided by othe fims o was invented by the R&D laboatoy of the fim itself. The question we study is when and if it is optimal to launch this poduct. n case the fim decides to launch the new poduct we also analyze the optimal capacity choice with which the fim decides to ente

3 the innovative poduct maket. Since the new technology is aleady pesent, ou model does not analyze the innovation pocess concening the invention of the technology. Besides, adopting the new technology the fim has the option to exit the maket at any point in time, i.e. the fim can decide to exit the maket of the old poduct because it consides the potential of the new poduct maket not pofitable enough to invest and thus, decides to exit befoe launching the new poduct. The exit option is conseved beyond the time of the potential investment in the new poduct. Thus, the fim can also exit the maket of the new poduct ievocably at any time. Befoe taking the ievesible decision to exit o invest, the fim futhemoe has the possibility to tempoaily suspend poduction. This suspension option esults in thee possible scenaios egading the maket of the cuent technology. The fim might neve suspend poduction because it is optimal to exit the maket befoe aleady. n the second scenaio opeation might not be pofitable anymoe at all consideing the cuent poduct maket and the fim has to decide whethe to exit o invest in the innovative maket out of suspension mode. n the thid scenaio the fim once suspended, decides to exit the maket if demand continues to fall o esume poduction if demand inceases again in the futue. We find that the fim uses the option to suspend opeation only when uncetainty is high, and the paametes that make the second maket attactive ae high. Fo the specific case that the fim invests out of suspension we find that the chosen capacity level stays constant in diffeent values of the cuent maket capacity size as well as in the scaling paamete of the demand intecept γ. The eason fo this is that the fim gives up the same evenue steam independent of γ when investing in the new poduct since it does not poduce befoe investment. n fact, we find that the fim invests such that also afte the investment the evenue steam is the same independent of γ. Theefoe, the capacity choice fo the new maket is insensitive to γ. We intoduce ou model in Section 3 and deive the esult that the optimal policy of the consideed stopping poblem exits and is unique. n Section 4 we specify a model assuming that the fim has infinite capacity available. This means we do not conside the capacity decision. n this case the fim is fully flexible egading poduction and can downscale, incease o suspend poduction at any point in time. We use this model as a benchmak in the analysis of ou main model pesented in Section 3. Analyzing the investment timing decision of the benchmak model we find that the effect of uncetainty on the investment theshold is contay to the standad eal options esult non-monotonic. The investment theshold inceases in uncetainty fo low values while it deceases fo high values of uncetainty due to the existence of the exit option. A simila esult was obtained by Kwon 010 who analyzes a one-time oppotunity to invest in impoving a cuent technology with declining pofit steam. He finds that, if the technology boost is sufficiently lage, then the investment theshold deceases in demand uncetainty. Howeve, we show that in case the investment decision does not only involve timing but also the choice of the optimal capacity size, 3

4 the investment theshold is again monotonically inceasing in uncetainty. Regading the exit decision, the standad eal options esult says that the exit theshold is deceasing with uncetainty Dixit 1989, Kwon 010 since the value of waiting inceases. We find that this esult, howeve, does not hold anymoe when the investment decision involves timing and capacity choice. We show that fo ou famewok the effect of uncetainty on the exit theshold is non-monotonic whee the inceasing pat is caused by the effect of capacity choice. This pape is oganized as follows. We eview elated liteatue in Section. Ou model is pesented in Section 3 followed by a simplified vesion that seves as benchmak in the analysis. The compaative statics analysis of the optimal policies fo both models is conducted in Section. Ou main esults ae pesent in Section 6 and we conclude in Section 7. Related Liteatue WORK N PROGRESS 3 Model - Capacity, Timing The fim cuently poduces an established poduct. The quantity, which has to be detemined at each point of time, is q 1, the pice is p 1 and the invese demand function is given by p 1 = θ q 1, in which the pocess θ t follows the geometic Bownian motion dθ t = α 1 θ t dt + σθ t dz t, with constant dift α 1 that is assumed to be negative and volatility σ. We distinguish between two types of cost. On the one hand the fim faces a fixed cost F. On the othe hand it has to incu unit poduction costs being equal to c. Due to the latte featue it can be optimal to tempoaily suspend poduction, i.e. q 1 = 0 fo some time. The fim has the option to stat poducing an innovative poduct which equies an investment in poduction capacity. The capacity of the new poduct is denoted by K. The investment cost is sunk and equal to δk. Denoting the pice and the quantity of the new poduct by p and q, espectively, at the moment of the new poduct launch the fim s demand function changes into: p = γθ q. 1 4

5 The fim poduces up to full capacity, i.e. q = K, except in the cases that demand falls so low that the pice would tun negative. n that case the fim tempoaily suspends poduction, i.e. q = 0. Because this innovative maket gows faste than the old one, we assume a diffeent speed of development. n paticula the dynamics of θ now become dθ = α θdt + σθdz. with α > α 1. n fact, we can define the stochastic pocess θt as follows dθt = αtθtdt + σθtdz, with α 1 fo t < τ, αt = α fo t τ, whee this τ will be specified late. The cost stuctue fo the new poduct also changes afte the new poduct launch. Wheeas the fixed cost still equals F, thee ae no vaiable cost. We motivate this by obseving that in the digital wold the unit cost of a poduct is most of the time vey small o negligible. n case the fim decides to invest in the new poduct, the fim chooses the optimal timing as well as the optimal size of the capacity investment. t can be the case that the new maket is not pofitable enough fo an investment to be undetaken. Since the old maket is deceasing ove time it can be optimal fo the fim to execise the option to exit the maket. We also allow fo the possibility to exit the maket afte the investment in the innovative poduct has taken place. Theefoe, the optimal stopping poblem can be stated as follows: τ1 Vθ 0 = sup E e t Π 1 θ 1 tdt + e τ1 max {0, τ 1 0 τ } max sup E e tτ1 Π θ t τ 1, K dt K θ 0 = θ 1 τ 1 δk θ 1 0 = θ 0 τ 1 {τ >τ 1 } τ 1 3 Hee, τ 1 denotes the fist time at which the decision make decides to invest in poduct o to exit the maket. τ denotes the time that the fim would decide to exit the maket of poduct, in case it has invested in the fist un. To detemine the value of investing in poject, we fist solve the subpoblem that is stated at the ight hand side of the maximization in equation 3. Consideing a specific cuent value fo θ 0 the net expected discounted pofit of investing in poject is given by: 5

6 τ V θ 0 = sup E θ0 τ = sup E θ0 τ τ 1 ττ 1 0 e tτ1 Π θ t τ 1 dt, 4 e t Π θ tdt, τ = sup E θ0 e t Π θ tdt τ 0 whee E θ denotes the expectation with espect to the pobability law Q θ of the pocess {θt; t > 0} stating at θ0 = θ R n. The optimal stopping poblem in 4 is a standad poblem. The instantaneous pofits in egion ae given by, Π θ = p q F = γθ K K F = γk θ K F. The fim poduces as long as the pice is positive p > 0 but will suspend poduction in case the pice tuns negative. Theefoe the pofit is given by γk θ K F Π θ = F fo θ > K γ, fo θ K γ. We denote the suspension theshold in maket by K γ = θ S. Taking into account that thee is an option to exit the maket, standad calculations see, e.g. Chapte 6 in Section of Dixit and Pindyck 1994 lead to the following expession fo the optimal value function V : γk α V θ, K = θ K +F + D 1 θ β4 fo θ > K γ, F + D θ β3 + D 3 θ β4 fo θ K γ, whee β 3 β 4 is the positive negative oot of the quadatic equation 1 σ ββ 1 + α β = 0. Hee we assume that the fim will exit out of the suspension egion. Howeve, it is also possible that the fim will exit befoe eve enteing the suspension egion i.e. θ S n this case the value function is given by V θ, K = γk θ K + F α = K γ < θ E with θ E denoting the exit theshold. + G 1 θ β4. n the following we have to distinguish those two cases. The exit theshold θ E fo the two cases as well as the specific expessions of the constant paametes D 1, D, D 3 and G 1, can be easily deived using the fact that the value function has to be continuous and smooth in the suspension theshold and applying value matching and smooth pasting at the exit theshold. The deivations as well as the explicit expessions fo the paamete values ae given in Appendix A.1. The exit thesholds fo the two diffeent cases ae given by θ E = 6 1 β4 β α F β 3 β 4 α γ K1 β 3, 5

7 fo case θ S > θ E and θ E = α γ β 4 K + FK, 6 β 4 1 fo case θ S < θ E. Now we can wite equation 3 as τ1 { } Vθ 0 = sup E e t Π 1 θ 1 tdt + e τ1 max 0, max V θ 0 = θ 1 τ 1, K δk θ 1 0 = θ 0. τ 1 0 K Now we conside the situation befoe the investment. The fim has essentially thee options. The fist is to invest in poduct while in poduction. The second is to suspend poduction and in the meanwhile invest in poduct. And the thid is to exit the maket. Let us fist detemine the cuent instantaneous pofits. The fim will suspend poduction when p 1 c = θ K 1 c < 0, i.e., when θ < θ S1 instantaneous pofit equals: Π 1 θ = F K 1 θ ck 1 + K 1 + F fo θ > c+k1, fo θ c+k1. = c+k1. The Denoting the exit theshold by θ E1 we can conside the following two cases: f θ E1 θ S1 then the fim will neve suspend poduction because it aleady has exited. f θ E1 < θ S1 thee exists a θ-inteval, whee it is optimal fo the fim to suspend poduction. n this egion the fim has two options: eithe to esume poduction if θ has inceased sufficiently o to exit the maket, which will happen when θ deceases even moe. n the following we will denote the invest theshold by θ. Figue 1 illustates the diffeent egions fo the thee options. The following poposition states that thee always exists the optimal policy fo the stopping poblem and specifies the optimal value function of the fim. Figue 1: Diffeent Cases Poposition 1 The optimal policy fo the stopping poblem of equation 3 always exists. The optimal value 7

8 function is uniquely given by V 1 θ fo θ D = θ E1, θ, Vθ = Ωθ othewise, whee we distinguish thee cases fo the function V 1.. f θ E1 given by c+k1, the value function V 1. is uniquely Fo case θ E1 V 1 θ = K 1θ ck 1 + K1 + F α 1 < c+k1 the value function is equal to + A 1 θ β1 + A θ β. V 1 θ = if θ c+k1 and equal to K 1θ α 1 ck1+k 1 +F + B 1 θ β1 + B θ β fo θ c+k1, F + B 3θ β1 + B 4 θ β fo θ < c+k1, V 1 θ = F + C 1θ β1 + C θ β, in case θ < c+k1. The value of the fim in the stopping egion is equal to Ωθ = max {0, V θ 0 = θ, K }. The optimal continuation egion is D = θ E1, θ. t is optimal to exit the maket when θ < θ E1 poduct when θ > θ. Othewise, it is optimal to continue opeations. and invest in the new n case the fim decides to invest in the new poduct instead of exiting, it does choose the timing as well as the size of this investment. Deiving this optimal investment decision we follow?. Fist, we ignoe the timing decision fo a moment and just take into account the deivation of the optimal capacity. Second, we deive the optimal timing of the investment o exit decision, espectively, taking the optimal capacity size fo given demand intecept θ in account. t is impotant to note that we wish to compute max K 0,γθ {V θ, K δk }. So if this maximum is not on the bounday we can just compute the zeo of Vθ,K δk K and then check that V θ,k δk K < 0. Othewise the maximum will be at the bounday. Poposition gives the optimal capacity decision as well as the equations that implicity give the exit as well as the investment decision fo the diffeent cases. To compute the optimal K we wite the expession of the deivate, fo the case that the maximum is not on the bounday. Note that we have to distinguish 6 cases. 8

9 Poposition The exit and investment thesholds fo the six diffeent cases, espectively, ae given implicitly by the following equations. Fo the pupose of eadability let s conside the following constants: a 1 = K 1 ; a = ck 1 + K1 + F ; a 3 = γ α 1 a 5 = γβ4 β 3 α β 3 β 4 α ; a 6 = γ β4 ; a 4 = α β 4 α γ β 4 1 β4 α 1 β 4 β 3 F β 3 β 4 α F β 3 β 4 Fo the fist thee cases it holds that θ E θ S. Theefoe if K is not on the bounday, K is implicitly given by the following equation: a 3 θ K + a 5 β 4 K 1β4 + a 6 β 4 β 3 K β 4 β 3 1 β 3 1 θ β4 = δ 7 The theshold fo those thee cases, is implicitly given by the solution of thee subsequent equations, espectively. 1. Case 1: n case of θ E1 > θ S1 and θ E θ S it holds that β 1 1 a 1 θ 1β + a 3 K a 1 θ 1β β 1 a θ β β 1 a 1 θ 1β1 β 1 β 4 + a 3 K a 1 θ 1β1 β β β 4 a K + F a 5 K β4 + a 6 K β4 a θ β1 1 a K + F a 5 K β4 + a 6 K β4 δk θ β β 3 + θ β4β = 0 8 δk θ β1 1 β 3 + θ β4β1 = 0 9. Case : n case of θ E1 < θ S1 < θ and θ E θ S it holds that c + K1 θ E1 = β 1 1 a 1 ck 1 + K β 1 c + K1 β 1 a K + F β 1 β 4 ck 1 + K1 β β β 4 a 5 K β4 + a 6 K β4 β 1 c + K1 1β + a 3 K a 1 θ 1β 1 β 3 θ β4β δk θ β 1β1 + a 3 K a 1 θ 1β1 c + K1 a 1 β1 a K + F a 5 K β4 + a 6 K β4 1 β 3 θ β4β1 + 1 β 1 ; 10 β 1 F δk θ β1 F β θ β1 + =

10 3. Case 3: n case of θ E1 < θ < θ S1 and θ E θ S it holds that β 1 β 4 θ E1 = K β 1 1a 3 K θ 1β β 1 + δk a 5 K β4 + a 6 K β4 1 β 3 θ β4β β 1a 3 K θ 1β1 β K β β 4 a 5 K β4 + a 6 K β4 1 β 3 1 β 1 F + δk θ β4β1 θ β + 1 β ; 1 F β θ β1 θ β1 + = 0 13 Regading the emaining thee cases it holds that θ E > θ S and theefoe if K is not on the bounday, it is implicitly given by the solution of the following equation: a 3 θ K β 4 F + K β 4 a 4 K + FK β4 θ β4 = δ 14 K β 4 1 The theshold fo those thee cases, is implicitly given by the solution of thee subsequent equations, espectively: 4. Case 4: n case of θ E1 > θ S1 and θ E > θ S it holds that β 1 1 a 1 θ 1β β 1 a 1 θ 1β1 + a 3 K a 1 θ 1β β 1 β 1 β 4 + a 3 K a 1 θ 1β1 β β 4 a θ β γk 1 α β 4 β a θ β1 γk 1 α β 4 a K + F δk θ β a 4 K + FK 1β4 a K + F a 4 K + FK 1β4 δk θ β1 θ β4β = 0 15 θ β4β1 =

11 5. Case 5: n case of θ E1 < θ S1 < θ and θ E > θ S it holds that c + K1 θ E1 = β 1 1 a 1 ck 1 + K 1 c + K1 β 1 β β 4 1β + a 3 K a 1 θ 1β β a K + F γk 1 β 1 β 4 a 4 K + FK 1β4 α β 4 c + K1 β 1 a 1 ck 1 + K 1 c + K1 β γk 1 α β 4 K + F K α γ θ β4β δk θ β 1β1 + a 3 K a 1 θ 1β1 β1 a K + F β 4 β 4 1 1β4 θ β4β1 1 β 1 ; 17 β 1 F δk θ β1 F β θ β1 6. Case 6: n case of θ E1 < θ < θ S1 and θ E > θ S it holds that K θ E1 = β 1 1a 3 K θ 1β β 1 + δk γk 1 β 1 β 4 K + F 1β4 α β 4 1 θ β4β α β 4 K γ β 4 1 β 1 F β β 4 γk 1 α β 4 K + F K α γ β 1a 3 K θ 1β1 β K 1β4 β 4 θ β4β1 β δk = 0 18 θ β + 1 β ; 19 θ β1 F β θ β1 = Analysis of Benchmak Model n this section we pesent a model that will seve as a benchmak fo the analysis of the model intoduced in Section 3. We now assume that the fim has infinite capacity. Theefoe, including a paamete K is not necessay. n ode to stat poducing an innovative poduct, the fim has to pay sunk cost. All othe assumptions emain unchanged. Theefoe the optimal stopping poblem does not diffe much fom the pevious case: τ1 Vθ 0 = sup E e t Π 1 θ 1 tdt + e τ1 max {0, τ 1 0 τ sup E e tτ1 Π θ t τ 1 dt θ 0 = θ 1 τ 1 } τ 1 {τ >τ 1 θ 10 = θ 0, 1 } τ 1 Hee, τ 1 denotes the fist time at which the decision make decides to invest in poduct o exit the maket. τ denotes the time that the fim would decide to exit the maket of poduct, in case it has invested in the fist un. 11

12 As befoe, V is given by: τ V θ 0 = sup E θ0 e t Π θ tdt. τ 0 The instantaneous pofits in egion ae given by Π = p q F = γθ q q F. n this model, we compute the optimal quantity to poduce. That quantity equals q = γθ, which esults in a pofit of Π θ = γ θ F. 4 Taking into account that thee is an option to exit the maket, standad calculations, given in Appendix A.3, lead to the following expession fo the optimal value function V : V θ = γ θ 4 α σ F + Aθβ4. whee β 4 is again the negative oot of the quadatic equation 1 σ ββ 1 + α β = 0. Afte simila computations as in the pevious model, we obtain the following expession fo θ E : F θ E = γ 4 α β4 σ. β 4 Now we conside the situation befoe the investment. Let us fist detemine the cuent instantaneous pofits. Since thee is no estiction in the poduction quantity, maximizing the pofit function w..t to the optimal output quantity, we deive that: q 1 = θ c. Fom this expession we see that the fim will suspend poduction, povided that it has not exited aleady, wheneve θ is below c. The instantaneous pofit, theefoe, equals: θc F fo θ > c Π 1 θ =, F fo θ c. Denoting the exit theshold by θ E1 we can conside the following two cases: f θ E1 c will neve suspend poduction because it aleady has exited. f θ E1 < c then the fim thee exists a θ-inteval, whee it is optimal fo the fim to suspend poduction. n this egion the fim has two options: eithe to esume poduction if θ has inceased sufficiently o to exit the maket, which will happen when θ deceases even moe. θ S = c denotes the suspension theshold. The following poposition states that thee always exists the optimal policy fo the stopping poblem and specifies the optimal value function of the fim. 1

13 Poposition 3 The optimal policy fo the stopping poblem of equation 1 always exists. The optimal value function is uniquely given by V 1 θ fo θ D = θ E1, θ, Vθ = Ωθ othewise, whee we distinguish two cases fo the function V 1.. f θ E1 c then the function is equal to while fo case θ E1 V 1 θ = < c V 1 θ = θ 4 α 1 σ cθ α 1 + c 4 K + A 1θ β1 + A θ β, the function is equal to θ 4α 1σ cθ α 1 + c 4 K + B 1θ β1 + B θ β fo θ c, K + B 3θ β1 + B 4 θ β fo θ < c, if θ > c and equal to V 1 θ = K + C 1θ β1 + C θ β, if θ < c. The value of the fim in the stopping egion is equal to Ωθ = max 0, V θ 0 = θ. The optimal continuation egion is D = θ E1, θ. t is optimal to exit the maket when θ < θ E1 in the new poduct when θ > θ. Othewise, it is optimal to continue opeations. and invest n ode to deive the two thesholds θ and θ E1 we apply the value matching and smooth pasting conditions which leads to the following systems of equations, stated in Poposition 4, that implicitly define the thesholds fo the diffeent cases. Poposition 4 Consideing the following constants, b 1 = 4 α 1 σ ; b = c α 1 ; b 3 = c 4 F ; b 4 = γ 4 α σ ; b 5 = F, the investment and exit theshold fo the thee cases, espectively, ae implicitly given by the following equations. f θ E1 > c the thesholds θ and θ E1 ae implicitly given by β b 1 θ β1 b 1 b 4 θ β1 b β 1 θ 1β1 θ 1β1 + b 3 β θ β1 θ β1 β 1 b 1 θ β b 1 b 4 θ β β b 5 + θ β1 + Aθ β4β1 β β 4 = 0, 3 b β 1 1 θ 1β θ 1β + b 3 β 1 θ β θ β β 1 b 5 + θ β + Aθ β4β β 1 β 4 =

14 Fo the case that θ E1 < c < θ it holds that the thesholds ae defined by the following equations system. β1 1 c θ E1 = β b 1 b 1 b 4 θ b 5 β β1 1β1 β1 c c b β 1 θ 1β1 + b 3 + b 5 β θ β1 1 β θ β1 + Aθ β4β1 β 1 β β 4, 5 β 1 b 1 c β b 1 b 4 θ β β 1 θ β b β 1 1 c 1β θ 1β b 3 + b 5 β 1 c β θ β + Aθ β4β β 1 β 4 b 5 β 1 θ β = And fo the case that θ E1 < θ < c the following equations implicitly define the thesholds. θ E1 = 1 b 4 β θ β1 β θ β1 b 5 β b 4 β 1 θ β β 1 θ β 1 + A β β 4 θ β4β1 β 1, 7 + A β 1 β 4 θ β4β b 5 β 1 θ β = Compaative Statics n this section we conduct a compaative statics analysis of the value functions V, V in maket 1 and, espectively as well as the exit theshold θ E. We fist establish the convexity of V which leads to the compaative statics with espect to σ. Poposition 5 The optimal etun function V is convex fo both cases. The poof is stated in Appendix B. Next, we examine the compaative statics of V. with espect to α, σ and θ 0. Poposition 6 The optimal etun function V is non-deceasing in α, in σ and θ 0 fo both cases. f θ 0 > θ E then V is stictly inceasing in α. These esults lead us to the compaative statics egading the exit theshold with espect to α and σ, stated in the following poposition. Poposition 7 The exit theshold θ E then θ E is stictly deceasing in α. is non-inceasing in α, in σ and θ 0 fo both cases. f θ 0 > θ E Based on the expessions fo θ E, in equations 5 and 6 fo the fist model and fo the benchmak model, we can futhe infe that: 14

15 Remak 1 The exit theshold θ E is invesely popotional to γ in both cases. Remak n the fist model, fo the case θ S < θ E i.e. exit out of poduction, fo K abitaily small o lage, θ E is abitaily lage. Fo the othe case, if β 3 >, θ E is inceasing with K, othewise θ E deceases with K. Now fo an analysis of the value function of maket 1, V.: Poposition 8 The optimal etun function V is convex fo both cases. Poposition 9 Fo all θ R +, the value function of the fim V is nondeceasing in α 1, α and σ. WRTE SOMETHNG HERE Poposition 10 The pobability that the fim is investing athe than exiting, i.e. theshold θ is hit befoe θ E1, is given by the pobability that P = α 1 1 θ0 σ θ E1 1 α θ σ θ E1 1 Poposition 11 Let T E = inf{t : θ t = θ E θ 0 = θ } be the time that the fim exits the second maket once it has invested. Then its expected value is given by Poposition 1 E T E = θ ln θ E 1 if α σ α < 1 σ, othewise. Let T denote the time until the company decides, optimally, to invest in a new technology o to exit the maket which one occus fist. Then 6 Results ET = 1 1 ln σ α 1 θ0 θ E1 1 θ0 1 θ θ E1 1 α 1 σ α 1 1 σ θ E1 6.1 Effect of inceasing uncetainty on theshold θ ln. θ E1 The standad eal options esult says that the investment theshold goes up with inceased uncetainty eflecting the value of waiting. 30 Tables and 1 show fo the consideed investment decision, a diffeent esult that holds. The investment theshold fist goes up with uncetainty but deceases fo high values of uncetainty. The latte is caused by the existence of the exit option. The value of this option inceases 15

16 Table 1: Benchmak Model. Paamete Values: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 0., γ = 1., F =, = 100. Thus θ S = c = 5. σ θ θ E θ E Case 1 P invest θs x 1.56% 35.4% 44.18% 55.1% 71.97% ET E θ0=θ Table : Benchmak Model. Paamete Values: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 0., γ = 1., F =, = 100. Thus θ S = c = 5. σ θ θ E θ E Case: 3 P invest θs 58.90% 66.88% 73.10% 79.71% 88.57% x ET E θ0=θ

17 with uncetainty and when investing the fim gains this option to exit. As a esult the value of investment inceases with uncetainty and theefoe the fim wants to invest soone. A simila esults was obtained by Kwon 010. Howeve, he used a Bownian motion with dift to model the uncetainty in the pofit steam. As a esult it is not clea what the elation is between the investment timing with the theshold. Modeling the uncetainty with a geometic Bownian motion allows us to give insight about the effect of uncetainty on the theshold as well as the timing given that the fim invests. See the expession fo the expected time in Poposition 1. Table Howeve, when the investment decision not only involves timing but also capacity size has to be detemined the investment theshold is again monotonically inceasing in uncetainty. When uncetainty is high the fim invests in a lage capacity. This implies that the exit theshold fo the second poduct is highe and thus the second poduct will be poduced duing a shote time. This educes the value of the investment so that the fim has less incentive to invest and it invest late. 6. Effect of inceasing uncetainty on exit thesholds Fom standad eal options liteatue it is known that the exit theshold is deceasing with uncetainty see e.g. Dixit 1989 and Kwon 010. The intuition is that the value of the option to esume poduction o investment in the innovative poduct is highe when uncetainty is lage and theefoe, the fim wants to keep these options alive. Howeve, Tables 3 and 4 show that in ou famewok the effect of uncetainty on the exit theshold is non-monotonic. The inceasing pat is caused by the effect of the capacity choice. Due to the inflexible poduction stuctue the fim is foced to poduce up to capacity all the time. When uncetainty goes up the capacity size K and thus poduction is lage. This leads to consideable losses when θ is low. This induces the fim to exit befoe the θ is too low. The deceasing pat is due to the eason aleady given, i.e. the value of keeping the options to esume poduction and/o investing in the innovative poduct alive inceases with uncetainty. 6.3 Effect of old maket poduction capacity The decision of the fim to invest in the new maket depends stongly on the capacity size in the old maket. Wok in Pogess 6.4 When does suspension not occu? When the value of the option to stat again poduction is low, the fim will not use the suspension option but instead exit opeations. This happens in cases that uncetainty is low and the dift is sufficiently low. n addition in case of consideing suspension vesus exit in the fist maket suspension does not occu when 17

18 Table 3: Paamete Values: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, γ = 1., K 1 =, δ = 10, F =, i.e. θ S1 = c+k1 = 3, θ0 = θ +θ E1. σ K θ θ E θ E Case ET P 41.73% 4.74% 43.14% 41.71% 39.34% 39.67% ET E Table 4: Paamete Values: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, γ = 1., K 1 =, δ = 10, F =, i.e. θ S1 = 3, θ0 = θ +θ E1. σ K θ θ E θ E Case ET P 44.6% 44.75% 44.80% 44.49% 4.43% ET E

19 Table 5: Paamete Values: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, γ = 1., σ = 0.1, δ = 10, F =, θ0 = θ +θ E1. K K θ θ E θ E Case ET P 4.0% 39.36% 39.80% 4.74% 44.96% 44.99% ET E Table 6: Paamete Values: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, γ = 1., σ = 0.1, δ = 10, F =, θ0 = θ +θ E1. K K θ θ E θ E Case ET P 4.93% 4.30% 43.51% 44.75% 44.75% 44.75% ET E 19

20 Table 7: Paamete Values: α 1 = α = 0.0, = 0.1, c = 0, = γ = 1, σ = 0.1, δ = 10, F =, θ0 = θ +θ E1. K K θ θ E θ E Case ET P 33.10% 14.69% 10.36% 9.08% 8.53% 8.9% ET E the option to invest in the new poduct is not attactive, i.e. γ is low and α as well as σ ae also low. 6.5 Constant capacity when investing out of suspension Tables 8, 9 and 10 show that the capacity level is insensitive to γ when the fim invests out of the suspension egion, i.e. Cases 3 and 6. This can be explained as follows. nvesting out of the suspension egion implies that the fim does not poduce befoe investment. Theefoe, the fim gives up the same cuent evenue when investing in the innovative poduct. Next, we want to ague that the fim invests such that also afte the investment the evenues ae the same. We do this by showing that the fim chooses the investment theshold such that the pice as well as the pice dynamics stay the same. Fom the table it can be obtained that the value of γθ stays constant in γ. The latte chaacteistic implies that the output pice at the moment of investment fo the innovative poduct is the same in all situations. To conside the dynamics, we deive dp = α γθdt + σγθdz 31 fom Equations 1 and. Hence the conclusion is that also the pice dynamics stay the same fo diffeent γ. We conclude that in all situations the investment of δk geneates the same evenue steam in all situation. This explains why the chosen capacity level K is also the same. The same easoning explains why the capacity stays constant when changing K 1 while investing out of suspension, see Table 6 fo K 1 =,.5, 3. 0

21 Table 8: Model fom poposition 1: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, σ = 0.1, K 1 =, δ = 10, F =, i.e. θ S1 = 3, θ0 = θ +θ E1. γ K θ θ E θ E Case ET P 7.77% 44.98% 45.0% 45.0% 44.99% 45.1% ET E Table 9: Model fom poposition 1: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, σ = 0.1, K 1 =, δ = 10, F =, i.e. θ S1 = 3, θ0 = θ +θ E1. γ K θ θ E θ E Case ET P 4.45% 44.75% 44.76% 44.78% 44.80% 44.73% ET E 1

22 Table 10: Model fom poposition 1: α 1 = 0.0, α = 0.01, = 0.1, c = 1, = 1, σ = 0., K 1 =, δ = 10, F =, i.e. θ S1 = 3, θ0 = θ +θ E1. γ K θ θ E θ E Case ET P 41.3% 44.7% 44.7% 44.7% 44.74% 44.70% ET E Table 11: Paamete Values: α 1 = α = 0.0, = 0.1, c = 1, = 1, γ = 1., σ = 0.1, δ = 10, F =, θ0 = θ +θ E1. K K θ θ E θ E Case ET P 41.54% 37.5% 36.44% 40.09% 44.% 45.14% ET E

23 7 Conclusions WORK N PROGRESS A Additional Model Details A.1 Maket n ode to deive the constant paametes D 1, D, D 3 and G 1 of the value function and the exit thesholds fo the two cases, we study the decision to exit the maket. Denoting the exit theshold by θ E, we can wite down the following value matching and smooth pasting conditions fo the case θ E < θ S : whee V, θ θ=θe = 0, 3 V, θ θ = 0, 33 θ=θe V,1 θ K θ= = V, θ K γ θ=, 34 γ V,1 θ θ = V, θ K θ= θ, 35 K γ θ= γ V θ, K = γk α θ K +F + D 1 θ β4 fo θ > K γ F + D θ β3 + D 3 θ β4 fo θ K γ =: V,1 θ, =: V, θ. Equations 77 and 78 account fo the fact that the value function has to be smooth in θ = K γ. Solving equations 75 to 78 one can easily deive the exit theshold θ E D 1, D and D 3 : θ E = D 1 = D = 1 β4 β α F β 3 β 4 α γ K1 β 3, β 3 α α β 3 β 4 γβ4 K β4 + D 3, β 4 α α β 3 β 4 γβ3 K β3, D 3 = θ β4 E β 3 β 3 β 4 F. 36 and the expessions fo the paametes Fo the second case, i.e. θ E > θ S, the value matching and smooth pasting conditions ae given by V θ θ=θe = 0, 37 V θ θ = θ=θe 3

24 Solving this equation system one obtains A. Maket 1 θ E = α γ β 4 G 1 = γk α 1 β 4 β 4 1 α K + FK, γ β 4 K + F 1β4. β 4 1 K n ode to deive the constant paametes and the investment as well as the exit theshold θ and θ E1 we use the value matching and smooth pasting conditions. Those ae given fo the thee diffeent cases in the following. A..1 Case 1 - θ E1 > θ S1 and θ E θ S The value matching and smooth pasting conditions fo case 1 ae given by: V 1 θ θ=θe1 = 0 39 V 1 θ θ = 0 40 θ=θe1 V 1 θ θ=θ = V θ θ=θ δk 41 V 1 θ θ = V θ θ=θ θ 4 θ=θ Equations 8 and 83 lead to the following expessions fo the paametes A 1 and A : A 1 = a 1β 1θ 1β1 β 1 β A = a 1β 1 1θ 1β β β 1 a β θ β1 a β 1 θ β Fom equations 84 and 85 alongside the fist two we obtain equations 8 and 9 which implicity give the investment and exit theshold θ and θ E1. 4

25 A.. Case - θ E1 < θ S1 < θ and θ E θ S Fo case the coesponding value matching and smooth pasting conditions ae given by: V 1, θ θ=θe1 = 0 43 V 1, θ θ = 0 44 θ=θe1 V 1,1 θ c+k θ= 1 V 1,1 θ θ θ= c+k 1 = V 1, θ c+k θ= 1 = V 1,θ θ θ= c+k V 1,1 θ θ=θ = V θ θ=θ δk 47 V 1,1 θ θ = V θ θ=θ θ 48 θ=θ Equations 86 though 89 leads to the following expessions fo the paametes B 1 B 4 : B 1 = B = a 1 β 1 a 1 β 1 1 β B 3 = θ β1 F β β 1 β B 4 = θ β 1 F β 1 β 1β1 c+k 1 ck + 1+K β 1 c+k 1 β1 c+k 1 + B 3 β β 1 1β ck + 1+K β β1 1 c+k 1 + B 4 β 1 β The equations 10 and 11 ae obtained fom 90 and 91, whee the fist explicitly gives an expession fo θ E1 and the latte an implicit expession fo θ. A..3 Case 3 - θ E1 < θ < θ S1 and θ E θ S The value matching and smooth pasting conditions fo this case ae given by V 1 θ θ=θe1 = 0 49 V 1 θ θ = 0 50 θ=θe1 V 1 θ θ=θ = V θ θ=θ δk 51 V 1 θ θ = V θ θ=θ θ 5 θ=θ As befoe, fom the fist two equations 9 and 93 esult in the following expessions fo the paametes C 1 and C, espectively. 5

26 β C 1 = θ β1 F β β 1 β C = θ β 1 F β 1 β The equations 1 and 13 ae obtained fom 94 and 95, whee the fist explicitly gives an expession fo θ E1 and the latte an implicit expession fo θ. Fo the othe thee cases we have the same value matchings and smooth pastings, the only change is on V, theefoe the values of A, B and C do not change. Wheeas the expessions fo θ and θ E1 accodingly, but ae obtained like in the pevious cases. change A.3 Maket n ode to deive the constant paamete A we study the decision to exit the maket, we can wite down the following value matching and smooth pasting conditions: V θ = γ θ 4 α σ F + Aθβ4. V θ θ=θe = 0, 53 V θ θ = θ=θe Solving equations 96 to 97 one can easily deive the exit theshold θ E paamete A: A.4 Maket 1 θ E = F γ 4 α σ. F A = θ β4 E β 4 β4 β 4 and the expessions fo the A.4.1 Case 1 - θ E1 > c Value matching and smooth pasting fo this case give: V 1 θ θ=θe1 = 0 55 V 1 θ θ = 0 56 θ=θe1 V 1 θ θ=θ = V θ θ=θ 57 V 1 θ θ = V θ θ=θ θ 58 θ=θ 6

27 Equations 98 and 99 lead to the following expessions fo the paametes A 1 and A : A 1 = b 1β θ β1 A = b 1β 1 θ β b β 1θ 1β1 β 1 β b β 1 1θ 1β β β 1 + b 3 β θ β1 + b 3 β 1 θ β Fom equations 100 and 101 alongside the fist two we obtain equations 3 and 4 which implicity give the investment and exit theshold θ and θ E1. A.4. Case - θ E1 < c < θ Value matching and smooth pasting fo this case give: V 1, θ θ=θe1 = 0 59 V 1, θ θ = 0 60 θ=θe1 V 1,1 θ θ= c V 1,1 θ θ θ= c = V 1, θ θ= c = V 1,θ θ θ= c 61 6 V 1,1 θ θ=θ = V θ θ=θ 63 V 1,1 θ θ = V θ θ=θ θ 64 θ=θ Equations 10 though 105 leads to the following expessions fo the paametes B 1 B 4 : B 1 = B = β1 1β1 c b 1 β c b β 1 + β b 3 + b 5 β1 c + B 3 β 1 β β 1β β c b 1 β 1 c b β c β1 b 3 + b 5 + B 4 β β 1 B 3 = θ β1 β β β 1 b 5 B 4 = θ β β 1 β 1 β b 5 The equations 5 and 6 ae obtained fom 106 and 107, whee the fist explicitly gives an expession fo θ E1 and the latte an implicit expession fo θ. A.4.3 Case 3 - θ E1 < θ < c The value matching and smooth pasting conditions fo this case ae given by 7

28 V 1 θ θ=θe1 = 0 65 V 1 θ θ = 0 66 θ=θe1 V 1 θ θ=θ = V θ θ=θ 67 V 1 θ θ = V θ θ=θ θ 68 θ=θ As befoe, fom the fist two equations 9 and 93 esult in the following expessions fo the paametes C 1 and C, espectively. C 1 = θ β1 β β β 1 b 5 C = θ β β 1 β 1 β b 5 The equations 7 and 8 ae obtained fom 110 and 111, whee the fist explicitly gives an expession fo θ E1 and the latte an implicit expession fo θ. B Poofs B.1 Poof of Poposition 1 Now assume that V 1. in that poposition is a candidate fo the optimal value function, with thee cases. n the following we veify that V 1. it indeed satisfies all the sufficient conditions fo being the optimal value function specified in Theoem of?. Oksendal s φ., f. and g. ae hee given by V 1., Π 1. and V θ, K K, espectively. And we have that V = R + 0, D = θ, θ, theefoe D = {θ E1, θ }. Conditions iii, iv, viii and ix hold tivially because θ follows a geometic Bownian motion. V 1. is continuous diffeentiable in V since we impose the value matching and smooth pasting conditions in D, alongside θ S1 see equations 8 though 95, which elates with i. Futhemoe, V 1. is twice continuously diffeentiable except at D {θ S1 }. Since V 1. is a polynomial in θ, the second ode deivatives of V 1. ae finite nea D, which checks condition v. Moeove, we intoduce the patial diffeential opeato L applied to the pocess {θt; t 0}:L = t + αθθ θ + 1 σ θ θ. But since the time-dependence of the etun function is only thoufh the discount facto e t, the infinitesimal geneato can be eplaced by: L = + α 1 θ θ + 1 σ θ θ To obtain ii we use simila easoning as?. To achieve vi and vii we conside one case of V 1, while fo the othe cases simila calculations will apply. 8

29 V 1 θ = K 1θ α 1 ck1+k 1 +F + B 1 θ β1 + B θ β fo θ c+k1 F + B 3θ β1 + B 4 θ β K 1 θ + ck 1 + K1 + F LV 1 θ = F fo θ < c+k1 fo θ c+k1 fo θ < c+k1 Theefoe condition vii holds since LV 1 θ + Π 1 θ = 0. Fo condition vi we poceed similaly as?. B. Poof of Poposition All the calculus needed ae stated in Appendix A.. B.3 Poof of Poposition 3 The poof is simila to the poof of Poposition 1, theefoe we omit it. B.4 Poof of Poposition All the calculus needed ae stated in Appendix A.4. B.5 Poof of Poposition 5 Fist we show that Π is convex fo both cases. Fo the case Model - Capacity, Timing the pofit function is given by: γk θ K F fo θ > K γ Π θ =, F fo θ K γ. Let θ x = xθ xθ, whee x 0, 1. f θ 1, θ > < K γ case θ 1 K γ f θ x K γ : f θ x > K γ : θ then: Π θ x xπ θ xπ θ F xf + 1 xγk θ K F K γ θ Π θ x xπ θ xπ θ then the convexity obviously holds. Fo the γk xθ xθ K F xf + 1 xγk θ K F θ 1 K γ 9

30 Fo the Benchmak model, the pofit function is given by Π θ = γ θ F. 4 which is obviously convex. Taking into account that in both cases the pofit function Π. is convex fo evey θ 1, θ and x 0, 1, we obtain the following inequality: τ V θ x = sup E θx e t Π θ tdt, τ 0 τ = sup E θx e t Π θ x e τ 0 τ x sup E θ1 e t Π θ 1 e τ 0 α σ α σ t+σz t dt, t+σz t dt τ + 1 x sup E θ e t Π θ e τ 0 α σ t+σz t dt, = xv θ xv θ, which concludes ou poof. 69 B.6 Poof of Poposition 6 Let > 0 and denote by V θ, α the value function V with the dependence on the dift of the pocess hee denoted by α. τ V θ, α = sup E θ e t Π θe τ 0 τ sup E θ e t Π θe τ 0 Whee inequality 70 follows fom the fact that e α σ α + σ t+σz t dt, t+σz t dt, 70 V θ, α α σ +σz t < e α + σ +σz t with pobability 1 and Π is non-deceasing in θ. The inequality becomes stict if τ > 0 i.e. θ 0 > θ E. Moeove, as τ is the optimal stopping time fo the poblem with dift α then it is suboptimal fo the poblem with dift α +, which poves inequality 71. Concening the non-deceasing behavio of V as a function of the volatility σ, we efe to?, page 73, whee in a note he efes that fo convex contact functions the option pice, when the stock pice follows a geometic Bownian motion, is non-deceasing in the volatility. This esult holds in ou case as V is convex see Poposition 5. Finally, V is non-deceasing in θ 0 because Π is also non-deceasing in θ. 30

31 B.7 Poof of Poposition 7 Noting that θ E = inf{θ : V θ > 0} the esult follows in view of the last poposition. Since V inceases with the incease of α, σ o θ 0, its gaphic will ise, and theefoe zeo will slide to the left, deceasing with the incease of any of those vaiables. B.8 Poof of Poposition 8 n the following we show that the pofit function in egion 1, Π 1., is a convex function. Fo the case Model - Capacity, Timing the pofit function is given by: Π 1 θ = F K 1 θ ck 1 + K 1 + F fo θ > c+k1, fo θ c+k1. Let θ x = xθ xθ, whee x 0, 1. f θ 1, θ > < c+k1 then the convexity obviously holds. Fo the case θ 1 c+k1 θ it holds that if θ x c : Π 1 θ x xπ 1 θ xπ 1 θ F xf + 1 xk 1 θ ck 1 + K 1 + F c + K 1 θ f θ x > c : Π 1 θ x xπ 1 θ xπ 1 θ K 1 xθ xθ ck 1 + K 1 + F xf + 1 xγk θ K 1 F θ 1 c + K 1 which shows the convexity fo the emaining cases. Fo the Benchmak model, the pofit function is given by: θc F fo θ > c Π 1 θ =, F fo θ c. Let θ x = xθ xθ, whee x 0, 1. f θ 1, θ > < c case θ 1 c θ it holds that then the convexity obviously holds. Fo the 31

32 if θ x c : Π 1 θ x xπ 1 θ xπ 1 θ θ c F xf + 1 x F c θ if θ x > c : Π 1 θ x xπ 1 θ xπ 1 θ θ xθ1 + 1 xθ c c F xf + 1 x F θ 1 c which shows the convexity fo the emaining cases. n a next step, we define τ F θ 1 = 0 e t Π 1 θ 1 t dt θ 1 0 = θ 1. Then F θ 1 is a convex function in θ 1 by the same easoning that we used fo inequality 69. Finally, taking into account that the maximization of V ove K peseves the convexity of the function, as well as the maximum and the sum of two convex functions, then τ { } Vθ 1 = sup E θ1 e t Π 1 θ 1 tdt + e τ max 0, max V θ 1 τ, K K, τ 0 K whee K is δk in the fist model and in the Benchmak model, is also a convex function. B.9 Poof of Poposition 9 By poposition 6, V is nondeceasing in α, and since α affects only maket, V is also nondeceasing in α. Denote by Vθ, α 1 the value function V with the dependence on the dift of the pocess in maket 1, hee denoted by α 1. Let > 0 such that α 1 + < α. 3

33 τα1 Vθ0, α 1 = sup E θ τ α1 0 + e τ α 1 max τα1 sup E θ τ α1 0 + e τ α 1 max e t Π 1 θ0e α 1 σ t+σz t dt { 0, max V θ τ K α1, α K } e t Π 1 θ0e α 1+ σ { 0, max V θ τ K α1e τ α 1, α K } t+σz t dt 7 73 Vθ0, α whee in 7 we use the fact that Π 1 is non-deceasing. We showed befoe that V is inceasing with the initial value θ 0, which poves 73 of the inequality. Finally in 74 the sub-optimality of τ α1 is used. n ode to pove the behavio of V as a function of σ, we follow Theoem 4 of?. Conside dx = α x, which is an inceasing function on R + ; then all the conditions of Theoem 4 of? ae satisfied. 1. Let νθ 0 = E θ0 e τ fθ τ, whee in ou case f denotes the etun of stopping at τ which includes the etun of max{0, V }. Since max{0, V } is non-deceasing in σ, as poved befoe, we conclude based on Theoem 4 that V is a non-deceasing function of the volatility paamete σ. B.10 Poof of Poposition 10 Let θt denote the demand level at time t, with t < τ 1, so that the dift coefficient of the diffusion equation of θ is α 1. Theefoe it follows that θt is given by: θt = θ0 exp } {α 1 σ t + σw t θt > θ W t > 1 σ ln θ θ0 θt < θ E1 We wish to have θt > θ befoe θt < θ E1 : W t < 1 σ ln θe1 θ0 1 σ 1 σ α 1 σ α 1 σ By theoem 4.1 of?, we have that if Yt is a Wiene Pocess, if γ 1 > 0, γ < 0, δ 1 = δ 0, then the pobability that Y t γ 1 + δ 1 t fo a smalle t than any t fo which Y t γ + δ t is: Fo ou case we have that 1 n?, the function d is denoted by θ P = eγδ1 1 e γ1γδ1 1 t t 33

34 γ 1 = 1 σ ln θ > 0 θ0 γ = 1 σ ln θe1 < 0 θ0 δ 1 = 1 α 1 σ σ Theefoe: So e γ1γδ1 e γδ1 { θe1 = exp ln θ0 { θ = exp ln θ0 / θ θ0 P = α 1 1 θ0 θ E1 α 1 θ θ E1 1 α } 1 σ 1 = α } 1 σ 1 = σ 1 σ 1 θe1 θ0 θe1 θ α 1 σ 1 α 1 σ 1 B.11 Poof of Poposition 11 Note that { T E = inf t : z t + 1 σ σ α t = 1 σ ln θ θ E which is the fist passage time of a Bownian motion with dift 1 σ σ α } thought a level 1 σ ln θ θ E. f σ α > 0, as θ > θ E then the expected value is finite and the esult follows in view of Poposition 8.5 of?. Othewise it is infinite, as the state 1 σ ln θ θ E is null-ecuent. B.1 Poof of Poposition 1 The esult follows in view of the example of section 10.9 of?, with A given by α 1 and B given by σ, as the egion Ω in ou case is just the inteval θ E1, θ a time homogeneous egion. A Additional Model Details A.1 Maket n ode to deive the constant paametes D 1, D, D 3 and G 1 of the value function and the exit thesholds fo the two cases, we study the decision to exit the maket. Denoting the exit theshold by θ E, we can wite down the following value matching and smooth pasting conditions fo the case θ E < θ S : 34

35 whee V, θ θ=θe = 0, 75 V, θ θ = 0, 76 θ=θe V,1 θ K θ= = V, θ K γ θ=, 77 γ V,1 θ θ = V, θ K θ= θ, 78 K γ θ= γ V θ, K = γk α θ K +F + D 1 θ β4 fo θ > K γ F + D θ β3 + D 3 θ β4 fo θ K γ =: V,1 θ, =: V, θ. 79 Equations 77 and 78 account fo the fact that the value function has to be smooth in θ = K γ. Solving equations 75 to 78 one can easily deive the exit theshold θ E D 1, D and D 3 : θ E = D 1 = D = 1 β4 β α F β 3 β 4 α γ K1 β 3, β 3 α K α β 3 β 4 γβ4 β4 + D 3, β 4 α α β 3 β 4 γβ3 K β3, D 3 = θ β4 E β 3 β 3 β 4 F. and the expessions fo the paametes Fo the second case, i.e. θ E > θ S, the value matching and smooth pasting conditions ae given by Solving this equation system one obtains A. Maket 1 θ E = α γ V θ θ=θe = 0, 80 V θ θ = θ=θe β 4 G 1 = γk α 1 β 4 β 4 1 α K + FK, γ β 4 K + F 1β4. β 4 1 K n ode to deive the constant paametes and the investment as well as the exit theshold θ and θ E1 we use the value matching and smooth pasting conditions. Those ae given fo the thee diffeent cases in the following. 35

36 A..1 Case 1 - θ E1 > θ S1 and θ E θ S The value matching and smooth pasting conditions fo case 1 ae given by: V 1 θ θ=θe1 = 0 8 V 1 θ θ = 0 83 θ=θe1 V 1 θ θ=θ = V θ θ=θ δk 84 V 1 θ θ = V θ θ=θ θ 85 θ=θ Equations 8 and 83 lead to the following expessions fo the paametes A 1 and A : A 1 = a 1β 1θ 1β1 β 1 β A = a 1β 1 1θ 1β β β 1 a β θ β1 a β 1 θ β Fom equations 84 and 85 alongside the fist two we obtain equations 8 and 9 which implicity give the investment and exit theshold θ and θ E1. A.. Case - θ E1 < θ S1 < θ and θ E θ S Fo case the coesponding value matching and smooth pasting conditions ae given by: V 1, θ θ=θe1 = 0 86 V 1, θ θ = 0 87 θ=θe1 V 1,1 θ c+k θ= 1 V 1,1 θ θ θ= c+k 1 = V 1, θ c+k θ= 1 = V 1,θ θ θ= c+k V 1,1 θ θ=θ = V θ θ=θ δk 90 V 1,1 θ θ = V θ θ=θ θ 91 θ=θ Equations 86 though 89 leads to the following expessions fo the paametes B 1 B 4 : 1β1 c+k a 1 β 1 1 ck + 1+K β1 β 1 c+k 1 B 1 = + B 3 β β 1 1β c+k a 1 β ck + 1+K β β1 1 c+k 1 B = + B 4 β 1 β β B 3 = θ β1 F β β 1 β B 4 = θ β 1 F β 1 β The equations 10 and 11 ae obtained fom 90 and 91, whee the fist explicitly gives an expession fo θ E1 and the latte an implicit expession fo θ. 36