Appendices * for. R&D Policies, Endogenous Growth and Scale Effects

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1 Appndics * for R&D Policis, Endognous Growth and Scal Effcts by Fuat Snr (Union Collg) Working Papr Fbruary 2007 * Not to b considrd for publication. To b mad availabl on th author s wb sit and also upon rqust from th author.

2 A-1 Appndix A : Proof of Stability for Comptitiv Equilibrium I charactriz th transitional dynamics of th conomy by gnrating an autonomous systm for th quations of motion for c and d, whr c is consumption pr capita and d D/N is R&D difficulty pr capita. During th transition, th labor markt conditions, fr-ntry in R&D and optimal rnt protction activity conditions must hold. I rstrict attntion to th domain d(t) > 0, and λ(1 s) > c(t) > 0 which nsurs (t) > 0. Not from quation (13) that υ / υ = D /D. Substituting for X(ω, t) from (21) into (7) using = (ω, t) and X = X(ω, t) and th masur on of industris, it follows that D /D = [(δ +δ A )s / γd] + (μ μ A ). Combining (19) with (21), wγx(t)/n(t) can b writtn as wγx(t)/n(t)= δa (1 φ )s/(1 + μ)γ. Solving quation (11) for r(t) and substituting th rsulting xprssion into (4), using th xprssions for υ / υ and wγx(t)/n(t) immdiatly implis: 1 c( λ 1) δs s( δ + δ A ) c = c + ( c,d )[ 1 ( μ μ A )]] ρ. d λa (1 φ ) γ ( μ + (1 / ( c,d )) γ whr (c, d) = [1 s (c/λ)]/da coms from (20). Obsrv that dd / dc is indtrminat. On th c= 0 othr hand, it is possibl to draw som infrncs on th movmnt of c around c = 0. Mor spcifically, if μ μa < 1 thn d c /dc > 0, implying that starting from any point on c = 0 curv, an incras in c lads to c < 0. If μ μa > 1 thn d c /dc rmains ambiguous. To find th quation of motion for d simply substitut (c, d) into d / d = D /D n using th xprssion for D /D from abov. This implis: ( )s ( ) c d δ + δ A μ μ A = + 1 s nd a γ. λ Obsrv that dd / dc < 0 and that d d /dd < 0. Starting from any point on th d =0 lin, an d = 0 incras in d rndrs d <0, and an dcras in d rndrs d > 0.

3 A-2 Givn th ambiguous rlationships coming from c =0, thr ar multipl possibilitis. Howvr, w know that in th rlvant domain, th intrsction of c =0 and d =0 gnrats a uniqu point. Hnc, using a graphical approach, it is straightforward to analyz th local stability of this nonlinar systm. Whn c = 0 is upward sloping, th systm has a stabl nod if d c /dc > 0 and a stabl focus othrwis. Whn c = 0 is downward sloping, th systm has a stabl nod indpndnt of th sign of d c /dc. This rsult holds rgardlss of c = 0 bing flattr or stpr than d =0. Thrfor, w conclud that th systm is locally stabl, xhibiting ithr a stabl nod or a stabl focus. Figur Appndix A plots th dmarcation curvs c =0 and d =0 undr th bnchmark valus for th paramtrs. Obsrv that c =0 is upward sloping and givn μ μa < 1 it follows that d c /dc > 0. In this cas, th systm is saddl path stabl. Numrical simulations imply that for a wid rang of rasonabl paramtrs th pictur dpictd in Figur Appndix A rmains valid. Appndix B: Optimal R&D Policy: Solution, Uniqunss and Stability Th social plannr s problm is to { } 0 max -( ρ-n)t { Φ( t )log λ + log[1 s a d( t ) ( t )]}dt (A1) subjct to th stat quation Φ = (t), d = [(δ + δa)s/γ] + (μ μ A )(t)d(t) nd(t), th initial conditions Φ(0) = 0, d(0) = d 0 > 0, and th control constraint, (1 s)/a d(t) (t) 0 for all t. Th currnt valu Hamiltonian for this problm can statd as: H = Φ logλ + log[1 s a d] + θ + η[(δ + δ A )s/γ + (μ μ A )d nd]. For an intrior solution, th first ordr condition is: Η a d = + θ + η( μ μ A )d = 0. (A2) (1 s a d ) Th costat quations ar

4 A-3 H θ = ( ρ n ) θ = ( ρ n ) θ log λ Φ (A3) H a η = ( ρ n ) η = ( ρ n ) η + η[ ( μ μ A ) n]. (A4) d 1 s a d I solv th optimal control problm for a balancd growth quilibrium in which all ndognous variabls grow at constant rats (although not ncssarily th sam rat) and invstmnt in R&D is positiv > 0. It follows from (7) that for D /D to b constant, X(t)/D(t) and must b constant as wll. Givn X /X = n by (21), for X(t)/D(t) to b constant, D /D = n must hold. This in turn implis that d = D(t)/N(t) is constant. Substituting R(t) = D(t) into (20) implis that c must b a constant. Lt g θ = θ / θ. Constancy of gθ rquirs θ b a constant, which in turn implis θ / θ = 0. Lt g With and d constant, constancy of g η rquirs η b a constant, which in turn implis η = η / η. η / η = 0. Imposing θ = 0 and η = 0 on (A3) and (A4) implis θ = logλ/(ρ n) and η = a /[1 s a d)((μ μ A ) ρ)]. Imposing d = 0 yilds d = (δ + δa)s/[γ(n (μ μ A ))]. Not that (20) collapss to c/λ = 1 s a d. Substituting for θ, η and d into (A2) using c/λ = 1 s a d givs (31). Th RD SO and LM conditions, givn by (31) and (27) rspctivly, dtrmin th optimal balancd growth lvls ~ and c ~. To s uniqunss, considr plotting th RD SO and LM curvs in (, c) spac rstricting th domain to λ(1 s) > c > 0 and n/ (μ μ A )> > 0. For th RD SO quation: (dc/d) > 0. Morovr, as 0, c (c0) SO = λsa (ρ n)( δ + δ A )/ (logλ)nγ, and as max = n/ (μ μ A ), c. For th LM quation: (dc/d) LM > 0. Furthrmor, as 0, c λ (1 s), and as max = n/ (μ μ A ), c. Hnc, for a uniqu quilibrium, th intrcpt of th LM curv must b strictly CE highr than that of th RD curv: λ(1 s) > (c0) logλ > [sa (ρ n)(δ + δ A )] / [(1 s)nγ]. SO Obsrv that this is quit similar to th uniqunss condition for comptitiv quilibrium which was λ(1 s) > c 0 λ 1 > [sa (1 φ )(ρ n)( δ + δ A )]/[(1 s)nγ]. To analyz stability, I invok th transvrsality conditions: RD SO

5 A-4 lim (ρ n)t θ(t)φ(t) = 0. t lim (ρ n)t η(t)d(t)= 0. t Th transvrsality condition for θ and th costat quation imply that θ(t) = logλ(ρ n) for all t > 0 [s Grossman and Hlpman (1991), p. 71 and 103]. To s this formally, not that th solution for θ = (ρ n)θ log λ is θ(t) = θ0 (ρ n)t [ logλ ( (ρ n)t 1)]/ (ρ n), whr θ 0 is th initial valu for θ, which I assum xists but is unknown. Substituting this into th transvrsality condition implis (ρ n)t θ(t)φ(t) ={θ 0 [log λ/(ρ n)]} Φ(t) = 0. Givn lim Φ(t) =, th only possibl solution to this t limit is θ0 = log λ/(ρ n). Th solution to θ thn implis log λ θ ( t ) = ρ n lim t Substituting this into th transvrsality condition and using th L Hopital s rul implis that lim (ρ t n)t θ(t)φ(t) = [log λ/(ρ - n)](t)/, which quals zro for a finit lvl of (t) which follows from th constraint (1 s)/a d(t) (t) 0. Substituting for θ(t) = log λ/(ρ - n) into (A2) givs: 1 1 s 1 = ( η,d ) = d a log λ /[ d( ρ n )] + η( μ whr (η,d)/ η > 0, (η,d)/ d < 0. Substituting (η,d) and θ(t) into th costat quation for η and μ A, ) th quation of motion for d implis th following autonomous systm: a η = η[ ρ ( μ μ A ) ( η,d )] + [(1 s ) / ( η,d )] a d ( )s d δ + δ A = + [( μ μ A ) ( η,d ) n]d. γ

6 A-5 At th stady-stat for d ~ > 0, w nd n - (μ - μ A ) > 0. So, w rstrict attntion th nighborhood of th stady-stat whr < n/(μ - μ A ) holds. Obsrv that dη /dη and thus( dd / dη ) cannot b signd unambiguously. On th othr hand, around th stady-stat d d /dd < 0 and d d /dη > 0; thus, ( dd / dη ) d = 0 > 0. In othr words, d =0 is upward sloping, and starting from d =0 an incras η= 0 (dcras) in d rndrs d < 0, causing d to dcras (incras). Givn that a uniqu stady-stat quilibrium xists, w can xamin all of th possibilitis. First, whn η = 0 is upward sloping, η = 0 may b stpr of flattr than d = 0 and dη /dη mayb positiv or ngativ. Hnc, with η =0 upward sloping, thr ar four possibilitis. It can b shown graphically that th systm is saddl path stabl in ithr cas. Scond, whn η = 0 is downward sloping, dη /dη may b positiv or ngativ. Thus, withη = 0 thr ar two possibilitis. It can b shown graphically that th systm is saddl-path stabl whn dη /dη >0 and xhibits a stabl focus whn dη /dη < 0. Appndix C: Idntifying th Marginal Impact of Innovation I now driv an xprssion for th wlfar impact of a marginal innovation, following closly Grossman and Hlpman (1991, pp ) and Sgrstrom (1998, pp ). I considr a situation in which an xtrnal agnt bcoms succssful in innovating a highr quality product in industry ω at tim t = 0. I thn invstigat th impact of this vnt on th wlfar of all conomic agnts othr than th xtrnal agnt. To do this, I prturb th comptitiv quilibrium solution by at tim t = 0 and invstigat th impact on th discountd wlfar for th priod (0, ). I xclud th wlfar of th xtrnal agnt from th analysis bcaus th fr-ntry in R&D condition implis that for any ntrprnur ngagd in R&D, th R&D costs must b xactly balancd by th xpctd discountd rwards from R&D. Not that with th masur on of structurally-idntical industris, it again follows that D(ω,t) = D(t), (ω,t) = (t), X(ω,t) = X(t) for all ω and t.

7 A-6 Lt E(t) = c nt dnot th aggrgat consumr xpnditur. Thus, (30) can b rstatd as: U = 0 ( ρ n )t E( t ) Φ ( t )log λ + log dt nt λ To driv th wlfar impact of an incrmntal innovation, I diffrntiat th abov: du -( ρ n )t -( ρ-n)t 1 de( t ) = log λdt dt E( t ) ( t ) Th first trm quals logλ/(ρ n) and captur th consumr surplus xtrnality. Th scond trm capturs th businss staling and intrtmporal R&D spillovr xtrnalitis. Th succssful innovation by th xtrnal agnt lads to th rplacmnt of th incumbnt firm in industry ω, rsulting in a loss of stram of monopoly profits and lowr incoms for its stockholdrs. In addition, th marginal innovation raiss th difficulty of futur rsarch, rsulting in mor rsourcs bing divrtd to innovation activitis. Both of ths ffcts lad to lowr consumption xpnditur, which ar compoundd by multiplir ffcts. I now xplicitly idntify th businss staling and intrtmporal R&D xtrnalitis. Not that aggrgat xpnditur quals aggrgat incom minus aggrgat savings: E(t) = [sw + (1 s)] nt + π(t) a D(t), whr th first trm masurs th labor incom from spcializd and non-spcializd labor, th scond trm masurs th aggrgat profit incom (xcluding that of th xtrnal agnt), and th third trm masurs th aggrgat invstmnt in R&D. Diffrntiating E(t) with rspct to Φ and noting /d = 0 givs: de( t ) nt dw dπ dd( t ) = s + a. ( t ) Rcall that by combining (13) and (19), which hold both in and out of th stady-stat, w hav drivd w = a δ(1 φ )/γ(1+μ) thus dw/ = 0. Nxt, I considr dπ/. This trm capturs th dclin in aggrgat xpnditur associatd with th loss of profits in industry ω du to th marginal

8 A-7 innovation. In this modl, th ffctiv rplacmnt rat taks into account th rnt protction costs of incumbnt firms and quals (1 +η()). Sinc th arrival of innovations is govrnd by a Poisson procss whos intnsity quals, th arrival of ffctiv rplacmnt incidnts is also govrnd by a Poisson procss whos intnsity quals (1 +η()). It follows from th proprtis of th Poisson procss that th ffctiv duration of monopoly powr is xponntially distributd with paramtr (1 +η()). In th vnt of no furthr innovation btwn tim 0 and tim t that is, btwn th tim th xtrnal agnt innovats and th tim that signifis th currnt priod with probability (1 +η())t, no furthr innovation occurs and th conomy forfits monopoly profits at tim t of amount (λ 1)E(t)/λ. In addition thr is a multiplir ffct bcaus th loss of profits in industry ω will induc a fall in aggrgat xpnditur, which will translat into lowr incoms and xpnditurs in othr industris. Th combind chang in total profits quals: π λ 1 = E( t ) λ Nxt, considr a d( d ( 1+ η( )) D( t )) λ 1 de +. λ, which capturs th xpnditur dclin associatd with th highr rsourc rquirmnt in R&D du to marginal innovation. Th first stp is to find th solution to th diffrntial quation (7). Not that D(ω,t) = D(t), (ω,t) = (t) and X(ω,t) = X(t) for all ω and t. In addition, X A (t)= X(t) and A (t)= (t). Th solution to (7) is thn givn by: + t Φ ( t )( μ μ ) ( δ δ A )s ( t )( ) ns-( - ) ( s ) = A Φ μ μ D( t ) D + A μ μ A Φ 0 ds, whr D 0 is th lvl of R&D γ 0 difficulty at tim 0. Diffrntiating this givs: dd( t ) ( μ μ A ) Φ ( t ) = ( μ μ A )D0. Substituting th xprssions for dπ/ and dd(t)/ into th de(t)/ xprssion and collcting trms givs:

9 A-8 de( t ) ( t 1 ) E( t ) ( 1+ η( )) 1 ( μ μ A ) Φ ( t ) = ( λ 1) λa ( μ μ A )D0. E Substituting th abov xprssion now into du/ using E = [λa D(t)(ρ + (1 + η()) n)]/(λ 1), which follows from (25) and (13), and calculating th intgral implis: du log λ λ 1 ( λ 1)( μ μ A ) = ρ n ρ + (1 + η( )) n ρ + (1+ η( )) n 0 D 0 -( ρ n )t + ( μ μ A ) Φ ( t ) D( t ) dt To valuat th intgral trm, I invok th stady-stat proprtis of th modl as in Sgrstrom (1998, p. 1309). At th stady-stat (t)=, Φ(t)= t and D(t) =D 0 nt. Substituting ths into th intgral trm and valuating givs (33). Appndix E: Effcts of paramtr changs on R&D policy W can us th simulation rsults to dtrmin ach paramtr s qualitativ impact on spillovrs that works indirctly by altring th innovation rat. Rcall that ths wr ambiguous and involvd multipl ffcts. Th on analytical rsult w had was that an incras in dcrass th BS ffct. Th simulations rsolv th ambiguity on th IS ffct, rvaling that an incras in dcrass th IS ffct [rcall that thr wr two compting forcs on IS: a highr rducs th BS componnt of th IS ffct and at th sam tim it incrass th spillovr factor in th IS ffct]. 32 Thus, I can stat th following Lmma 3: Numrical simulations imply that an incras in rducs th ngativ xtrnalitis BS and IS and thrby calls for a ris in φ SO. Thus, any paramtr chang that lads to a largr innovation rat pushs th R&D policy toward subsidy. Armd with Lmma 3 w can now shd furthr light on th simulations using th rsults from Proposition 1 and th analytical xrcis in th prvious sction. E.1. An incras in λ 32 Th drivations ar availabl from th author upon rqust. Th rsults hold for a wid rang of paramtrs.

10 A-9 A highr λ lads to an incras in (Proposition 1), which indirctly rducs th siz of th ngativ xtrnalitis (Lmma 3) and thus calls for an incras in φ SO. Ths must b wighd against th dirct ffcts though. A largr λ incrass both th CS ffct (a positiv xtrnality) and th BS ffct (a ngativ xtrnality) and at th sam xrts an ambiguous ffct on IS ffct (a ngativ xtrnality). Simulations imply that th wlfar gains ar ngativ whn λ is of small and larg magnitud, and positiv whn λ is of mdium magnitud. E. 2. An incras in μ A highr μ lads to an dcras in (Proposition 1), which indirctly incrass th siz of th ngativ xtrnalitis (Lmma 3) and thus calls for rducd R&D subsidy. In addition, thr ar th dirct ffcts. A largr μ incrass th BS ffct and th IS ffct, both of which ar ngativ xtrnalitis. Hnc th dirct ffcts also call for rducd R&D subsidis. Thrfor w can conclud that an incras in μ would imply a fall in φ SO. D.3. An incras in μ A A highr μ A lads to an incras in (Proposition 1), which indirctly rducs th siz of th ngativ xtrnalitis (Lmma 3) and thus calls for an incras in R&D subsidy. In addition, th dirct ffcts complmnt this impact. A largr μ A dcrass th BS and IS ffcts, both of which ar ngativ xtrnalitis. Thrfor w can conclud that an incras in μ would imply an incras in φ SO. E.4. An incras in δ A A highr δ A lads to an dcras in (Proposition 1), which indirctly incrass th siz of th ngativ xtrnalitis (Lmma 3) and thus calls for a rduction in R&D subsidy. Again thr ar also th dirct ffcts that rinforc this mchanism. A largr δ A incrass th magnituds of ngativ xtrnalitis th BS and IS ffcts. Thrfor w can conclud that an incras in δ A would imply a fall

11 A-10 in th optimal R&D subsidy rat. Th ffcts for δ ar th sam, simply working in th opposit dirction. It is straightforward to dtrmin th optimal R&D policy ffcts associatd with th rst of th paramtrs using Proposition 1, Lmma 3 and (33). Tabl 1 and Figur 2 summariz th rsulting outcoms.

Appendices * for. R&D Policies, Endogenous Growth and Scale Effects

Appendices * for. R&D Policies, Endogenous Growth and Scale Effects Appendices * for R&D Policies, Endogenous Growth and Scale Effects by Fuat Sener (Union College) March 28 For the paper published in the Journal of Economic Dynamics and Control 32 (12), 28: 3895-3916.