STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS

Transcription

1 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS MARK C. DUGGAN Abstact. In today s economc clmate, enegy s at the foefont of publc attenton. Renewable enegy s a feld whose technology s constantly changng. Fms must adapt to the eve changng maket n ode to keep up wth the compettos ates of eseach and development R&D. We exploe how dffeent ndusty scenaos affect the optmal level of R&D fo each fm. Some of the popetes that affect funds devoted to R&D nclude the numbe of fms, the elatve costs, compettve substtutable goods, and the method used to fnd the compettve equlbum Counot o Betand. Ove tme, each fm wll make dffeent decsons based upon each fm s poducton pces as a esult of R&D, thus we beak a dynamc game nto stages wth espect to poducton pce. We exploe how the dffeent aspects of the ndusty hut o pomote R&D, usng stochastc pocesses, along wth othe tools of pobablty and game theoy. Ths compettve maket poblem n R&D can be modeled as a dynamc game, n patcula, we consde Counot and Betand competton. In ode to detemne what decson each fm makes, we consde dffeent stages based on the level of technology of each fm. Technology mpovement s modeled as a Posson pocess, whee the amount of tme fo a fm to make an mpovement s a andom vaable. We obseve that ths pocess has the Makov popety: the popety that the condtonal pobablty dstbuton of futue states of the pocess depends only upon the pesent state. In ode to undestand what detemnes how much effot fms put nto eseach, snce, n geneal, thee ae no closed fom solutons, numecal expements wll be un usng vaous softwae MATLAB n patcula. The solutons ae found consdeng a fnte numbe of stages and usng the game theoy concept of Nash equlbum on each stage. Date: June 7, 3. Thanks go to the Mateals Reseach Laboatoy at UCSB fo the suppot va the Reseach Intenshp n Scence and Engneeng, and to my eseach advso Pofesso Mchael Ludkovsk of the Statstcs Depatment at UCSB.

2 MARK C. DUGGAN Contents. Intoducton 3. Theoetcal Backgound 3.. Game Theoy & Competton 3.. Stochastc Pocesses 5 3. A Sequence of Inceasngly Complcated Reseach Scenaos A Smple Scenao: Counot Monopoly Counot Duopoly: Sngle Fm Reseach 3.3. Counot Duopoly: Dual Reseach Counot Substtuton 8. When to Say No to R&D 8.. Lnea 8.. Exponental 5. Fgues 6. Futhe Reseach 3 Refeences 3

3 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 3. Intoducton We buld on the example by Ludkovsk and Sca [3. Ths example descbes competton between a poduce of an exhaustble esouce ol, natual gas, etc. wth a geen poduce that has access to an nexhaustable esouce, such as sola o wnd. Rathe than explong the detals of exploaton fo moe of an exhaustable esouce, we focus on the detals of R&D to educe the cost of poducng some nexhaustable esouce. We wll consde a pont pocess that counts the numbe of poducton cost eductons, as a esult of successful technology mpovements, ove tme. In ode to detemne the appopate amount of effot o fundng fo eseach, we ely on economc game theoy, n patcula the theoes of Counot and Betand. Befoe gettng nto the detals of jagon and concepts, we wll dscuss the deas behnd the pocess n whch the fm s able to educe ts cost. A fm chooses to devote some amount of esouces effot to devote to R&D, n hopes of educng ts poducton cost, n ode to maxmze ts pofts. We suppose that eseach successes happen ncementally. Moe pecsely, eseach successes ae epesented by a pont pocess, whee a success coesponds to a cost educton. We suppose that ths pocess s memoyless: that f a success has not occued n the past t tme, then the fm faces the same scenao as t dd t tme ago. Ths means that unde constant effot, the expected tme untl the next success s the same fo anywhee n T, T +, whee T j s the tme of a success. Futhemoe, constant effot must be an equlbum soluton wthn one of these ntevals, as the game has not changed dung the tme between consecutve successes nteaval tme, and so we suppose that effot s constant wthn each nteval, because the equlbum solutons ae often unque, focng constant effot to optmze. Fom ths we see that n T, T + we have a pont pocess that s dependent upon the effot of a fm dung that nteaval tme. So, costs can be loweed step by step by means of eseachng. Fms must decde between puttng n moe effot now to eap nceased pofts late o makng as much money as they can now wth mnmal effot put nto eseach. At each step, the optmum quantty each fm chooses s found by usng Counot s model of competton n ode to fnd a Nash equlbum, a soluton such that devaton fom ths soluton by ethe fm would esult n educed poft fo that fm.. Theoetcal Backgound Ths chapte ntoduces the concepts and defntons n economcs and pobablty that ae necessay to undestand the detals of ths pape... Game Theoy & Competton. Defnton. A game s descbed by mutually awae atonal playes, those playes possble decsons stateges, and a set of payoffs fo each playe, condtoned on the decsons made by all playes. In the context of esouce makets, playes wll be efeed to as fms, and payoffs ae gven n the fom of a eal numbe of dollas. A playe s atonal f hs pefeences defne a total odeng on R,.e. a fm pefes moe money to less money and has the ablty to pedct hs compettos decsons, because hs compettos ae atonal as well.

4 MARK C. DUGGAN Defnton. A Nash equlbum, often called an equlbum, s a set of stateges fo each fm, such that no fm has ncentve to change hs stategy,.e. wee one fm to change hs stategy, hs eanngs would be less than n equlbum. Defnton. In Competton, fms poduce some quantty, q, of a good at some magnal cost, s, and sell t to the publc at a pce, P, wth poft π qp s C, whee C s the sum of costs that ae not dependent on the quantty poduced, n ths case the cost of eseach. We wll also use net gans less poducton cost ρ qp s π + C. Defnton. The equlbum pce s the pce that pevals n the maket, the pce at whch fms sell goods. Wee some fm to chage moe, he would sell nothng. Altenatvely, wee a fm to chage less than the equlbum pce, hs compettos would sell nothng n the case whee the fm can supply enough to satsfy the ente demand, o clea the maket; ths loweng of pces would necessaly cause hm to make less money, by defnton of equlbum. Defnton. Goods ae not always dentcal, so we can consde substtute goods: a pa of goods such that an ncease decease n the pce of one good nceases deceases the demand fo the othe good. Ths elatonshp can be descbed by the goods coss-pce elastcty of demand. Fo goods and wth quanttes q [, q [ and pces P [, P [, the elastcty s δ, q[ P [ P [ q [, o moe ntutvely, the ato: pecent change of q [ ove pecent change of P [. Defnton. In Counot s Model of Competton,,,,... fms espond to a pce functon by smultaneously choosng a quantty, q [,, to poduce. The pce functon s nvesely popotonal to quantty. Fo the duaton of ths pape, we wll use the pce functon P Q, whee Q n q[. It s supposed that consumes wll puchase all goods suppled. In geneal, havng moe fms dves the pce down, as a lage quanty of goods wll be poduced, but each fm wll poduce fewe goods. Ths setup would encouagee colluson nto a catel, so that the model becomes lke that of a monopoly, because ths s paeto effcent the hghest total poft, summng all fms pofts fo the fms; howeve, ths s not possble, when we assume that the game s compettve allows no coopeaton. The equlbum pce and quanttes Nash equlbum can be found by solvng the system of equatons q [ q [ P Q s [ q [ q [ Q s [ Q q [ s [ { s [ q [ } j max q[j,, whee s [ s the cost of poducton fo fm. Notce how each fm s optmum quantty s dependent on the othe s. As mentoned, ths soluton s a Nash equlbum, meanng that devaton by ethe pat would cause fo educed pofts. Fo futhe dscusson of equlbum see [. Notce that the stategy space s { q [ q [ }, but we can clealy lmt the stategy space to quantty beng between and, snce poducng moe than wll always esult n negatve poft snce poducton cost s always nonnegatve.

5 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 5 Defnton. In Betand s Model of Competton, fms espond to consume demand by smultaneously choosng pces. Notce that when fms poduce dentcal goods, the equlbum pce s the magnal cost of poducton of the fm wth the second lowest pe-unt poduct cost magnal cost of all fms. If fms have dffeent magnal costs, then the fm wth the lowest magnal cost can set the pce below all othe fms magnal costs, blockadng them fom the maket, meanng they cannot poduce, because they would lose money by dong so. Ths fm wants to have the hghest pce possble that blockades all othe fms fom the maket, so they set the pce to the second lowest magnal cost of all fms. All fms wth the same magnal cost, wll compete and set the pce at magnal cost, and evey fm makes zeo poft. Ths s because f any fm wee to ase ts pce, t would be undecut by anothe fm; altenatvely, f a fm lowes ts pce below magnal cost t would lose money. Compason between Betand and Counot. The Betand model pedcts that f thee ae two o moe fms, then pce s dven down to the magnal cost of the fm wth second lowest magnal cost, wheeas n the Counot model, pces appoach the same value magnal cost, as the numbe of fms appoaches nfnty. Defnton. Dscountng s a method to detemne the pesent value of futue payoffs. We consde the contnuous tme dscount e t. Fo example, whethe a fm pefes a dolla today t o two tomoow t can be detemned by calculatng whethe e o e s lage, that s, the fm pefes a dolla today, f > log... Stochastc Pocesses. Defnton. A stochastc pocess o andom pocess s a famly of andom vaables ndexed by some set T. We consde a famly of andom vaables {Nt : t }, whee Nt counts the numbe of successes these esult n cost eductons ndexed by the set [ T, T, T..., T N, the tmes at whch these successes occu. Ths example s a specal case of stochastc pocess: a pont pocess. Defnton. A Makov chan s a collecton of andom vaables such that gven the pesent, the futue s condtonally ndependent of the past; ths popet s known as the Makov popety, o memoyless popety. We assume that ou pocess satsfes the stong Makov popety, whch states that fo the Posson pocesss X, wth some stoppng tme T, XT + s XT s s a Posson pocess statng fom tme and ndependent of stoppng tme T and ndependent of hstoy X T.

6 6 MARK C. DUGGAN Defnton. A Posson pocess s a Makov pocess whch counts the numbe of events and the occuance tme of each event. The tme between each pa of consecutve events s dstbuted exponentally and ndependently of each othe. Two occuences have neglgble pobablty of happenng smultaneously. A Posson pocess s sad to have ntensty λ f PNt j λtj e λt fo j,,,..., j! whee Nt s the cumulatve numbe of occuances by tme t [. That s, Nt has Posson dstbuton wth paamete λt, whch we abbevate as Nt Posλt. Notce that the expected tme between successes s the ecpocal of the ntensty. 3. A Sequence of Inceasngly Complcated Reseach Scenaos In ode to buld up to a moe complcated model of competton between two fms who can each educe poducton costs by eseachng, fst we consde a sequence of models each addng a new laye of complexty. 3.. A Smple Scenao: Counot Monopoly. Model. A sngle fm poduces some good that has poducton cost s, and pce, P q, whee q s the quantty poduced by that fm. In ths scenao, fo the fm to maxmze poft, t must optmze wth espect to quantty. Ths yelds a poft of q qp s q q s s q q s. q P q s s s s s. Defnton. A fm chooses some amount of effot, a, o fundng fo eseach and development R&D, n ode to educe the cost of poducton; ths effot has an assocated cost C. We wll mostly use the cost of effot functon Ca a +ka, whee k s efeed to as the effot weght. In geneal, we wll consde a Posson pocess N wth ntensty λa that counts the numbe of cost eductons successes, and wth success tmes [ T, T, T,... T N. Poposton. Suppose T Expλa, then [ e T E λa +, whee E[X s the expectaton o expected value of the andom vaable X. Poof. Notate F T t e λt, the cumulatve dstbuton functon of T. Then E [ e T As a esult we see that [ e T E e t F T tdt E [ e T e t e λt dt λa λa+ λa +. λa λa +.

7 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 7 Buldng on Model : Effot. Now consde a fm that poduces a good at ntal cost st, chooses some quantty to poduce qt and can choose some amount of effot to put nto eseach, at, to educe the poducton cost fom s to s to... to s N. Defne the dscounted poft functon [ π N s E e t qtp qt st Cat dt s s, whch detemnes the dscounted poft a fm eceves fom tme zeo onwad, whee N possble avals ae consdeed. We wll dop the condton notaton and emembe that st s. To maxmze ths, take the supemum at each tme, t, wth espect to quantty and effot; defne [ V N s sup π N s sup E qt [,,at [, t R + q,a e t qtp qt st Cat dt. If we consde the example above Model, and suppose the fm decdes to put n no effot, then the value functon smplfes to [ E e t q P q sdt q P q s s. Consde the smplest case that nvolves possble postve effot, N ; one chance fo mpovement. Now let us consde V s, whch we can smplfy by splttng the ntegal at the fst success tme, T, and notcng that t s optmum to put n no effot afte T, snce thee s no oom fo cost educton. Notce that the optmum effot s fxed between T we use T to notate an ntal tme, when no successes have occued and T, snce nothng changes dung that tme, due to the memmoyless popety. Thus between any two avals T, T + when pce s s, we wll call the optmum effot a and the optmum quantty q maxmzng π N s. To compactfy notaton, defne ρqt qtp qt st, poft dsegadng cost of effot, and ts optmum ρ k sup q qtp qt s k, and defne poft,πqt, at ρt Cat, as well as ts optmum π k ρ k Ca k. 3. snce V s [ T [ sup E e t ρqt Cat dt + sup e t ρqt Cat dt q,a t T q,a t T by the Makov popety [ T E e t ρ Ca dt [ e T E [ + sup E e T q,a t ρ Ca + E [ e T V s. λa + ρ Ca + λa π λa + + λa λa + ρ λa + V s,, e t ρqt Cat dt

8 8 MARK C. DUGGAN V s e t ρ Ca dt e t ρ dt ρ. Fo P q, we pevously found fnd the quantty that optmzes poft: q s and ρ s. Then to fnd a, the optmzng effot value, fnd the supemum of Equaton 3. wth espect to a, emembeng π s a functon of a. Ths one step pocess can be genealzed to an N step pocess. See fgues and on the followng page fo an example of a soluton to ths pocess wth a lnea cost scheme and an exponental cost scheme.

9 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 9 Fgue. The fgue shows the optmum effot the effot that maxmzes the value functon, gven poducton cost of a fm wth a monopoly at each poducton cost fo vaous step szes N fo a lnea cost scheme wth constants k, λ,.5,,.. Ths soluton s found usng Equaton 3.. Fgue. The fgue shows the optmum effot of a fm wth a monopoly at each poducton cost fo fm, s l e zl, l,... N fo the dffeent z values gven wth constants k, λ,.,,.3. Ths soluton s found usng Equaton 3..

10 MARK C. DUGGAN Model. As the pevous model alluded, we consde a fm that can make N mpovements. Consde a patton S of the nteval [s, s N, {s, s,..., s N }, whee s < s <... < s N. Then a fm wll choose optmum q, a that maxmze V N s, whle the pce emans s. T Expλa s the tme of the th pce educton fo N, and we wll defnet though thee s no th aval. Ove tme, we have at tme, cost s s untl a eseach beakthough at T, at whch tme the poducton cost s educed to s. The fm can contnue puttng n effot to acheve cost eductons, untl they decded to stop puttng n effot at T N, whee the poducton cost s s N. As a esult of the pevous model, we see that V N s l V N s l π l + λa l V N s l+, fo l,,..., N, whee V N s N+ :, snce λa l + [ sup E a,q t sup a,q t T E e t ρqt Cat dt s s l [ T [ e t ρqt Cat dt + sup E e t ρqt Cat dt a,q t T T afte splttng the ntegal, we use the stong Makov popety fo s, [ T [ E ρ l Ca l e t dt + sup E e T e t ρqt Cat dt a,q t T [ e T E ρ l Ca l + E [ e T V N s. then we use Poposton to evaluate expectatons, λa l + ρ l Ca l + λa l λa l + V Ns π l + λa l V N s l+ λa l + fo l,,... N. Notce that no matte how we defne V N s N+, so long as t s fnte, V N s N π N +λa N V N s N+ λa N + π N, snce a N ; so we wll defne V N s N+, fo convenence. In ode to fnd V N s, we fst solve fo V N s N, then V N s N, etc., as a esult of the ecusve natue of V N s l. Obseve that whle the poducton cost s s, at s fxed at some level, a l, because t s an equlbum soluton; wee the fm to make an adjustment dung ths tme, t would be less than optmal. The same holds fo q l. We have aleady dscussed how to fnd each q l, usng the methods of Counot, and n Model 3, we wll be moe pecse n how to fnd each a l. Lemma. Success & Decay Rates. Fo a sngle fm wth the oppotunty to educe poducton costs fom s to s to... to s N, the amount of effot a t that optmzes poft s the same n the followng scenaos: The fm has success ate λ, decay ate. The fm has success ate λ λb, and decay ate b, fo some constant b >. Futhemoe V N s m bv N s m fo all m, whee V N s m s the value functon fo the fm wth success ate λ, and decay ate.

11 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS Poof. Fst ecall that fo any m and less than N +, V N s m λa m + π m + λa m λa m + V s m+ whee π m and a m maxmze V N s m. We wll poceed by nducton on poducton cost, s m. Begn wth the base case, m N: bv Ns N b π N b π N b π N V Ns N. Now consde the geneal case, π bv Ns m b sup a λ a + + λ a λ a + V s m+ π sup b a λ a + + λ a λ a + bv s m+ π sup b a bλa + b + bλa bv s m+ bλa + b π sup a λa + + λa λa + V s m+ V N s m. Notce that snce we can facto b fom both supemums, they ae ove the same quantty. Thus the optmum effot s the same n both cases. The esult follows by nductve easonng. Because of ths dependency, we wll tend to let λ, n ode to compactfy notaton. It s mpotant to note that n the case of two fms, t may be easonable to eque that [ and [ ae the same, whch would necesstate keepng λ and sepaate. 3.. Counot Duopoly: Sngle Fm Reseach. Model 3. Now we consde competton between two fms, whee Fm has the oppotunty to educe ts cost of poducton by means of R&D, but Fm has fxed poducton cost s [. Suppose Fm has ntal cost of poducton s [ and N successes can educe hs cost of poducton to s [ { Then, consde a patton S of the nteval [s [, s[ N, s [ N s [, s[ },..., s[ N N., whee s [ < s[ <... <. Ths s an N + stage game. The quanttes each fm chooses s the { same as n } the classcal Counot model: Dung [T, T +, Fm chooses quantty q [ s max [ q [, and Fm { } chooses quantty q [ s max [ q [, fo moe detal on how we ave at these solutons see the defnton of Counot s model of competton. If one fm has poducton cost too hgh to ente the maket + s [k s [j <, t poduces no goods and the othe fm chooses the quantty as n the monopoly case q [k s[k. Supposng q [j >, j q [, q [ + s [ s [, + s[ s [. 3 3

12 MARK C. DUGGAN Fom ths we conclude that P +s[ +s [ 3 and that ρ q P s [ + s[ s [ 3 + s [ + s [ s [ 3 + s [ s [ N 9 q. Now we want to fnd a. We stat by fndng V N s [ N ρ, then solvng by fndng a N that maxmzes V N s [ N and so on, usng 3. V N s [ l V N s [ N λa N + ρ N Ca N + λa N ρ N a N +, λa l + ρ l Ca l + λa l a l + V Ns [ l+, n ode to fnd the optmum amounts of effot at each stage. Ths noton s extends natually to Counot competton wth m fms, whee Fm s the only fm wth the choce to put effot nto eseach. In Equaton 3., we can fnd a l by dffeentatng V N s [ l wth espect to a l n ode to fnd ctcal ponts we wll use the cost funton Ca a + ka : V N s [ a l a l l λa l + ρl a l ka λa l l + a l + V N s [ l+ λa l + a l k λρ l a l ka l λa l + λa l + a l k λρ l a l ka N + λv N λa l + a l k + λ + a l + λ V N s [ l+ ρ l k + λ λ V N s [ l+ + λa l + λa l a l + V N s [ l+ ρ l s [ l+.

13 3.3. Counot Duopoly: Dual Reseach. STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 3 Lemma. Fo two fms, wth success ate λ [ and λ [ espectvely, the pobablty that Fm s fst aval occus befoe Fm s fst aval s P T [ < T [ a [, λ[ a [, λ[ + a [, λ[, whee a [h [, s the optmum amount of effot fo Fm h between the ntal tme, t T T [ and t mn{t, T }, fo h,. Poof. T [ T [ s the dffeence of two exponental dstbutons wth paametes a [, λ[ and a [, λ[, so t has PDF fx And thus P T [ > T [ T [ T [ a [, λ[ a [, λ[ [ a [, λ[ +a e a, λ[ x,λ[ a [, λ[ a [, λ[ [ a [, λ[ +a e a, λ[ x,λ[ P T [ T [ a [, λ[ fo x, fo x < > T [ T [ fxdx a [, λ[ + a [, λ[. Model. Now we consde competton between two fms, whee both fms have the oppotunty to educe the costs of poducton by means of R&D. Suppose fm h has ntal cost of poducton s [h, whee N h successes can educe hs cost of poducton to s [h the nteval [ s [h, s[h N h, chooses quantty q [,j max { { s [h, s[h,..., s[h N h } s [ q [,j, N h. Then, consde a patton S of, whee s [h < s [h <... < s [h N h. Dung [T, T +, Fm } { } and Fm chooses quantty q [,j max s [ j q[,j,. Notce that each fm need not be at the same step, that s fo poducton pces s [, s [ j, j and need not be the same. Supposng q [h >, [ + s q [, q [ j j s [, + s[ s [ j f + s [ j s [ > and + s [ s [ j >. 3 3 Othewse the fm wth the lesse poducton cost teats the maket as n the monopoly case, and the othe fm efans fom poducng, so q s,. Fom ths we conclude that P [ o P [ and that +s [ j +s[ 3 +s[ ρ [ q [ P [ s [ + s[ j s [ 3 [ + s j + s [ s [ + s[ 3 j s [ 9 N q [.

14 MARK C. DUGGAN Now we want to fnd a. We stat by fndng V [ s [ N, s [ N ρ[ N,N. We wll consde when the fms ae at the th and jth cost eductons, but by conventon we wll notate ths as s [ and s [, because of the memmoyless popety. Also note, when we consde PT [, we suppose that < T [ the pevous aval tme fo each fm was T [ and T [ espectvely, so that we ae cosdeng the next aval fo each fm. { } Fst we notce that mn T [, T [ s the mnmum of two exponental dstbutons wth paametes λ [ and λ [, so mn T [, T [ Exp λ [ + λ [. In ode to fnd the maxmum amout { } of effot fo the fm to choose, consde the value functon we wll pove that ths s the coect one soon [ [ [ mn{t V [ s [, s[ E π [,T }, e t dt + P T [ < T [ [ T E [e V [ s [, s[ + + P T [ > T [ [ T E [e V [ s [, s[ π [, λ [ a [, + λ[ a [, a [, λ[ a [, λ[ + a [, λ[ a [, λ[ a [, λ[ + a [, λ[ λ [ a [, λ [ a [, + V [ s [, s[. λ [ a [, λ [ a [, + V [ s [, s[ Notce that ths ecuson depends on a gd of possble poducton costs n [s [ N, s [ [s[ N, s [ usually [, [,, athe than just a lne, as n pevous examples. Lemma 3. Fo two fms wth the oppotunty to eseach to educe the poducton costs fom s [ and s[ to s [ to... s [, espectvely, we can fnd the optmum amount of effot to s [ to... s [ N a [,j and a[,j N fo each fm to put nto eseach by maxmzng the value functon π [ V [ s [, s[, + a[, λ[ V [ s [, s[ + a [, λ[ V [ s [, s[, a [, λ[ + a [, λ[ + wth espect to effot and quantty, whee λ [k s the success ate ntensty fo fm k that s T k T k [k Expλa, whee T s the tme of the th success fo fm k, and s the decay facto. Poof. We poceed as n n the fst model, splttng up the ntegal by aval tmes because poducton cost, st; quantty, qt; pce, P qt; effot, at; and cost of eseach Cat ae all fxed between avals but ths tme, avals fo ethe fm must be consdeed. In the followng equatons, we wll dop the supescpt ndcatng to whch fm the vaable s assgned, snce they ae all vaables fo fm untl we add supescpts back n. P T [j > T [k s evaluated by Lemma. E [e T [j s evaluated as n Poposton, as s E [e mn { T [ [,T }, snce

15 { } mn T [, T [ V [ s [, s[ sup q,a t R + E sup q,a t mn{t [ [,T + sup STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 5 Exp λ [ + λ [. [ e t qtp qt st Cat dt s[ t s [, s[ t s [ } E E q,a t mn{t [ [,T } [ [ [ mn{t,t [ by the law of total pobablty, [ E π [, [ [ mn{t,t } + P T [ < T [ + P T [ > T [ E [ π [, } e t qtp qt s [, Cat dt e t qtp qt st Cat dt mn{t [ [,T } e t dt sup q,a t mn{t [ [,T sup q,a t mn{t [ [ [ mn{t,t } [ [ [ T E e E + P T [ > T [ [ [ T E e E [ e t dt } E E [,T } e t qt + T [ sup q,a t mn{t [ [,T } e t qt + T [ [ e t qtp qt st Cat dt T [ T [ < T [ [ e t qtp qt st Cat dt T [ > T [ T [ + P T [ < T [ P qt + T [ P qt + T [ by the Makov popety and the towe ule, [ E π [, [ [ mn{t,t } + P T [ < T [ + P T [ > T [ π [, E sup q,a t sup q,a t { mn e + P T [ > T [ E [e T [ e t dt [ [ T E e [ [ T E e T [ λ [ a [, + λ[ a [, π [, a [, λ[ a [, λ[ + a [, λ[ } [,T + P T [ < T [ sup q,a t mn{t [ [,T } [ [ st + T Cat + T st [ dt T [ < T [ [ [ st + T Cat + T st [ dt T [ < T [ e t qtp qt st Cat dt T [ < T [ e t qtp qt st Cat dt T [ < T [ T [ < T [ E [e a [, λ[ a [, λ[ + a [ λ [ a [, + λ[ a [, V [ s [, s[, λ[ λ [ a [, + λ[ a [, + V [ T [ T [ < T [ λ [ a [, + λ[ a [, λ [ a [, + λ[ a [, + V [ s [, s[. V [ s [, s[ + s [, s[

16 6 MARK C. DUGGAN π [ V [ s [, s[, + a[, λ[ V [ s [, s[ a [, λ[ + a [, λ[ + + a [, λ[ V [ s [, s[. Let us consde a smple case, whee two fms each have the oppotunty to make one eseach beakthough, educng the poducton cost fom s [ s [ > / to s [ s [. Thus when one fm makes a beakthough, they have a monopoly, blockadng the othe fom poducton. We detal the fms value functons at each step below. Fo fm k, whee P,j q [,j q[,j. V [k s [, s[ V [k, π[k, ρ[k, 9, π [ V [ s [, s[, + a[, λ[ V [ s [, s[ a [, λ[ + [ ka ρ [, a [, a [ ρ [, + a[, λ, + a[,, λ[ + ρ[ [, k a [, λ[ + ρ[, λ[ a [, Ths s maxmzed when V [ s [, s[ a [ λ [ a [, + λ ρ[ [, k a [, λ [ ρ [, + a[, λ ρ[ [, k λ [ a [, + λ [ ρ [, k + λ [ ρ [, kλ[ a [, λ[ a [, λ [ ρ [, + ρ[ a[ [,, λ k a [, λ [ ρ [, ρ[, λ [ a [, a [, k a [, λ[ a [, [ + a, + λ[ ρ [, ρ[, + k + +λ [ ρ [, ρ[, λ [ k f λ [ ρ [ λ [ f λ [ ρ [, ρ[,, ρ[, a [, > λ [ k. λ [ k

17 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 7 whch s maxmzed when a [, snce the poblem s symmetc. π [ V [ s [, s[, + +a[, λ[ V [ s [, s[ a [, λ[ + ρ[, + λ[ a [, ρ [, λ [ a [, +, + +λ [ ρ [, ρ[, λ [ k f λ [ ρ [ λ [, ρ[, f λ [ ρ [, ρ[, > λ [ k, λ [ k π [ V [ s [, s[, + a[, λ[ V [ s [, s[ + a [, λ[ V [ s [, s[ a [, λ[ + a [, λ[ + Dffeentatng gves V [ s [, s[ a [, a [, π [, + a[ + k + V [ s [ + a [, λ[ V [ s [, s[ 3.3 a [,, λ[, s[ ρ[, +λ[ a [, a [ V [ s [, s[ k+v [ s [,s[ λ [ a [, + ρ [, + a [, λ[ a [, λ[ + a [, λ[ +, + k V [ s [ ρ [, + kv [ + f k V [ s [, s[ < a [ f k V [ s [, s[ k V [ s [,s[, λ[, s[ ρ [, +a[, + ρ[ λ [, k a [, λ[ + s [, s[ +a [, λ[ V [ s [,s[ V [ s [, s[ V [ s [, s[ V [ a [, λ[ V [ s [, s[ s [,s[ a [,. +ρ [, + ρ [ V [ s [, s[, + ρ [,, and a symmetc equaton holds fo a [,. Then we solve fo effot, a[l,, fo each fm, l,, usng these two equatons. Ths s a Nash equlbum as well. Recall that the quanty we fnd at each stage s a Nash equlbum, whch we fnd at each step sepeately, makng each step a smple Counot game. When each fm s n a gven stuaton they choose a quantty poduced, as pevously pescbed, and they also choose an effot to put nto eseach, but ths effot depends

18 8 MARK C. DUGGAN on the opponents effot; thus a Nash equlbum must be found by solvng the pa of equatons gven by Equaton 3.3 and ts countepat. 3.. Counot Substtuton. Model 5. Pevously, we assumed goods wee dentcal, o pefect substtutes; howeve, ths s not always the case, so we consde fms sellng goods that ae substtutes,.e. P [ q [ j δq[j, whee δ s the coss-pce elastcty of demand, as defned n., between any two dffeent fms often denoted δ,j fo elastcty between good and good j, but hee we assume δ,j δ k,l fo all j, k l. Consde just two fms now, and let us use the notaton of the pevous model. If one fm has poducton costs too hgh to ente the maket δ + δs [j s [k no goods and the othe fm chooses the quantty as n the monopoly case q s q [h >, q [, q [ j [ δ + δs j s [ δ, δ + δs δ δ+ δ s [ +s [ j [ s [ j. >, they poduce. Supposng. Notce that ths Fom ths we conclude that P [ and that ρ [ δ q [ changes the ρ l π l Ca l functon, but no othe pat of V N s l π l+λa l V N s l+ λa l +, so the geneal pocess emans the same.. When to Say No to R&D.. Lnea. We exploe unde what condtons t s not wothwhle to contnue puttng effot nto eseach. Suppose a fm s decdng whethe to stat eseachng, n the monopoly case. Then the fm wll choose not to eseach f fo all a,

19 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 9 V s n a > V s n a ρ > ρ C λa + + λa λa + V s n+ ρ n+ ρ > ρ C λa + + λa λa + λa + ρ > ρ C + λaρ n+ aλρ λρ n+ > C a < C λ ρ n+ ρ If we let C a + ka, then λ ρ n+ ρ k < a. Notce that f the condton must be satsfed fo all a, then t wll suffce fo t to be satsfed fo a ρ n+ ρ < k. sn+ s n < k, whee we defne λ/. Notce that f k, ths condton s always satsfed. Othewse, we can consde a few dffeent cases. Fo the smplest case, the lnea case, s n+ s n N s n s n N, gves us the condton sn+ s n < k. sn + N s n k > N + N s n N s n > N + kn s n N > N + kn n < + Ns + kn In concluson, the fm wll not eseach when n < + Ns + kn. Notce that f N < + Ns + kn kn Ns < N < s + s + k the fm wll not eseach at all. < k.

20 MARK C. DUGGAN.. Exponental. Fo the exponental case, e zn+ e zn e z s n+ s n e z fo some z >, gves us the condton:. Notce that we get equalty when sn+ s n > k. sn e z s n > k. s n e z + s n e z e zn e z + e zn e z e zn z e zn + e zn e zn z k >. k >. e zn z e zn + e zn e zn z > k. k >. o equvalently e zn s n e z ± e z + e z k, e z n z log e z ± e z + e z k. λ e z Thus the condton s satsfed n between the two ctcal values pluggng n exteme values of and takng the lmt as n makes the left hand sde of the Inequalty. vey small. 5. Fgues Counot Duopoly, Sngle Fm Reseach, Lnea Cost

21 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS Fgue 3. Shows optmum effot choces fo Fm, gven dffeent poducton costs fo Fm and constants s [, N, k, λ,, 5,.5,,.. Notce that total effot s maxmzed when Fm s poducton costs ae aound, and mnmzed when Fm s poducton costs ae. Notce the dp n the ed cuve aound s. and n the geen cuve aound s.6. These dps ae a esult of changng fom a duopoly to a monopoly; Fm s blockaded fom competng. To emphasze these dps, see Fgue.

22 MARK C. DUGGAN Counot Duopoly, Sngle Fm Reseach, Lnea Cost Fgue. Shows optmum effot choces fo Fm, gven dffeent step szes ecall costs mpove lnealy n ths case, N cost educton fo each success and constants s [, s[, k, λ,,.6,.5,,.. Fgue 5. The fgue shows the optmum effot of a fm wth who competes wth a fm wth fxed poducton cost.6 at each poducton cost fo fm, e zl, l,... N wth constants k, λ,.,,.3.

23 STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 3 6. Futhe Reseach A few thngs that could be moe closely obseved ae: Ctcal R&D ponts When does a fm decde neve to eseach? When does a fm decde to stop eseachng? Those these questons ae mentoned, a complete answe s not yet clea. How pattonng of poducton costs affects optmum effot Can we choose an effot a and fnd a poducton cost patton {s, s,..., s N } such that the optmum effot s a? 3 Refnements of cost pattons If a fm has poducton cost patton S, and a second fm s cost patton s a efnement of S, then the expected poft of the second fm s geate than o equal to the expected poft of the fst fm. Refeences [ A. Counot, Reseaches on the Mathematcal Pncples of the Theoy of Wealth, 838, [ G. Gmmett and D. Stzake, Pobablty and Random Pocesses, Oxfod Unvesty Pess,. [3 M. Ludkovsk and R. Sca, Exploaton and Exhaustblty n Dynamc Counot Games, Euopean Jounal on Appled Mathematcs 33, Depatment of Mathematcs, Unvesty of Calfona, Santa Babaa, CA 936. E-mal addess: makduggan@umal.ucsb.edu