R&D Investment and Technology Adoption

Transcription

1 R&D Investment nd Technology Adoption Veen Hgspiel 1, Kno J.M. Hismn 1,2, nd Cládi Nnes 3 1 CentER, Deptment of Econometics nd Opetions Resech, Tilbg Univesity, Post Office Box 90153, 5000 LE Tilbg, The Nethelnds 2 Consltnts in Qntittive Methods, Post Office Box 414, 5600 AK Eindhoven, The Nethelnds 3 Deptment of Mthemtics nd CEMAT, Institto Speio Técnico, Av. Rovisco Pis, Lisbo, Potgl Jne 2, 2010 Abstct We stdy model of technology doption. The fim s pofit is depending on the technology tht it ses. The fim holds the option to pgde its cent technology with new one. Technology pogess is modeled with Poisson pocess. The fim cn eithe ely on the extenl R&D mket o it cn setp its own R&D cente. The ivl te of new technologies in the extenl R&D mket is constnt nd given. Wheneve the fim chooses to setp its own R&D cente it cn optimly set the ivl te of the new technologies. Howeve, the setp costs e incesing in the selected ivl te. We identify the pmete vles fo which its optiml fo the fim to setp its own R&D cente nd fo those pmete vles we detemine the optiml ivl te. Moeove, we detemine the optiml technology doption tigge in ll sittions. We end the ppe with comptive sttic eslts. 1 Intodction In most technology doption models the technologicl pogess is exogeneosly given. In this ppe we extend the litete on technology doption by ssming tht the fim tht dopts the new technology cn inflence the speed of technology doption by setting p n R&D cente. We investigte the sittions in which its beneficil fo fim to se this oppotnity. Fthemoe, we investigte the impct on the speed of technology doption nd the technology level tht is choosen by the fim. Plese note tht this is wok in pogess. The ltest vesion cn be downloded fom 1

2 2 Model We conside isk netl vle mximizing fim. The fim disconts with te > 0. The pofit flow t time t is denoted by π t (θ t ), whee θ t denotes the technology efficiency level tht the fim ses t time t. The fim initilly podces with technology θ 0. We ssme tht the pofit flow of fim is incesing in the technology efficieny pmete, i.e π θ > 0. The fim cn pgde its technology once, the pgde costs e eql to I. New technologies ive ove time ccoding to Poisson pocess with te λ. We ssme tht the efficiency level of ech new technology is highe thn its pedecesso. So we hve tht with pobbility λdt, dθ = 0 with pobbility 1 λdt. Fthemoe, we ssme tht the fim cn inflence the speed of ivl of new technologies by setting p n R&D cente. Let the setp costs fo sch cente be eql to exp(λ) nd the nning cost pe peiod c. The model is solved in two steps. In the fist step the optiml technology doption sttegy is detemined given the choosen level λ. The second step ses the eslt of the fist step to detemine the optiml level of λ. (1) 2.1 Optiml Technology Adoption Given tht we know the choosen level λ we cn se the eslts fom Hismn (2001, Chpte 2). The vle of the fim fte pgding to technology θ is denoted by V (θ). We ssme tht the fim closes the R&D cente fte it hs pgded its technology. So we hve tht V (θ) = t=0 π t (θ)e t dt = π(θ). (2) Fom Hismn (2001, Chpte 2) we know tht thee exists theshold level θ fo which it holds tht the fim shold pgde its technology the fist moment tht θ is lge o eql thn θ. In o model we will hve tht the theshold level is fnction of λ, i.e. θ (λ). In ode to deive this theshold we hve to investigte the egion fo which we know tht the fim will invest fte the next technology ivl, i.e. the egion {θ θ (λ) < θ θ (λ)}. Let F (θ,λ) denote the vle of the fim befoe the fim hs pgded its technology, then the following Bellmn eqtion shold hold fo F It holds tht Sbstittion of (4) into (3) nd ewiting gives 1 F (θ,λ) = π(θ 0 ) c+ lim E[dF (θ,λ)]. (3) dt 0 dt E[dF (θ,λ)] = λdt(v (θ+) F (θ,λ))+o(dt). (4) F (θ,λ) = π(θ 0) c +λ + λ (V (θ +)). (5) +λ 2

3 At the theshold vle θ (λ) the fim is indiffeent between investing ight wy nd witing fo the next technology. In othe wods t the theshold the following vle mtching condition mst hold F (θ (λ),λ) = V (θ). (6) Sbstittion of the deived eqtions nd ewiting gives the following eqtion tht implicitly defines the theshold θ (λ) π(θ 0 ) c +λ + λ ( π(θ ) (λ)+) = π(θ (λ)). (7) +λ Fom Hismn (2001, Chpte 2) we know tht the nmbe of needed technology jmps n, the expected technology doption time nd its vince e eql to θ n (λ) θ 0 =, (8) E[T ] = n λ, (9) V[T ] = n λ 2. (10) In the egion {θ θ θ (λ) } the sme Bellmn eqtion s (3) mst hold. The diffeence is tht in this egion we hve tht we know tht fte the next technology ivl the fim will not invest fo se. So tht Sbstittion of (11) in (3) nd ewiting gives E[dF (θ,λ)] = λdt(f (θ +,λ) F (θ,λ))+o(dt). (11) F (θ,λ) = π(θ 0) c +λ + λ (F (θ+,λ)). (12) +λ Following the soltion methods in Hismn (2001, Chpte 2) the soltion of the diffeence eqtion (12) cn be deived. We hve tht F (θ,λ) = ( ) λ θ (λ) θ ( π(θ (λ)) π(θ ) 0) c +λ 2.2 Optiml Speed of Technology Aivl + π(θ 0) c. (13) Knowing the eslts of the pevios sbsection the fim will tht level fo the speed of ivl tht mximizes its vle. In mthemticl wods the fim solves ( mx F (θ,λ) exp(λ) 2). (14) λ 0 Sbstittion of eqtion (13) into eqtion (14) gives ise to the following fist ode condition, which implicitly defines λ + ( λ +λ ( λ +λ ) θ (λ) θ ) θ (λ) θ ( π(θ (λ)) π(θ (λ)) θ π(θ 0) c θ (λ) )( λ(+λ) θ (λ) θ + 1 ( ) λ θ log (λ) +λ exp(λ) = 0. (15) 3 )

4 We cn solve (15) nmeiclly if we hve n expession fo θ (λ) G(λ,θ) = π(θ 0) c +λ + λ ( π(θ+) +λ. Define G(λ,θ) s ) π(θ) +I (16) So tht G(λ,θ (λ)) = 0 nd we hve tht θ (λ) = G(λ,θ) G(λ,θ) θ θ=θ (λ) = ( ) π(θ 0) c π(θ (λ)+) (+λ) 2 (+λ) 2 λ π(θ (λ)+) (+λ) θ 1 π(θ (λ)) θ (17) We cn deive λ fte sbstitting (17) into (15) nd solving the eslting eqtion nmeiclly. 3 Reslts 3.1 Exmple We se the sme pmetes s in Hismn (2001, Chpte 2), so we tke π = 200θ 2, = 0.1, = 0.1, I = Fthemoe, we tke = 600 nd c = 20. In cse the fim does not setp n R&D cente we ssme tht λ = 1. In the ltte cse we cn deive with eqtion (7) tht θ = 2.703, so tht 18 jmps e needed nd the expected dopion time is 18 yes with stndd devition of yes. The vle of the fim is eql to In cse the fim does setp n R&Dcente we find tht the fim shold set the optiml speed oftechnology ivl eql to λ = The vle of the fim then eqls The eslting investment tigge is eql to θ (λ ) = 5.093, so tht 71 jmps e needed nd the expected doption time is yes with stndd devition of yes. The setp costs fo the R&D cente e We conclde tht fo these pmete vles it is optiml to setp the R&D cente s the vle of the fim with the R&D cente is highe thn the vle of the fim withot the R&D cente. In Fige 1 the egion in which it is optiml fo the fim to setp n R&D cente in the (,c) spce is dwn. We conclde tht the highe e the setp costs fo the R&D cente, the lowe the nning costs c mst be nd vice ves. Figes 2-5 show the optiml size λ, the optiml technology doption tigge θ (λ ), the expected technology doption time E[T ], nd the nmbe of needed jmps n, espectively, s fnction of the setp costs. The nning costs e tken sch tht the fim is exctly on the indiffeence bondy between setting p n R&D cente nd not setting p n R&D cente. The highe e the setp costs, the lowe will be the optiml size λ s eslt the optiml technology doption tigge θ (λ ) will be lowe. The plot fo the expected technology doption time shows two effects. In genel highe setp costs implies lowe speed of technology ivl nd lowe technology doption tigge, which e two opposing effects on the expected technology doption time. A lowe speed of technology ivl eslts in highe expected technology doption time. Howeve, lowe technology doption tigge deceses the expected technology doption time. It tns ot tht in the pesented cse 4

5 the fome effect domintes so tht in genel the tend is tht the expected technology doption time is incesing. Fthemoe, we see tht cve tht epesents the expected technology doption time cn decese in the shot n. This is csed by the fct tht the nmbe of needed jmps is step fnction nd in between the steps we hve tht λ deceses so tht the expected technology doption time inceses ntil the nmbe of needed jmps is edced. At tht moment the expected doption time jmps down little nd stts to ise gin Do not setp c 1000 Setp Fige 1: Region in the (,c) spce in which it is optiml fo the fim to setp n R&D cente Λ Fige 2: Optiml size λ on the indiffeence bondy s fnction of. 5

6 Θ Λ Fige 3: Optiml technology doption tigge θ (λ ) on the indiffeence bondy s fnction of E T Fige 4: Expected doption time E[T ] on the indiffeence bondy s fnction of. 6

7 n Fige 5: Nmbe of needed jmps n on the indiffeence bondy s fnction of. 4 Conclsion In this ppe we stdied the combined R&D investment nd technology doption poblem of the fim. Refeences Hismn, K. J. M. (2001). Technology Investment: A Gme Theoetic Rel Options Appoch. Klwe Acdemic Pblishes, Dodecht, The Nethelnds. 7