Technical Memorandum Number 815. Bigger Slice or Larger Pie? Optimal Marketing Strategies for New Firms. John Angelis Moren Lévesque

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1 Techncal Memorandum Number 815 Bgger Slce or Larger Pe? Optmal Marketng Strateges for New Frms by John Angels Moren Lévesque June 26 Department of Operatons Weatherhead School of Management Case Western Reserve Unversty 33 Peter B Lews Buldng Cleveland, Oho 4416

2 Bgger Slce or Larger Pe? Optmal Marketng Strateges for New Frms John Angels a, * and Moren Lévesque b a Weatherhead School of Management Case Western Reserve Unversty 19 Eucld Avenue Cleveland, OH Phone: Fax: ohn.angels@case.edu b School of Engneerng Unversty of Waterloo 2 Unversty Avenue West Waterloo, Ontaro, Canada N2L 3G1 Phone: ext 5367 Fax: levesque@uwaterloo.ca Abstract Just as ther establshed counterparts do, new frms that poneer nascent ndustres strve to ncrease proft by acqurng the sales of rval frms. But these new frms can also ncrease sales by attractng new customers to ther nascent ndustry. Ths artcle nvestgates ths tradeoff n marketng expendture. By dervng condtons under whch an ncrease n the sales of ts rval can augment a frm s proft n a sngle-perod model, we hghlght how new frms can dstngush themselves. Extensve numercal analyss on a mult-perod framework suggests that an ncrease n a frm s unt proft margn, effectveness n ganng new sales, or ntal sales level, but a decrease n sales decay, may cause a postve spllover for ts rval. Ths research thus provdes gudance to new frms attemptng to fnd a balance between growng ther ndustry (.e. the pe) and competng on market share (.e. the slce). Keywords: Marketng, Nascent Industry, Expendture Strateges, Optmal Control, Dfferental Games * Correspondng author We are thankful to Bo Carlsson, Lsa Mallart, Matthew Sobel and Danel Solow for ther nvaluable comments and suggestons. John Angels gratefully acknowledges fnancal support from the Weatherhead School of Management Summer Research Fundng at Case Western Reserve Unversty. 1

3 1. Introducton Investng n advertsng based on product characterstcs and product comparsons can allow a new frm to acqure exstng ndustry sales unts belongng to other new frms. The targetng of another frm s sales to gan market share.e. ncrease the slce of the pe can be qute drect. An example of such targetng s Mcrosoft s campagn aganst AOL n the Internet servce provder (ISP) market. AOL had become the market leader among ISPs by offerng users an easy-to-use product. MSN then offered, among other benefts that targeted AOL users, an applcaton for AOL users who chose to swtch to MSN that transferred the e-mal addresses and calendar nformaton from the AOL program to the MSN program (Hu 22). But a new frm can also beneft from nvestng to create customers.e. ncrease the whole pe. Such a customer has not yet been attracted to any frm and s consdered an unrealzed customer for the ndustry untl he becomes a customer to one of the frms. Unlke n establshed ndustres, entrepreneurs must nform unrealzed customers about ther new ventures and also convnce prospectve customers that they are legtmate. Aldrch and Fol (1994) refer to ths as cogntve legtmaton, whereby the new frm needs the spread of knowledge among unrealzed customers to become wder so that more unrealzed customers wll trust that frm and become realzed customers. To do so, entrepreneurs often use awareness or nformatonal advertsng (Dorf and Byers 25). Sometmes the nvestment n attractng unrealzed customers may be ndrect and amed at changng ndustry condtons or government regulatons. For nstance, AOL, MSN, and other ISP frms made nvestments to remove threats to ndustry sales growth, such as choosng to fle ont lawsuts n an attempt to slow down Internet spam communcaton. Brandenburger and Nalebuff (1996) refer to ths smultaneous need to compete and cooperate as co-opetton. What makes nascent competton nterestng s the need for competng new frms to buld prmary demand (.e. ncrease the overall customer demand for the ndustry) by nvestng to attract unrealzed customers. Once a frm makes an unrealzed customer aware of ts product, all nascent ndustry members can compete for the busness of that now realzed customer. Thus, frms may potentally beneft from the spllover of other frms ncreased prmary demand as well as ther own. In fact, most entrepreneurs wll never be able to ndependently create a new market smply by buldng ther companes (Goldsten 24: B2). The rado-frequency dentfcaton tag (RFID) ndustry, for nstance, had been around snce the 2

4 198 s wth relatvely low sales. Industry sales grew rapdly when several new frms agreed to common standards, whch made t possble for Wal-Mart and the Department of Defense to mandate RFID tag usage throughout ther supply chan. Orders for hundreds of mllons of tags for Wal-Mart s supplers wll now be fulflled by many dfferent manufacturers wthn the RFID ndustry. New frms certanly are aware that market share actvtes alone (ncreasng ther slce of the pe va sales acquston) wthout any market growth actvtes (ncreasng the sze of the pe va sales attracton) s lkely to lead to a decrease n the sze of the market. The decrease occurs because realzed customers depart the ndustry due to ther own forgetfulness or dssatsfacton wth some aspect of the ndustry, and no new customers are replacng them. In the acton pursut games ndustry, Bernsten (24) argues that because communcaton has been focused almost exclusvely on product sales, ths gnores the fact that mprovng the envronment for makng sales IMPROVES [emphass by the author] sales [ ] [U]ltmately, there won t be a market unless all of us learn a lttle more about good PR (p 14). Consequently, optmal promotonal strateges should not only evaluate the proper expendture amount, but also how much to spend on unrealzed as compared to realzed customers. We ntentonally use the term expendture rather than advertsng because we want to consder the full range of nvestment actvtes that affect purchases. As outlned n Aldrch and Fol (1994), legtmzng actvtes that help grow sales nclude common standard-settng, nfluencng thrd-party actors, or governmental lobbyng not ust tradtonal advertsng. We therefore ask how a new frm attemptng to maxmze proft should choose between nvestng to attract unrealzed customers, whch also ncreases total ndustry sales, and nvestng to acqure realzed customers, whch only ncreases the new frm s sales. The selecton of the tme frame between product launch and sgnfcant sales penetraton allows us to complement the lterature on new product marketng, as well as on market share games n olgopoles. The gap n the entrepreneural marketng lterature s artculated by Morrs et al. (22), who comment that an unanswered queston s whether optmal levels of entrepreneural marketng exst and, f so, what parameters nfluence those optmal levels. Soberman and Gatgnon (25) also draw attenton to the lack of marketng research drected towards understandng the relatonshp between compettve dynamcs and market evoluton. Studes on market share games n olgopoles model only one advertsng expendture varable, 3

5 whch thus assumes the effect of a new frm s advertsng to be the same on all customers. Fruchter (1999) ponts out that her olgopoly advertsng model would be more accurate f t would specfy the dfference n effcency between acqurng a customer from a rval frm and attractng unrealzed customers. Thus, we address a gap n the lterature by modelng customers accordng to whether they are realzed or as yet unrealzed. The remander of the artcle s organzed as follows. We revew relevant lterature and then formulate a model where the expendture for attractng unrealzed sales unts s modeled as a separate decson than that of the expendture for acqurng sales unts from the frm s rval. We derve optmal expendture strateges for two competng frms n a sngle-perod and a multperod (dfferental game) model. For the latter, we use numercal and graphcal methods to further our understandng of ndustry growth. Senstvty analyss of the optmal expendtures s the emphass of ths numercal analyss. We conclude wth a dscusson on extensons and opportuntes for future work. 2. Lterature Revew We focus on the lterature on market share games n olgopoles. Frst, we revew studes that deal wth spllover benefts occurrng smultaneously wth competton among frms. Then, we hghlght the work on contnuous-tme monopoly settngs and olgopoly settngs (dfferental games) where customer segmentaton occurs. Wrather and Yu (1979) drectly model spllover benefts to other frms n an olgopoly settng. In ther framework, a frm can nvest n product expendtures (whch have spllover benefts for other frms) or brand expendtures (whch are predatory n nature and acqure the sales of other frms). The obectve of both frms s to maxmze sales. They derve equlbra and comment on ncentves for cooperaton; however, there s no dynamc component to ther model. In a mult-perod, dscrete-tme olgopoly settng, Fredman (1983) sets up a game where multple frms smultaneously decde on ther quantty sold and advertsng level. A parameter s used to model an ndustry level of cooperatve or predatory behavor. However, that parameter s the same for the entre ndustry and unaffected by advertsng expendtures or tme. Also, Fredman (1983) does not dstngush between unrealzed and realzed sales, as we do. In Krshnamurthy s (2) sngle-perod game, nstead, frms choose generc advertsng (whch expands the market) and brand advertsng (whch ncreases frm market share). He assumes that 4

6 generc advertsng s cooperatve and focuses on when cheap-rdng occurs. Krshnamurthy et al. (23) argue that cooperatve generc advertsng s lmted to ndustres subect to government regulaton or experencng sales declne. Nether phenomenon s lkely for a nascent ndustry. The Lanchester model of combat frst suggested for advertsng applcatons n Kmball (1957) assumes that a frm acqures compettors sales unts va ts advertsng expendtures and loses ts own sales unts va compettors advertsng expendtures. Most Lanchester models consder ndustry sales to be of fxed sze. On the other hand, Vdale and Wolfe (1957) examne a stuaton where there are unrealzed sales and, as a result, ndustry sales are not fxed. They model the growth and decay of the sales of a monopolst who can attract new sales from unrealzed sales by ncreasng advertsng expendtures. The wanng effect of advertsng expendtures s modeled by a decay factor that acts on realzed sales of the frm. Scholars have combned the concepts of the Lanchester and Vdale-Wolfe advertsng models to relax ther lmtatons. The Vdale-Wolfe model s augmented by Seth (1973), who expands the modelng of a monopoly and Deal (1979), who ncludes competton n a duopoly. Some of the more recent combnatons of the two models are Fruchter (1999), who adds growth to a Lanchester model, and Wang and Wu (21), who nclude decay and growth n a Lanchester formulaton. However, ths lterature stll represents advertsng actvty by one varable that equally mpacts all sales not belongng to the frm. (Extensve revews on dfferental games that use advertsng expendture as decson varable and sales or market share as state varable appear n Erckson 1995, 23 and Fechtnger et al ) There has recently been ncreased nterest n customer segmentaton n dfferental games, usng dfferent varables to represent nvestments n retanng the frm s own customers, acqurng customers from another frm, and attractng new customers to the ndustry. For multple frms and fxed ndustry sales, Fruchter and Zhang (24) model expendtures for retanng and acqurng customers. Instead, Hartl and Kort (25) model expendtures for retanng and attractng customers n a sngle frm context. Because of our focus on nascent ndustres, we treat retenton only n passng. The mult-perod duopoly model that most closely resembles our work s that of Bass et al. (25a, 25b), whch models brand and generc advertsng for two frms n an nfnte and fnte horzon context, respectvely. The generc advertsng of one frm drectly benefts the other ust the same as f the other frm would have done the generc advertsng. Ther model assumes that the market sze expands lmtlessly wth 5

7 generc advertsng and thus that ndustry sales are non-decreasng over tme. For the nfnte horzon model, they present as an extenson a model that assumes that the market has a celng. We model a market wth a celng and sales decay n a fnte horzon context. We also dffer from ths work n that our expendture toward unrealzed customers s not a publc good type of generc advertsng. Instead, the other frm only benefts from a frm s expendtures that mpact unrealzed customers by acqurng the newly realzed customers of that frm. Our modelng contrbuton to the olgopoly lterature s twofold. Frst, we are able to represent the compettve components of the Lanchester model and the market growth and decay components of the Vdale-Wolfe model. To our knowledge, no other work has yet ncluded the decay component as well; we show that sales decay has a strong effect on expendture polces and profts. Second, we separate the sales of the rval frm from unrealzed sales n a mult-perod context. As stated above, most prevous research models expendture as one varable. However, for a new frm, the one-varable modelng approach assumes that frm expendtures prmarly amed at acqurng another frm s sales have the same mpact on unrealzed sales, and frm expendtures meant to attract unrealzed sales have the same mpact on another frm s sales. Such an assumpton may hold true n mature ndustres, where even customers who have yet to buy are famlar wth the product and an establshed frm gans sales only at the expense of other frms. However, ths assumpton s lkely to be volated n a nascent ndustry. Realzed customers are typcally early adopters and nnovators (Moore 2), whle most unrealzed customers need to be educated about the product (Dorf and Byers 25) and overcome legtmacy concerns about the ndustry (Aldrch and Fol 1994) before becomng realzed customers. Also, a new frm can gan sales from attractng unrealzed customers, whch does not lower the sales of any other frm. The separaton n sales necesstates usng two expendture varables for dstnct nvestments n realzed and unrealzed sales (as done n Wrather and Yu 1979 and n Bass et al. 25a, 25b). 3. Optmal Control Model 3.1. Control Varables Let t represent contnuous tme and T be the length of the plannng horzon. Frm chooses to nvest n acqurng the sales of another frm at a cost ; ths nvestment s n share expendtures. Symmetrcally, frm chooses to nvest n attractng new customers va the unrealzed group of customers at a cost w ( t) a ( t) ; ths nvestment s n growth expendtures. Growth 6

8 expendtures can also be thought of as a frm s nvestment n speedng up product dffuson to potental customers. As the nascent ndustry grows to maturty, the total sales of the frms approach the market celng. As a result, there are fewer and fewer unrealzed customers to be acqured over tme. Yet, to ncrease ndustry sales, frm must attract unrealzed customers to tself va w ( t) customers of frm va a ( t)., whereas to turn realzed ndustry sales nto sales, frm must nvest to acqure 3.2. Law of Moton for Sales We focus on a duopoly of frms and. Let t, and (t) be that of frm. Hence, S S ( t) be the number of unts sold by frm at I( t) = S ( t) S ( t) + (1) s the number of unts sold by the entre ndustry at t. We were nspred by the Lanchester model (Kmball 1957) and Vdale and Wolfe (1957) for our formulaton of the law of moton of S ( t). The share of realzed ndustry sales held by each frm vares over tme as a result of share expendtures a ( t). The effectveness of share expendtures depends on the frm as well as product and market condtons. We denote the effectveness of share expendtures as a constant ρ (,1]. Frm s share expendtures have an mpact of value ρ f ( a ( t)) on acqurng the sales of frm, whereas t loses ts own sales at a rate determned by the share expendtures of frm. Thus, each dollar of share expendtures that mpacts the other frm s sales va f ( a ( t )) yelds a ρ fracton of the other frm s realzed sales. The functon f( a( t )) s postve and ncreasng n ts argument. Frm benefts from frm s growth va ρ f ( a ( t)). These combned effects on frm s sales are ρ f ( a ( t)) S ( t) ρ f ( a ( t)) S ( t). (2) Industry sales reach market saturaton as frms buld prmary demand. Let m be the fxed number of unts at whch the market saturates, whch corresponds to the maxmum possble sales for the ndustry (the assumpton that the market celng can be known s also used by Vdale and Wolfe 1957, Deal 1979, Fruchter 1999, and Wang and Wu 21). Industry growth can only occur by attractng unrealzed sales unts that no frm has yet acqured, specfcally m I() t such 7

9 unts. We allow frm attracton of unrealzed customers to depend on the mpact of ts growth expendtures, as do Vdale and Wolfe. Effectveness of growth expendtures for frm s denoted as a constant β (,1]. Frm s expendtures have an mpact of value β g( w( t)) on attractng unrealzed sales unts to frm. Thus, each dollar of growth expendtures that mpacts unrealzed sales va g ( w ( t)) yelds a β fracton of unrealzed sales. The functon g ( w ( t)) s postve and ncreasng n ts argument. There are also non-expendture related factors that affect sales. Let [1 ] [,1] be the sales decay parameter for frm ; t represents the proporton of sales unts δ that depart the frm (.e. become unrealzed) due to factors such as forgetfulness and product and market characterstcs. These combned effects on frm s sales are β g ( w ( t))[ m S ( t) S ( t)] [1 δ ] S ( t). (3) From combnng (2) and (3), the change n frm s sales over tme can be expressed as S( t) = ρ f( a( t)) S( t) ρ f ( a( t)) S( t) + βg( w( t))[ m S( t) S( t)] [1 δ] S( t). (4) The four terms n (4), respectvely, are the gans n sales unts acqured from competton, losses n sales unts due to competton, gans n sales unts due to the entry of unrealzed sales unts nto the ndustry, and losses n sales unts due to the ext of prevously realzed sales unts from the ndustry. We note that f () and g() can be chosen so that equaton (2) s the Lanchester equaton that descrbes frm competton for exstng sales, and equaton (3) s the Vdale-Wolfe 1 model descrbng growth and decay of sales Obectve Functon Let π be frm s unt proft margn, whch s assumed fxed over tme. Followng the lead of Deal (1979), frm s obectve s to select on the fnte horzon [,T] a stream over tme of share expendtures and of growth expendtures that maxmze ts proft (net of advertsng costs) plus a valuaton of the frm s end-of-horzon sales. Formally, T a(), t w() t t [, T] P = Max [ π S () t a () t w ()] t dt + rs ( T), (5) 1 The orgnal 1957 model uses [m I(t)]/m as opposed to m I(t). We can add 1/m by adustng β accordngly. 8

10 where r s the salvage value frm places on end-of-horzon sales. For a frm solely concerned wth proft, r equals zero. However, a frm that cares about maxmzng ts end-of-horzon sales level more than proft would set r at a relatvely hgh postve level. Smlar to Wang and Wu (21), we do not nclude a dscount factor n the obectve functon. Ths allows for the model recommendatons for nvestment over tme to be more nterpretable. Also, because we are nterested n the nascent perod of the ndustry, T would be relatvely small, makng dscountng unessental. Frm s complete optmal control formulaton ncludes (4), (5) and the followng condtons. Intal sales are set to a constant, S () = S, whereas end-of-horzon sales S (T ) are free and not requred to meet a pre-set level (smlarly for frm by symmetry). The control varables are assumed to be nonnegatve wth a ( t) [, ) and ( t) [, ). We dscuss how S () t, S (t), and m S () t S () t are kept nonnegatve as we solve the sngle-perod model. w 4. Sngle-Perod Analyss In ths secton we characterze the optmal levels of share and growth expendtures for a scenaro wth two frms where each makes a one-tme decson. We nvestgate the sngle-perod model for two reasons. Frst, t helps buld nsght for the mult-perod settng, as comparatve statc results cannot be obtaned n that settng. Second, the solutons for the sngle-perod model requre lttle nformaton relatve to that of the mult-perod one and could thus concevably be used as a decson support tool n practce. The solutons of most dfferental games (and ours) requre knowng the effectveness and decay parameters for the other frm(s) and, as a result, emprcal verfcatons (e.g. as n Erckson 1992 and Wang and Wu 21) have tended to focus on mature commodty olgopoles (e.g. Anheser Busch vs. Mller, Coke vs. Peps) wth a wellknown advertsng hstory. Further, senstvty analyss from a sngle-perod settng s lkely to provde nsghts as to what adustments should be taken n a mult-perod settng when a parameter change only affects the results of a sngle perod e.g. a frm s future sales may be unaffected by a decrease n sales decay due to current weather. As a specal case of the prevous problem formulaton we therefore solve for optmal expendtures n a sngle-perod context wth no salvage value. Because there s no sales overlap from perod to perod, there s no Nash equlbrum; the optmal expendtures of the frm do not 9

11 change the optmal expendtures of ts rval. As suggested earler, ntal values for sales of frm and are denoted as S and S. Because there are no upper bound constrants on share and growth expendtures and ths s a one-shot decson, end-of-perod sales could result n nfeasble total ndustry sales or sales acquston from a rval frm (.e. S + S > m, S < or S < ). Infeasble solutons are prevented by scalng the effectveness parameters, unt proft margn and sales decay approprately (emprcal lterature cted above offers applcatons n practce). Frm must thus choose expendture levels that solve P = Max π S a w a, w wth S = S + ρ f ( a ) S ρ f ( a ) S + β g ( w )[ m S S ] [1 δ ] S. (6) The second order condtons for optmalty s met (and there exsts a unque optmal soluton) f f and g are strctly concave n ther respectve parameter. 3 These condtons result n dmnshng returns from the expendture varables. We select exponent functons (as e.g. Erckson 1985) to further our analyss. These functons are desrable because by varyng the value of the exponent t s possble to swpe an nfnte set of possble curves, makng t easer to transform our theoretcal model nto an emprcally testable framework. Thus, f a 1 ( ) a γ 1 = and g ( w) = w θ, (7) where γ, θ (,1) represents the elastcty of share and growth expendtures, respectvely, for frm. The relatonshp between elastcty and effectveness for expendtures s that elastcty measures the frm s operatonal ablty, and effectveness measures the frm s marketng ablty. In other words, elastcty measures how much of the frm s nvestment s drectly appled to realzed and unrealzed sales, and effectveness measures the fracton of realzed and unrealzed sales that are acqured and attracted by the frm s nvestment. Ths yelds the optmal soluton 2 2 The formulaton n (6) can be constraned by requrng that ρ f (a ) δ and β g (w ) + β g (w ) 1, followed by usng the Karesh-Kuhn-Tucker condtons to solve for optmal expendtures. If the constrants are not bndng, optmal expendtures are obtaned by usng the frst-order condtons on the obectve functon wth respect to expendtures. If a constrant s bndng, because of the lnear costs n the obectve functon optmal expendtures are f or, (*) ( a * * ) = δ / ρ g ( w ) = [1 β g ( w )]/ β dependng on whch constrant s bndng. However, note that mplementatons of (*) mean that frm competton s over, as ether S = or S + S = m, respectvely. Thus, we focus on the unconstraned solutons of the expendture varables for subsequent analyss. 3 2 Negatve second order dervatves and f ( a ) w a g ( w ) a w = yelds a negatve defnte Hessan. 2 = 1

12 1 1 a [1 ] = πρ S γ γ and w [ ][1 = πβ m S S θ θ]. (8) Therefore, share expendtures depend on how many ndustry sales do not currently belong to frm, and growth expendtures depend on how many unrealzed sales stll reman. Because of ths dependence on ndustry behavor n aggregate, the strategy of frm as t pertans to the other frm only depends on ts ntal sales level. We analyze how the soluton n equaton (8) s affected by changes n key model parameters by performng senstvty analyss, whch s summarzed n Table 1. Results are obtaned by takng the partal dervatves of the expendture varables n (8) wth respect to each parameter, holdng all other parameters constant and assumng, as needed, that < S + S < m. We note that the condtons for concavty are suffcent to derve va comparatve statc analyss the results n Table 1 on optmal share and optmal growth expendtures 4. Insert Table 1 about here In dscussng our fndngs, we refer to frm as the frm and to frm as ts rval. We frst consder frm parameters that have no effect on the rval s performance the frm s growth expendture effectveness and sales decay. Then, we explore the mpact of the frm s parameters that do affect the rval s performance, namely unt proft margn, share expendture effectveness, and ntal sales levels. 5 Effects of the frm s growth expendture effectveness ( β ). An ncrease n β, whch also means an ncrease n the frm s number of sales that can be realzed, also causes an ncrease n total expendtures, end-of-perod sales, and proft but only for the frm. In contrast to Krshnamurthy (2), the rval does not ncrease proft because growth expendtures are not cooperatvely spent. 4 Comparatve statc analyss for the general functons f and g s done usng the Implct Functon Theorem (e.g. 2 Currer 2). Snce all cross-dervatves equal, gven parameter μ for frm, a P a μ = and 2 2 μ P a 2 w P w μ =. The results hold when f and g are ncreasng concave functons for both frms. 2 2 μ P w 5 For both frms, an ncrease n m, the ndustry celng, means an ncrease n the number of sales that can be realzed, and thus causes an ncrease n ther growth expendtures, end-of-perod sales, and proft. 11

13 Effects of the frm s sales decay ( 1 δ ). An ncrease n ( 1 δ ), however, does not affect the frm s total expendtures or those of ts rval, but t does mean that the frm can retan fewer realzed sales and loses both end-of-perod sales and proft (although the rval s end-ofperod sales and proft are unchanged). The expendture fndng s strkng: frms should ether spend more to offset the effects of sales decay or spend less because they cannot retan the sales they gan. We nvestgate ths further n the mult-perod settng. Effects of the frm s unt proft margn ( π ). An ncrease n π leads to the frm spendng more on both attractng unrealzed sales and acqurng sales of ts rval. Ths fndng s supported n the sngle-perod model of Monahan (1987). Snce an ncrease n the frm s total expendtures (not due to any sales losses) leads to ncreases n ts end-of-perod sales, one would also expect an ncrease n ts proft. However, ths s not necessarly the case as exhbted n Table 1 wth a non-monotonc relatonshp between and proft. Further, ts rval s proft and end-of-perod sales decrease f the frm obtans hgher profts for each unt sold. For example, f the frm announces a maor breakthrough n producton that results n cost savngs, then unt proft margn for the frm ncreases, and ts rval should be prepared for the frm to spend more to acqure ts sales. Effects of the frm s share expendture effectveness ( ρ ). If some element of the frm s product or advertsng ablty that affects exstng sales unts mproves, causng an ncrease n ρ, then the frm should always spend more to attract sales unts from ts rval. Ths uncondtonal fndng s n contrast to Monahan (1987). In Monahan s market share attracton game, a frm s market share depends on the rato of ts advertsng mpact to the sum of advertsng mpact for both frms. Because there were zero unrealzed sales, he found that the frm may decrease expendtures f t already s superor to ts rval n the rato of unt proft margn or share expendture effectveness. An ncrease n ρ ncreases end-of-perod sales and proft for the frm because of the extra sales and revenue from share expendtures. An ncrease n ρ, however, does not affect the rval s expendtures, but ts rval s proft and end-of-perod sales decrease f the frm mproves the effectveness of ts share expendtures. The proft results complement the fndngs of Krshnamurthy (2). π 12

14 Effects of the frm s ntal sales level ( ). We consder at length the relatonshps between ntal sales levels and both expendtures and frm performance because these S relatonshps are counter-ntutve and several are non-monotonc. If S ncreases, then the frm should keep constant spendng on acqurng the rval s sales unts and decrease spendng to attract unrealzed ones. The decrease occurs because the pool of unrealzed sales has grown smaller and, as a result, less s spent on pursung those sales. If sales decay s low, then the frm may ncrease end-of-perod sales and proft from a hgh. Because ths s a sngle-perod model, f sales decay s hgh, the frm may have an ncentve to have low ntal sales levels. But what happens to the expendtures and proft of the frm f ts rval experences a change n ts ntal sales? We next dentfy the condton that determnes whether total expendture ncreases or decreases. Specfcally, an ncrease n ntal sales for the rval results n the frm ncreasng ts share expendtures but decreasng ts growth expendtures. The ratonales for these fndngs are ntutve. The rval s found to have more ntal sales unts than the frm expected. Because all other varables, ncludng ntal sales of the frm, are consdered constant for the senstvty analyss, ths rval sales growth must be due to rval attracton of unrealzed sales. Therefore, the frm transfers some of ts expendtures from unrealzed to realzed sales unts. If the elastcty for growth and share expendtures are the same (.e. γ = θ) e.g. due to departmental overlap n S β overhead and other costs we can state a relatvely smple condton ( > [ m S S] ρ under whch an ncrease n ntal sales for ts rval leads to an ncrease n the frm s total expendtures. A hgh elastcty of expendtures makes t more dffcult for the condton to be met f the frm s better at attractng unrealzed sales than acqurng sales from ts rval (.e. ts effectveness rato β ρ exceeds 1) and easer to meet the condton otherwse ( β ρ < 1). If the effectveness rato of growth to share expendtures s less than one, an ncrease n the rval s ntal sales level yelds larger total expendtures, end-of-perod sales, and proft for the frm f the rval s ntal sales ( S ) exceed unrealzed sales ( m S S ). In a Vdale-Wolfe sngle-perod model wth ndrect frm competton, an ncrease n ts rval s ntal sales always lowers the frm s proft. In a Lanchester sngle-perod model where S 1 1 γ ) 13

15 unrealzed sales are always zero, an ncrease n ts rval s ntal sales always lowers the frm s end-of-perod sales. We balance these results by showng that an ncrease n ts rval s sales may actually ncrease the frm s sales (and proft). Ths may seem counter-ntutve at frst sght, but consder a frm that s neffcent n attractng unrealzed sales, yet effcent at convncng customers of ts rval to swtch to ts product. Due to ts small effcency rato, such a frm can beneft from ncreases n the sales of ts rval from beng able to acqure them. We verfed that the left-hand sde of the above nequalty s largest, and therefore that nequalty s most lkely to hold, when S = I, the ndustry level of sales. In other words, a frm s more lkely to proft from ncreased sales of ts rval f that rval domnates the ndustry n terms of number of sales. Ths observaton, along wth our fndngs, shows that the frm s less (more) lkely to proft from the growth of ts rval when ether that rval s sales or total ndustry sales are small (large, respectvely). We next explore the robustness of some of these fndngs n a mult-perod settng. 5. Mult-Perod Analyss 5.1. Optmal Expendture Strategy The approprate soluton method for an open-loop Nash equlbrum for the optmal control problem formulated n (4) and (5) s that of dfferental games. Usng Pontryagn s maxmum prncple, we maxmze the Hamltonan for frm as gven by H () t = π S () t a () t w () t + λ () t S () t + ϕ () t S () t, (9) wth boundary equatons λ ( T ) = r and ϕ ( T ) =. The mmedate proft of the frm s π S () t a () t w () t, as we saw n the analogous sngle-perod case. In a mult-perod settng, however, we must also consder the tme remanng untl the end of the plannng horzon and the fact that a frm s current sales can be acqured by ts rval n the future. The future value of a ϕ sales unt to frm s λ () t S () t + () t S () t, as co-state varable λ ( t) measures the future value of one addtonal sales unt for frm to frm s profts and co-state varable ϕ ( t) measures the future value of one addtonal sales unt for frm to frm s profts. Solvng the necessary condtons for the Hamltonan wth respect to the control varables yelds optmal expendtures 14

16 1 1 [ λ ϕ ] ρ γ and w * () () [ () ()][1 ] t = λ t β m S t S t θ θ a * () t = () t () t S ()[1 t ] γ. 6 (1) A closed form soluton for the control varables, whch are affected by future sales values, requres a closed form soluton for the co-state varables. However, the co-state varables cannot be solved n closed form, and thus no drect senstvty analyss can be conducted n order to make comparsons wth the sngle-perod model. Instead, we gan nsghts through numercal analyss on the behavor of the control varables, state varables, and proft over tme Numercal Analyss We use a soluton algorthm that dscretzes tme usng dfference approxmatons, and solves for the state and co-state varables usng forward and backward passes as descrbed n Erckson 23 (p 61-62). We dd extensve testng over a wde range of parameter values. Care was taken n choosng the parameters; large values of growth expendture effectveness or unt proft margn may result n soluton nstablty. To smplfy the soluton process, we confned the numercal analyss to the case where γ = γ = θ = θ = 12. For the senstvty analyss, the market celng m s standardzed by settng t equal to 1; consequently unt proft margn for one frm refers to the value of all sales to that frm. The effects of salvage value were found to be smlar to the results of Deal (1979) n that ncreasng the salvage value that a frm places on ts end-of-horzon sales ncreases ts total expendtures and end-of-horzon sales, but decreases ts proft. As assumed n Bass et al. (25b) and for ease of comparson, we use r = r =.The length of the plannng horzon affects the end-of-horzon sales values. A very short fnte horzon s domnated by the values chosen for ntal sales level and salvage value at the end of the horzon. However, an nfnte horzon s unrealstc as we nvestgate heren the nascent perod of the ndustry; ntal sales levels and salvage value should mpact end-of-horzon sales and proft. We chose a common plannng horzon of T = 6 perods to perform senstvty analyss. 7 Because fnancng s mportant for new frms, we also record frst perod earnngs (proft from t = to t = 6 We note that the optmal decson varables n (8) and (1) are nearly dentcal except for the unt proft margn. Proft s now a functon of sales future value over tme rather than the fxed unt proft margn, π. In equaton (1), the unt proft margn s the dfference n future proft from frm possessng a sale over and above that of the rval possessng that sale (λ (t) - ϕ (t)). In contrast, only the future proft for frm s ncluded n equaton (1) for growth expendture. Ths s because an unrealzed customer has zero value to the frm before beng realzed. 7 A dfferent tme horzon may yeld, for nstance, dfferent end-of-horzon sales for both frms, but the drecton of the senstvty relatonshps appears robust to the length of the tme horzon. 15

17 1). Smpson s 1/3 Rule, a calculus technque for approxmatng a defnte ntegral, was used to sum frst perod earnngs, proft, and expendtures. A samplng of the run results are offered n Table 2. The frst half of each run s when the frm has an advantage on the rval n growth effectveness, whereas the second half of each run s when the frm has an advantage on the rval n share effectveness... (Insert Table 2 about here).. The dscusson that follows focuses on how the dynamc dmenson changes some of the results from our sngle-perod model and prevous models n the lterature. We provde n Table 3 a summary of the conectures derved from our numercal analyss. To showcase the mpact of frms havng dfferent levels of growth and share effectveness, we choose parameters that allow us to demonstrate the full effects of parameter changes on share expendture, growth expendture, end-of-horzon sales, and proft. In dscussng our fndngs, we stll refer to frm as the frm and to frm as ts rval... (Insert Table 3 about here).. Effects of growth expendture effectveness ( β ). As per Run (1) n Table 2, an ncrease n β yelds ncreased end-of-horzon sales and proft for the frm, but unlke Wang and Wu (21) t does not yeld dmnshed total expendtures. From a comparatve statc analyss, Bass et al. (25a) show n an nfnte-horzon game that an ncrease n growth expendture effectveness for one frm ncreases only that frm s growth expendtures but ncreases proft for both frms. Instead, Run (1) shows that regardless of each frm s effectveness n expendtures, hgher growth expendture effectveness for the frm decreases the growth expendtures, but ncreases the share expendtures, of ts rval as there are fewer unrealzed sales than before and more realzed sales for the frm. There are also several non-monotonc results from an ncrease n growth expendture effectveness. In Run (1), we show how the share expendture effectveness of the rval affects 16

18 whether that rval ncreases end-of-horzon sales and proft when the frm ncreases growth expendture effectveness. In the frst half of Run (1), the rval possesses a share expendture effectveness advantage compared to the frm (.3 to.1). If the frm ncreases ts growth expendture effectveness, the rval exhbts hgher end-of-horzon sales and proft. The frm responds by ncreasng ts own share expendtures because of ts rval s sales growth. Snce the rval s sklled at acqurng the frm s sales, t benefts from the frm s mprovement. In the second half of Run (1), nstead, the frm possesses a share expendture effectveness advantage compared to ts rval (.3 to.1). If growth expendture effectveness for the frm s.5,.1, and.2, ts rval ntally exhbts lower end-of-horzon sales and proft and the frm decreases share expendtures. It s only when the frm s advantage n growth expendture effectveness (.3 as compared to.5 for the rval) s larger than ts advantage n share expendtures effectveness (.3 to.1) that the rval starts ncreasng ts end-of-horzon sales and proft. The result n Run (1) suggestng that the rval s proft can be decreased by an ncrease n frm growth expendture effectveness dsagrees wth the monotonc ncrease fndng of Bass et al. (25a). The ratonale s that the declne n unrealzed sales caused by the frm s mproved growth expendture effectveness benefts ts rval only f that rval possesses suffcent effectveness n share expendtures. In our model, there s not a guaranteed spllover from the frm s growth expendtures to the sales of ts rval, as there s n Bass et al. (25a, 25b). Thus, we demonstrate that an ncrease n a frm s ablty to attract new sales does not always beneft competng frms. We also note that n the frst entry of Run (1), where growth expendture effectveness for both frms s.5, both frms exhbt postve frst perod earnngs, but low proft. As growth expendture effectveness ncreases, the frm s frst perod earnngs become more and more negatve, although when the frm s growth expendture effectveness s.4, ths trend begns to reverse tself. Comparng the frst to the second half of Run (1), t s also more expensve n the frst perod f the frm s share expendture effectveness s relatvely hgh. Thus, frms wth expertse n rapd sales growth may face a hgher rsk of runnng out of cash. Effects of sales decay (1 ). As per Run (2), an ncrease n (1 δ ) yelds decreased δ end-of-horzon sales and proft for the frm, whch supports the work of Deal (1979), and lower growth and share expendtures, n contrast to Wang and Wu (21) who argue for an ncrease n expendtures. Further, f the frm experences hgher sales decay, ts rval decreases share expendtures but ncreases growth expendtures. However, unlke Deal (1979) who dd not have 17

19 separate effectveness measures for unrealzed and realzed customers, hgher sales decay for the frm may cause a decrease n end-of-horzon sales and proft for ts rval. The ratonale s that f the frm s rval prefers to acqure the frm s sales rather than to attract unrealzed customers, a drop n the frm s sales wll hurt t. We note that the reducton n expendtures caused by a hgher sales decay means that frst perod earnngs ncrease, but n the long run the frm loses proft. Graphcal llustratons of sales, proft, and expendtures over tme for the frm provde addtonal nsghts; these llustratons are for the frst half of Run (2). As per Fgure 1(a), low sales decay leads to hgher sales at every tme perod. By the end of the frst perod, reduced growth expendtures begn to harm the frm s proft, as per Fgure 1 (b), and by T = 6 the frm s most proftable and has the most sales wth lower sales decay. Share expendture curves, as n Fgure 1(c), keep ther shape, but are lowered as sales decay ncreases. The nverted U-shape s commonly observed when the rval ncreases sales, but when, nstead, t decreases sales or grows them slowly a monotoncally decreasng convex curve s most often obtaned. For growth expendtures, Fgure 1 (d) suggests that hgh sales decay lowers ntal spendng on growth expendtures. However, after t = 3, the reverse s true; the hgher the sales decay, the more the growth expendtures. In expermentng wth ths and other parameter sets over longer tme horzons, we fnd that growth expendtures and n some cases total expendtures may ncrease over tme f the frm experences a loss n sales. Ths s n contrast to the maorty of the studes that use dfferental games where they show a pattern of ntal hgh expendtures to rapdly acqure sales, followed by a monotonc decrease n expendtures over tme (e.g. Erckson 1995). Studes that show non-monotonc expendture behavor nclude Erckson (1985) when a market s rapdly expandng and Bass et al. (25a, 25b) for brand advertsng (smlar to our share expendture) when both frms ncrease sales over tme. We complement these studes by showng that frm growth expendtures can ncrease over tme, even when the frm s sales decrease..... (Insert Fgure 1 about here).... Effects of unt proft margn ( π ). As per Run (3), an ncrease n π yelds ncreased end-of-horzon sales and proft for the frm, whch s n contrast to our sngle-perod fndngs where proft decreases. The ratonale appears to be that when there s a carryover effect n sales, 18

20 the frm can ncrease ts proft no matter what the rval does. An ncrease n unt proft margn for the frm also ncreases expendtures for the frm, a result that s also confrmed by Wang and Wu (21). We were able to construct a few runs (not shown) where the rval decreased ts share expendtures, but Run (3) represents the more typcal rval reacton wth ncreased share expendtures and decreased growth expendture. Arguably, the rval s reactons could be because the frm s ncrease n share expendtures causes nvestng n growth expendtures to be less proftable for the rval, but the frm s ncrease n end-of-horzon sales causes nvestng n share expendtures to be more proftable for the rval. Surprsngly, n contrast to our sngle-perod model, an ncrease n unt proft margn for the frm may ncrease the end-of-horzon sales and proft of the rval, as seen n the second half of Run (3). The ratonale s that the rval may beneft because, n a multple-perod context, t can leverage ts greater expertse n acqurng the frm s sales rather than attractng unrealzed sales. Our numercal fndngs for the rval thus contrast those of Bass et al. (25a), who fnd that a frm s ncrease n unt proft margn decreases share expendtures and steady-state market share for ts rval. Therefore, we offer an altered recommendaton. When the frm s unt proft margn ncreases, rather than reactng by cuttng expenses, as n the Bass et al. model, ts rval should typcally ncrease share expendtures. That hgher unt proft margn for the frm results n hgher end-of-horzon sales for that frm and thus ts rval s ncrease n share expendtures may actually allow t to take away enough of the frm s sales to ncrease ts own end-of-horzon sales and proft despte the frm s ncrease n unt proft margn. Last, when unt proft margn ncreases n Run (3), frst perod earnngs tend to decrease and profts ncrease. For further analyss, we focus on the case where both frms have a unt proft margn of 1. If the frm s relatvely sklled at acqurng realzed customers then ts frst perod earnngs are postve, whereas f the frm s relatvely sklled at attractng unrealzed customers then t absorbs negatve frst perod earnngs (but ts proft s more than trple that of the rval). Thus, t appears that the frm that specalzes n targetng ts rval s less lkely to face negatve earnngs early, but s less proftable n the long term than the frm that specalzes n attractng new customers. We also note that a frm can grow quckly enough n the frst perod to experence the double beneft of ncreased frst perod earnngs and proft (e.g. we compare wth Run (1) when growth expendture effectveness for the frm ncreases from.3 to.4). 19

21 Effects of share expendture effectveness ( ρ ). As per Run (4), an ncrease n ρ yelds ncreased end-of-horzon sales and proft for the frm, whch s n contrast to Bass et al. (25a) who argue that decreased proft may also occur. As n Bass et al. (25a), however, the frm ncreases share expendtures and the rval decreases growth expendtures. The ratonale s that the rval, realzng that growth expendtures become less valuable when newly realzed sales are targeted more ntensely by the frm, should dmnsh ts growth expendtures. Run (4) shows non-monotonc responses n growth expendtures for the frm and share expendtures for the rval. It appears that the frm s ncrease n sales at ts rval s expense s eventually so rapd that ts rval ncreases share expendtures to gan some sales back. Bass et al. (25a) also fnd ambguous results for the frm s growth expendtures, but analytcally found that the rval should decrease share expendtures when the frm ncreases ts share expendture effectveness. Our result n Run (4) seems to ndcate that when there are few ntal realzed sales, the rval s share expendtures are mnmally affected by changes n the frm s share expendture effectveness. Thus, advertsng wars appear to be rare n our framework. Of the parameters n Table 3, share expendture effectveness s the only one whose ncrease suggests that the rval exhbts lower end-of-horzon sales and proft. The proft decrease s n contrast to the ambguous result Bass et al. (25a) derve. We also demonstrate n Run (4) that ncreases n share expendture effectveness are not as expensve for the frm n the frst perod as ncreases n the other parameters are. In fact, an ncrease n share expendtures for the frm has neglgble negatve effect on ts frst perod earnngs and ncreases ts rval s frst perod earnngs n the frst half of Run (2), where the frm s stronger n growth expendture effectveness. However, ncreases n proft for Run (4) tend to be less than that of Run (2) because sales are only added at the expense of the other frm. Effects of Intal Sales ( ). As per Run (5), a hgher yelds an ncreased proft for S the frm. We chose a relatvely hgher unt proft margn and lower sales decay to showcase a surprsng result: end-of-horzon sales may decrease f the frm exhbts hgher ntal sales. The decrease occurs f the frm faces a rval wth suffcently hgh share expendture effectveness, as seen n the second half of Run (5). In contrast to our sngle-perod model, the multple-perod framework does not suggest less proft from an ncrease n ntal sales. Ths observaton provdes one of the reasons why the mult-perod framework mproves on the logc of the sngle- S 2

22 perod one: the latter could not uncondtonally state that havng a hgh level of ntal sales was advantageous to the frm. Furthermore, an ncrease n the frm s ntal sales appears to decrease ts growth expendtures and that of the rval (because there are now fewer unrealzed sales), but ncrease the share expendtures of the rval. In smlar fashon to earler results, the frm may slghtly ncrease ts share expendtures f the rval s ncrease n share expendtures results n frm sales loss. We note that f the rval exhbts relatvely hgh share expendture effectveness, the rval uses share expendtures to ncrease ts end-of-horzon sales and proft. But f that rval exhbts relatvely low share expendture effectveness, t does not beneft n end-of-horzon sales or proft. 6. Implcatons and Future Work In ths artcle we synthesze the Lanchester (nspred by Kmball 1957) and Vdale Wolfe (1957) models for a nascent ndustry to show how frms should choose between share expendtures (customer acquston at the expense of a rval frm) and growth expendtures (attracton of unrealzed customers). In a sngle-perod settng, we showed that frms should ncrease expendtures as ther effectveness n those expendtures ncrease, and that a frm may beneft from the sales growth of ts rval. Therefore, frms n the mdst of a growng ndustry should not necessarly vew an ncrease n a rval s sales as a threat to ther proft. Spllover benefts, as n the RFID growth example mentoned n the ntroducton, may provde a net beneft to the frm. Based on numercal analyss, we also showed n a mult-perod settng that an ncrease n growth expendture effectveness, a decrease n sales decay, and an ncrease n unt proft margn or ntal sales level for the frm may have postve spllover effects for ts rval s end-of-horzon sales or proft. From the numercal analyss, ths s most lkely to be true f the rval exhbts hgh share expendture effectveness. Furthermore, Fgure 1 suggested that when the frm s sales decay ncreases, the frm ntally decreases growth expendtures and experences an ncreased proft, but over tme t eventually ncreases growth expendtures and exhbts a decreased proft. For the management team of a nascent frm, the general practcal mplcatons of our numercal analyss agree wth those of Bass et al. (25a). That s, a frm should pay close attenton to unrealzed customers at the begnnng of the nascent ndustry and not worry so much about the realzed customers of ts rval. In Table 2, the frm wth the greater effectveness n 21