Patent Examination Duration in an Endogenous Growth Model

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1 Paen Examinaion Duaion in an Endogenous Gowh Model Kiyoka Akimoo Takaaki Moimoo Absac We inoduce paen examinaion ino a sandad vaiey expansion and lab-equipmen ype R&D-based gowh model. Paen examinaion akes boh ime and cos. We examine he effecs of educing he paen examinaion duaion on paen pending vaieies, economic gowh, and welfae. The elaion beween he numbe of paen pending vaieies and paen examinaion duaion is inveed-u shaped, because educion in he paen examinaion duaion boh deceases he paen pending vaieies by ganing paens and inceases same by pomoing R&D. The educion in examinaion duaion pomoes economic gowh. Howeve, he elaionship beween examinaion duaion and social welfae is also inveed-u shaped because exemely sho paen examinaion equies massive esouces. Keywods: Paen office; Paen examinaion; Endogenous gowh; Innovaion. JEL Classificaion: O31; O34; O38. 1 Inoducion Innovaion and paening aciviy have been gowing all ove he wold. This gowh conibues o wold economic developmen. On he ohe hand, i causes some poblems a paen offices hough paen backlogs. Paen backlogs efe o he numbe of pending paen applicaions 1. Fims need o ake he following seps in he pocess of obaining a paen: filing he applicaion, equesing fo eseach, aking he examinaion, eceiving esuls and decisions fom he paen office, and finally obaining he paen. Thus, he lage he paen backlogs, he longe he duaion fom filing o obaining he paen. Gaduae suden; Gaduae School of Economics, Osaka Univesiy, 1-7 Machikaneyama, Toyonaka, Osaka , JAPAN; akimoo1015@gmail.com. Gaduae suden; Gaduae School of Economics, Osaka Univesiy, 1-7 Machikaneyama, Toyonaka, Osaka , JAPAN; qge020m@suden.econ.osaka-u.ac.jp. 1 The backlogs ae no well defined. Ohe definiions include unexamined applicaions and excess applicaions beyond he capaciy of he paen office. 1

2 Paen offices cope wih he backlogs, including he delay in obaining paens. Some offices se he fis acion (FA) ages and y o educe he acual he FA pendency. FA pendency is he aveage ime lapse beween he eques fo eseach and he peliminay decision by a paen office, which is used as an indicao fo backlogs. The annual FA age of he Unied Saes Paen and Tademak Office (USPTO) is 10 monhs by FY The USPTO achieved 16.4 monhs of FA pendency in Beween he end of FY2014 and ha of FY2015, FA pendency deceased by 1.7 monhs. The Japan Paen Office (JPO) achieved a long-em FA goal poposed in 2004 ha i would educe FA pendency o 11 monhs by FY2013 (he goal is called FA11). Afe achieving FA11, he JPO se a new age o educe FA pendency o less han 10 monhs by FY2023. Howeve, i emains unclea whehe he policy fo educing he paen examinaion duaion o deal wih he backlogs is bee fo economic gowh and social welfae, because less aenion has been paid o he paen examinaion duaion. Thus, his sudy invesigaes he effecs of educing he duaion of paen examinaion and deives he opimum duaion. We use he Rome (1987) s sandad vaiey expansion and lab-equipmen ype eseach and developmen (R&D) based gowh model. This is because final goods ae used fo R&D and he poducion of inemediae goods; fuhemoe, he vaiey of poducs is expanding hough R&D. We inoduce he paen examinaion duaion ino he sandad model. Ou model makes wo assumpions. Fis, a paen office exiss. The paen office examines each applicaion by fims who develop new vaieies of goods. The amoun of final goods used fo paen examinaion depends on he duaion of he examinaion. Tha is, educing he paen examinaion duaion equies moe esouces. Second, new enan fims mus pass paen examinaion by he paen office o obain he paen. Afe ha, hey sa poducing he paened poducs. Thus, hee is a lag beween invenion and poducion. In he dynamic geneal equilibium models of pevious lieaue on innovaion o paen policies, fims sa poducion as soon as hey succeed in innovaion. Tha is, hee is no lag because pevious sudies do no focus on paen examinaion. Ou model leads o he following esuls. The numbe of paen pending vaieies (o, paen backlogs) is an inveed U-shaped funcion of he paen examinaion duaion. Tha is, when he paen examinaion duaion is sufficienly long, educion of he duaion inceases paen backlogs. The shoe he paen examinaion duaion, he fase he economic gowh. If he examinaion duaion is sho, he expeced pofis fo new enans ae high; fuhemoe, he high expeced pofis acceleae R&D. On he ohe hand, i akes consideable esouces and coss o ealize sho examinaion duaions. Thus, exemely sho duaions hu he welfae of households. Moeove, we show he inveed-u shaped elaionship beween he paen examinaion duaion and social welfae hough numeical analysis. This sudy elaes o he lieaue on paen policies, especially macoeconomic lieaue based on R&D-based gowh models. We can divide he macoeconomic lieaue ino hee goups based on policies: paen lengh, paen beadh, and paenabiliy. On paen lengh, 2 The fiscal yea efes o he duaion fom Ocobe 1 o Sepembe 30 in he Unied Saes and fom Apil 1 o Mach 31 in Japan. 2

3 fo example, see Judd (1985), Iwaisako and Fuagami (2003), and Fuagami and Iwaisako (2007). On paen beadh, see Li (2001), Goh and Olivie (2002), and Chu and Fuukawa (2011). Thee exis a lage numbe of ohe seminal sudies on paen lengh and paen beadh. On he ohe hand, hee ae only a few sudies on paenabiliy, such as Hun (1999), Koléda (2004), O donoghue and Zweimülle (2004), Chu and Fuukawa (2013), and Kishi (2014). Howeve, as fa as we know, no sudy has been conduced on paen examinaion duaion. In addiion, ou sudy elaes o woks on inellecual popey ighs poecion in a Noh-Souh model. See Helpman (1992), Lai (1998), Glass and Saggi (2002), Kwan and Lai (2003), Tanaka e al. (2007), and Tanaka and Iwaisako (2014). The emainde of his pape is oganized as follows. Secion 2 pesens he basic model, mainly based on Rome (1987). Secion 3 descibes he dynamics of he model and seady sae. Secion 4 invesigaes he effec of educing he paen examinaion duaion on welfae. The esuls ae confimed in secion 5. Secion 6 povides concluding emaks. 2 Model Conside a vaiey-expansion and lab-equipmen ype R&D-based gowh model as in Rome (1987). The model includes a epesenaive household, a epesenaive final good fim, a coninuum of inemediae good fims, poenial enans o he inemediae goods seco, and a paen office. A epesenaive final good fim poduces final goods by using inemediae goods and labo in a compeiive make, wheeas inemediae goods ae poduced by using final goods in a monopolisically compeiive make. New vaieies ae developed by R&D using final goods. Mos of he seings of he model follow Rome (1987). In conas o he sandad vaiey-expansion and lab-equipmen ype endogenous gowh model, a newly developed vaiey canno be poduced in ou model unil he vaiey obains a paen. I akes ime and cos fo a paen office o gan a paen, and he cos is me by income ax. Fo simpliciy, paens ae assumed o no expie. 2.1 Households Households live infiniely, supply L unis of labo inelasically, and have a consan elaive isk avesion uiliy funcion: U = 0 ρ c1 σ e 1 d, (1) 1 σ whee c is consumpion, ρ > 0 is he discoun faco, and σ > 0 is he ae of elaive isk avesion o invese of he ineempoal elasiciy of subsiuion. Assume ha he discoun ae is high enough o saisfy ρ > (1 σ) a he equilibium so ha he uiliy will be welldefined. The budge consain is Ẇ = W + (1 τ )w L c L, (2) 3

4 whee W is he asse, is he inees ae, and w is he wage ae. Maximizing (1) wih espec o (2) yields he following Eule equaion: 2.2 Final Good ċ c = 1 σ ( ρ). (3) A epesenaive fim poduces final goods Y by using labo L and inemediae goods x. The poducion echnology is specified as A Y = L 1 α x ( j) α d j, (4) whee A is he numbe of vaieies poduced, ha is, he numbe of paened vaieies. Pofi maximizaion yields he following fis ode condiions: whee p is he pice of inemediae goods. 2.3 Inemediae Good 0 w = (1 α) Y L, (5) [ ] α 1 x ( j) p ( j) = α, (6) L Each paened inemediae good fim anslaes γ unis of final goods ino one uni of diffeeniaed inemediae goods. The pofi funcion of he inemediae good fim is π ( j) = max {p ( j)x ( j) γx ( j)}, (7) subjec o (6). The opimal pices, quaniies, and pofis ae p ( j) = p γ α, (8) ( α 2 x ( j) = x γ ( α 2 π ( j) = π (1 α)α γ ) 1 1 α L, (9) ) α 1 α L. (10) The opimal pices (8), quaniies (9), and pofis (10) ae he same fo all j [0, A ] and ime invaian. 4

5 2.4 R&D and Paen Claims New enans o he inemediae goods seco should fis develop new vaieies and, hen, apply paens. In ode o develop one vaiey, fims need o inves η unis of final goods in R&D aciviies. An enan ha has developed a new vaiey applies a paen claim o a paen office. The enan obains a value V in exchange fo he R&D coss. Theefoe, he pofis of new enan fims ae V η. As long as he pofis ae posiive, new enans ene he inemediae goods seco, and hus he numbe of developed vaieies, D, inceases. In conas, when he pofis ae negaive, hee ae no enans. Theefoe, =, if V > η, Ḋ [0, ), if V = η, = 0, if V < η. In equilibium, V η, Ḋ 0, and (V η) Ḋ = Paen Examinaion We assume ha he paen examinaion duaion is sochasically deemined accoding o he Poisson pocess, and ha he insananeous pobabiliy is µ. In ohe wods, a each poin of ime, he paen office awads paens fo a facion of µ of he paen pending vaieies, D A, ha is, Ȧ = µ(d A ). (11) The expeced paen examinaion duaion is 1/µ. I akes δ(µ) unis of final goods o paen one vaiey. Since he shoe paen examinaion duaion akes moe coss, δ(µ) is inceasing in µ, ha is, δ (µ) 0. The cos of he paen office is paid by labo income ax. Theefoe he govenmen budge consain is 2.6 No Abiage Condiion δ(µ)µ(d A ) = τ w L. (12) Fo simpliciy, assume ha he paen is no expied. Since he expeced paen examinaion duaion is independen of when a vaiey was developed, he discouned sum of expeced pofis is he same fo all paen pending (i.e., unpaened) vaieies, D A. Theefoe, he value of paen pending vaieies V is also he same fo all paen pending vaieies. The no abiage condiion abou paen pending vaieies equies ha hei value is jus he discouned sum of expeced pofis, ha is, V = µe µ(s ) 5 s z dz π u duds. (13)

6 The em s z dz π u du in (13) is he discouned sum of pofis when a fim obains a paen a s >, and he em µe µ(s ) is he pobabiliy ha a fim obains a paen a s >, given ha i does no do so a. The paened vaieies, A, do no face any unceainy, so he no abiage condiion abou paened vaieies equies ha he value of paened vaieies ae he discouned sum of pofis. 2.7 Make Cleaing V A = z dz π u du. (14) In his model, hee is a final goods make, a coninuum of inemediae goods makes of [0, A ), a labo make, and an asse make. The inemediae goods make cleas since each inemediae good fim behaves while consideing he demand (6). The labo make also cleas. The cleaing condiion of he final goods make is Y = c L + A The cleaing condiion of he asse make is 0 γxd j + ηḋ + δ(µ)µ(d A ). (15) W = V A A + V (D A ), (16) which is cleaed using Walas law focusing on he final goods make (15). 2.8 Equilibium In equilibium, x is symmeic fo all j [0, A ) fom (9), and hus he final good poducion can be wien as ( ) α α 2 1 α Y = A L. (17) γ The naional income is disibued o he inpu fo he inemediae good poducion, labo income, and pofi of inemediae good fims as follows: A 0 γxd j = α 2 Y (18) w L = (1 α)y A π = (1 α)αy (19) 6

7 The no abiage condiion abou paen pending vaieies (13) yields he dynamics of he inees ae as follows (see Appendix): ṙ = (µ + ) µπ η. (20) Since ṙ is inceasing in, deceases wih ime when is small and vice vesa; ha is, he dynamics of he inees ae (20) ae unsable. Since he inees ae is a jumpable vaiable, i iniially jumps o he seady sae defined as (µ + )/µ = π/η, o = (µ) µ 2 ( µη) π 1/2 1. (21) Moeove, noe ha d(µ) = 2 > 0 and lim dµ µ(µ+2) µ (µ) = π/η. Thus, he inees ae is inceasing in µ and conveges o π/η as µ goes o infiniy. 3 Dynamics and Seady Sae Inemediae good pices (8), inemediae good poducion (9), and pofis fo inemediae good fims (10) ae consan ove ime. The value of paen pending vaieies is also consan as long as R&D occus. The inees ae iniially jumps o is seady sae value (21). Wage (5) depends on final good poducion (17), which depends on he numbe of paened vaieies. The value of paened vaieies is deemined o mee he households budge consain (2) and o clea he asse make (16), depending on he inees ae (see Appendix). The dynamics of he es of he vaiables, numbe of developed vaieies D, numbe of paened vaieies A, and consumpion c ae invesigaed in his secion. These hee vaiables, D, A, and c, and hei diffeenial equaions ae anslaed ino wo new vaiables and hei diffeenial equaions. Then, a seady sae of he new vaiables is invesigaed. This seady sae coesponds o a balanced gowh pah on which he oiginal hee vaiables, D, A, and c, gow a he same consan gowh ae. Theefoe, he gowh ae a he seady sae is calculaed. 3.1 Dynamics Subsiuing (17) and (18) ino (15) yields (1 α 2 )A ( α 2 γ ) α 1 α L = c L + ηḋ + δ(µ)µ(d A ). (22) The lef hand side of (22) is he ne oupu of he final goods, ha is, he oupu of final goods minus he inpu fo he inemediae good poducion. The igh hand side is he demand of he final goods, ha is, consumpion, he inpu fo R&D, and he inpu fo paen examinaion. 7

8 Equaion (22) implies ha seveal paen pending vaieies (o paen backlogs) D A use a gea amoun of esouces ha can be used fo consumpion and R&D. Togehe wih (3) and (11), equaion (22) descibes he dynamics in case of posiive R&D. In addiion o hese hee diffeenial equaions, (3), (11), and (22), he following feasibiliy condiion mus hold: (1 α 2 )A ( α 2 γ ) α 1 α L c L + δ(µ)µ(d A ), (23) whee equaliy holds when Ḋ = 0. Feasibiliy condiion (23) equies Ḋ o be non-negaive 3. To educe he numbe of vaiables, define ω = D A, χ = c L A. (24) Because ω 1 = (D A )/A is he aio of paen pending vaieies o paened vaieies, ω epesens a measue of he numbe of paen pending vaieies (o paen backlogs). Subsiuing (11), (17), and (19) ino (22) yields he dynamics of ω as ω = 1 { µ[ηω + δ(µ)](ω 1) α } η α π χ. (25) Noe ha he em (1 α 2 )(α 2 /γ) 1 α α L is ewien as 1+απ in (25) by using (10). The dynamics α of χ can be obained fom (3) and (11) as { } 1 [ ] χ = χ (µ) ρ + µ µω. (26) σ Feasibiliy condiion (23) is ewien as χ 1 + α α π δ(µ)µ(ω 1). (27) In addiion, ω 1 mus hold fo all since D A fo all. Gaheing (25), (26), (27), and ω 1, a phase diagam of his economy can be depiced as in Figue 1. The diagonal aea in Figue 1 is no feasible. 3.2 Seady Sae In his model, he seady saes of ω and χ coespond o he balanced gowh pahs on which c, A, D, and Y gow a he same ae. The poin E in Figue 1 epesens he seady sae (o he balanced gowh pah), which is saddle pah sable (see Appendix). 3 See Appendix fo he dynamics in case of no R&D. 8

9 χ χ = 0 E ω = 0 O 1 ω ω Figue 1: Phase diagam Fom (25) and (26), he seady sae values of ω and χ ae Fom (3), he gowh ae a he seady sae is ω = [(µ) ρ], σµ (28) χ = µ[ηω + δ(µ)](ω 1) α π. α (29) g Ȧ = Ḋ = ċ A D c = 1 [ ] (µ) ρ. (30) σ Since (µ) is inceasing in µ, he gowh ae a he seady sae is also inceasing in µ, ha is, dg dµ = 1 d(µ) σ dµ > 0. (31) Theefoe, he shoe he paen examinaion duaion, he fase he gowh because of he highe expeced eun fom R&D. To ensue he posiive seady sae gowh, fis assume ha π η > ρ. (A1) This assumpion (A1) ensues posiive seady sae gowh when µ goes o infiniy; ha is, i ensues posiive seady sae gowh in he sandad vaiey expansion ype endogenous 9

10 gowh model. Then, hee is a lowe bound of µ, above which he posiive seady sae gowh is ensued since he seady sae gowh ae is inceasing in µ. Solving (µ) > ρ fo µ and applying (A1) yields ρ µ > µ 0 π ρη 1 > 0. (32) The inequaliy (32) specifies a lowe bound of Poisson aival ae, µ 0, ove which hee exiss a balanced gowh pah. In ohe wods, hee is no balanced gowh pah if he expeced paen examinaion duaion is oo long. 4 Compaaive Saics This secion invesigaes he effecs of educing he paen examinaion duaion, o inceasing µ. 4.1 Effecs on Seady Sae Values Diffeeniaing (28) wih espec o µ yields dω dµ = 1 σµ 2 ρ π/η ( π µη whee µ 1 saed. 4 π η ( ) πρη 2 1 > µ 0 since µ 1 µ 0 = 3 η+ϕ π ρ ( π ρ(η+ϕ) ) 1/2 0, if µ µ 1, < 0, if µ > µ 1, (33) ) 2 1 > 0. Then, he following poposiion can be Poposiion 1 (The seady sae aio of developed vaieies o paened vaieies). The seady sae aio of developed vaieies o paened vaieies, ω, is an inveed U-shaped funcion of he Poisson aival ae of paen examinaion, µ, and maximized a µ 1, which is geae han he lowe bound of he Poisson aival ae, µ 0. Poposiion 1 implies ha when he paen examinaion duaion is sufficienly long (µ < µ 1 ), a educion in he duaion (o an incease in µ) inceases he measue of paen backlogs, ω. In conas, when he paen examinaion duaion is sufficienly sho (µ > µ 1 ), he shoe he duaion (o highe he µ), he lesse will be he paen backlog, ω. Theefoe, educing he paen examinaion duaion does no always educe paen backlogs. This is because educing he duaion acceleaes boh R&D aciviies, Ḋ, and he paen examinaion, Ȧ. When he paen examinaion duaion is sufficienly long (µ < µ 1 ), R&D becomes moe acive han paen examinaion (Ȧ < Ḋ ), and hus ω inceases and vice vesa. Diffeeniaing (29) wih espec o µ yields dχ dµ = [ηω + δ(µ) + µδ (µ)](ω 1) µ [ 2η(ω 1) + η + δ(µ) ] dω dµ. (34) 10

11 Since dω /dµ 0 fo µ µ 1 and ω 1 fo µ µ 0, dχ /dµ is negaive a leas fo µ [µ 0, µ 1 ] and ambiguous fo µ > µ 1. Summaizing he above, when µ inceases, in Figue 1, he seady sae shifs o he lowe igh fo µ [µ 0, µ 1 ], and i shifs o he uppe o lowe lef fo µ > µ 1. The new seady sae is also saddle pah sable, and he economy conveges o i fom he lowe lef o uppe igh. Since ω is a sae vaiable and χ is a jump vaiable, when he seady sae shifs o he lowe igh, χ jumps o he lowe level. Tha is, a educion in he paen examinaion duaion makes χ jump o he lowe level fo µ [µ 0, µ 1 ], implying ha consumpion level a ha poin deceases. In conas, fo µ > µ 1, i is no obvious whehe χ jumps lowe o highe Effec on Welfae Befoe invesigaing he effec of educing he paen examinaion duaion on welfae, i is convenien o noe ha hee exiss he uppe bound of µ, o he minimum paen examinaion duaion. Since χ should be non-negaive fo all, he feasibiliy condiion (27) equies δ(µ)µ 1 + α α π ω 1. (35) Since he lef hand side of (35) is inceasing in µ and hee is ω on he igh hand side, hee is an uppe bound of µ depending on ω defined as δ(µ 2 (ω ))µ 2 (ω ) = 1 + α α π ω 1, (36) and dµ 2 /dω < 0. When µ goes o µ 2 (ω ), χ goes o 0 because hee ae no esouces o consume. This uppe bound is deceasing in ω. This means ha when hee ae a lo of paen pending vaieies, he paen examinaion duaion canno be educed oo much; howeve, when hee ae only a few paen pending vaieies, he duaion can be educed subsanially. Then, we define welfae as he indiec uiliy of he household. Subsiuing (3) and (21) ino (1) yields he equilibium (no only a he seady sae) welfae level wih posiive R&D as follows: U = W(c 0, (µ)) σc 1 σ 0 (1 σ)ρ (1 σ) 2 (µ) 1 ρ(1 σ). (37) Assume fis ha he economy iniially is in he seady sae E, and, second, ha he paen examinaion duaion is educed a ime 0. The effec of educing his duaion on welfae is dw(c 0, (µ)) dµ = dw(c 0, (µ)) dc 0 dc 0 dµ + dw(c 0, (µ)) d(µ) d(µ) dµ. (38) 11

12 Table 1: Baseline paamee values µ US L σ ρ α η δ γ / As discussed above, a leas fo µ [µ 0, µ 1 ], dc 0 /dµ < 0. Theefoe, he fis em of (38) is negaive a leas fo µ [µ 0, µ 1 ], wheeas he second em is posiive. Alhough i seems ha dc 0 /dµ becomes posiive fo highe µ, exemely high µ o exemely sho paen examinaion duaion hu he welfae. This is because fase paen examinaion akes moe cos. When µ goes o µ 2 (ω 0 ), χ 0 goes o 0 because hee ae no esouces o consume, ha is, dc 0 dµ µ=µ2 (ω 0 ) < 0, lim µ µ 2 (ω 0 ) c 0 = 0. Then, as µ goes o µ 2 (ω 0 ), he effec on he welfae (38) goes o negaive infiniy, because consumpion goes o 0 and he insananeous uiliy funcion saisfies Inada condiions. Theefoe, hee is an opimal µ beween µ 0 and µ 2 (ω 0 ) in ems of welfae. In addiion, he iniial jumps of χ 0 depend on iniial sae vaiable, and hus he opimal µ depends on ω 0. 5 Numeical Analysis To illusae he esul of he model egading he opimal µ, his secion conducs a numeical analysis. Alhough he model is highly sylized, hese numeical esuls help o undesand he esuls of he model. In ode o calculae he iniial jump of χ 0, we use he Relaxaion Algoihm of Timbon e al. (2008). In ime 0, he economy is a he seady sae and µ changes. Fo each change of µ, we calculae he ime pah of χ, which conveges o new seady sae given he iniial sae vaiable ω 0. Fom hose pahs, we obain he fis jump of χ 0 and calculae welfae (37). Then, he opimal change of µ can be obained. Fo calculaion, δ(µ) is specified as δµ ε. Baseline paamees ae given by Table 1. The populaion size is nomalized o 1. σ is chosen o ensue 2% seady sae gowh ae. ρ is 0.02, and α is 1/3. A paamee of R&D echnology, η, is nomalized o 1. The baseline invese of paen examinaion duaion, µ US, is suied o he aveage US final acion pendency fom 2013 (2.42 yeas) o 2014 (2.28 yeas). A paamee of paen examinaion cos, δ, is chosen o make δ(µ) he aveage aio of expendiue of he USPTO o he US R&D expendiue fom 2010 o 2013 when µ = µ US and ε = 1. A paamee of inemediae good poducion is se o ensue = We invesigae fou vesions of paamees and he iniial sae. The esuls ae given by Table 2. Columns (1)-(3) give he opimal µ fo diffeen ε, wheeas ohe paamees and iniial sae vaiable ae he same. The elasiciy of paen examinaion cos on he duaion is ε, and highe ε implies ha moe coss ae need o educe he paen examinaion duaion. 12

13 Table 2: Opimal invese of he paen examinaion duaion (1)ε = 1 (2)ε = 3 (3)ε = 5 (4)ε = 3, µ = Opimal µ Theefoe, fo highe ε, a longe paen examinaion duaion is desiable. In conas, column (4) pesens a esul fo when he iniial µ diffes, and hus he iniial sae vaiables also diffe, wheeas ohe paamees ae he same as in column (2). The iniial µ in column (4) is se o he Japanese final acion pendency a In column (4), he iniial paen examinaion duaion is shoe han in column (2). Theefoe, he iniial ω in column (4) is less han ha in column (2), and hus a shoe paen examinaion duaion is desiable. 6 Conclusion We consuc a vaiey expansion and lab-equipmen ype R&D-based gowh model in which paen examinaion akes ime and cos. Then, we invesigae he effec of educing he paen examinaion duaion on he numbe of paen pending vaieies (o paen backlogs), economic gowh on a balanced gowh pah, and welfae. We conside he aio of he numbe of paen pending vaieies o he numbe of paened vaieies as a measue of paen backlogs. Then, he elaion beween his measue of paen backlogs and paen examinaion duaion was found o be inveed U-shaped; ha is, a educed paen examinaion duaion inceases paen backlogs when paen examinaion is sufficienly long. The shoe paen examinaion, he fase is he economic gowh on he balanced gowh pah because hee is geae incenive fo R&D. Howeve, shoe examinaion is no bee fom he view-poin of welfae. I akes moe cos o examine paens soone. Theefoe, exemely sho paen examinaion equies enomous coss and hus he welfae. Moeove, he opimal paen examinaion duaion depends on he iniial sae. This is because when hee ae seveal paen pending vaieies a educing in he duaion equies moe esouces o gan hese paens, and hus educes iniial consumpion. These mechanisms of he model ae confimed by a numeical analysis, and we show ha hee exiss an opimal duaion beween he uppe and lowe bounds of he paen examinaion duaion. Appendix Dynamics of Inees Rae wih Posiive R&D The value of one vaiey ha is developed a and paened a s is V,s = s z dz π u du 13

14 The paen examinaion duaion is sochasically deemined accoding o he Poisson pocess wih he insananeous pobabiliy µ. Theefoe, he pobabiliy ha a paen claim has no been acceped duing he ime peiod (s ) is e µ(s ), and ha i is acceped a he end of his peiod is µe µ(s ). Theefoe, he expeced value of one vaiey ha is developed a is V = µe µ(s ) V,s ds = Diffeeniaing his wih espec o ime, we obain V = µ + = µ z dz π u du + µe µ(s ) s µe µ(s ) µ 2 e µ(s ) z dz π u duds z dz π u du + (µ + )V. s s z dz π u duds. z dz π u duds The pofi, π, is consan and he fim value, V, is consan η, as long as R&D occus. Theefoe, he non-abiage condiion is (µ + )η = µ Diffeeniaing boh sides wih espec o esuls in ṙ η = µπ + µ = µπ + (µ + )η ṙ = (µ + ) µπ η z dz πdu. z dz πdu The dynamics ae unsable. Theefoe, he inees ae iniially jumps o he seady sae defined as (µ + ) µ = π η. The Value of Paened Vaieies wih Posiive R&D Diffeeniaing (14) wih espec o ime yields V A = π + z dz π u du = π + V A. (39) 14

15 χ E χ 0 E O 1 ω ω 0 ω Figue 2: Phase diagam wih high iniial ω Subsiuing (5), (12), (15), and (16) ino (2) yields V A A + V A Ȧ + V(Ḋ Ȧ ) = [V A A + V(D A )] + (1 α)y δµ(d A ) [Y α 2 Y ηḋ δµ(d A )], V A A + (V A V)Ȧ = [V A A + V(D A )] (1 α)αy. Using (11), (39), and (19), his can be ewien as ( V A π)a + (V A V)µ(D A ) = [V A A + V(D A )] (1 α)αy, ( V A = 1 + ) η. µ This deemines he value of paened vaieies. Case of No R&D Since he economy conveges o he seady sae E fom he lowe lef o uppe igh in Figue 1, he economy canno ake a saddle pah conveging o he seady sae E when he iniial ω is lage enough. Figue 2 illusaes such a case. A doed aow epesens he saddle pah conveging o he seady sae E fom he uppe igh. In Figue 2, he iniial ω is oo lage o iniially ake he saddle pah. In ode o each he seady sae, he economy should sa wihou R&D, and iniial χ saisfies he feasible condiion (27) wih equaliy, ha is, he following no R&D condiion: χ = 1 + α α π δ(µ)µ(ω 1). (40) 15

16 The economy goes o he uppe lef along he no R&D condiion (40) (a dashed aow) unil i eaches poin E. Afe eaching E, R&D becomes posiive, and he economy sas fo he seady sae aking on he saddle pah (a doed aow). Tha is, when he iniial ω is sufficienly lage, hee is no R&D. When hee is no R&D, Ḋ = 0 and V < η. Theefoe,, V, and V A ae diffeen fom hose in case of posiive R&D. Since π is consan egadless of whehe R&D is posiive o no, V and V A ae deemined by fom (13) and (14). Theefoe, i is necessay o invesigae how is deemined in case of no R&D. In ode o each poin E, he economy should move along he no R&D condiion (27). Theefoe, he aio of he change of χ o he change of ω should be he same as he slope of he no R&D condiion (40): χ = dχ = δ(µ)µ. (41) ω dω Ḋ =0 Subsiuing (40) ino (25) and (26) yields Ḋ =0 ω Ḋ =0 = µω (ω 1), (42) [ ] [ ] 1 + α 1 χ Ḋ =0 = α π δ(µ)µ(ω 1) σ ( ρ) µ(ω 1). (43) Subsiuing (42) and (43) ino (41) and solving fo yields = σ [ 1+α π + µδ(µ)] µ(ω α 1) 1+α π δ(µ)µ(ω + ρ. (44) α 1) Equaions (42), (43), and (44) descibe he dynamics in case of no R&D. Since is no consan in case of no R&D, welfae is also diffeen fom (37). Le denoe he peiod when he economy eaches poin E in Figue 2. Subsiuing (3), (21), and (44) ino (1) yields he welfae level when hee is iniially no R&D, as follows: Sabiliy W e σ [(1 σ) ρ]ds d + σe σ 1 [(1 σ)(µ) ρ] ρ (1 σ)(µ) c 1 σ 0 1 σ 1 ρ(1 σ). The lineaized dynamic sysem in he neighbohood of seady sae E is ż = Hz, whee ( ) ω ω z = χ χ, H = ( 2µω + µ ( 1 δ η) 1 η µχ 0 ). 16

17 The maix H has one posiive chaaceisic oo and one negaive chaaceisic oo since Theefoe, he seady sae E is a saddle poin. de H = µ η χ < 0 Acknowledgemens We hank Koichi Fuagami and he semina paicipans a Oau Univesiy of Commece fo hei commens. Any eos ae ou esponsibiliy. Takaaki Moimoo acknowledge he financial suppo fom he Gan-in-Aid fo JSPS Fellows of he Japan Sociey fo he Pomoion of Science No. JP15J Refeences Chu, Angus C and Yuichi Fuukawa, On he opimal mix of paen insumens, Jounal of Economic Dynamics and Conol, 2011, 35 (11), and, Paenabiliy and knowledge spilloves of basic R&D, Souhen Economic Jounal, 2013, 79 (4), Fuagami, Koichi and Tasuo Iwaisako, Dynamic analysis of paen policy in an endogenous gowh model, Jounal of Economic Theoy, 2007, 132 (1), Glass, Amy Jocelyn and Kamal Saggi, Inellecual popey ighs and foeign diec invesmen, Jounal of Inenaional economics, 2002, 56 (2), Goh, Ai-Ting and Jacques Olivie, Opimal Paen Poecion in a Two-Seco Economy, Inenaional Economic Review, 2002, 43 (4), Helpman, Elhanan, Innovaion, imiaion, and inellecual popey ighs, Technical Repo, Naional Bueau of Economic Reseach Hun, Robe, Nonobviousness and he incenive o innovae: an economic analysis of inellecual popey efom, Technical Repo, Fedeal Reseve Bank of Philadelphia Iwaisako, Tasuo and Koichi Fuagami, Paen policy in an endogenous gowh model, Jounal of Economics, 2003, 78 (3), Judd, Kenneh L, On he pefomance of paens, Economeica: Jounal of he Economeic Sociey, 1985, pp

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