Intellectual property rights protection and endogenous economic growth revisited

Intellectual property rights protection and endogenous economic growth revisitedRubens P.Cysne n,David TurchickGraduate School of Economics,Getulio Vargas Foundation(EPGE/FGV),Brazila r t i c l e i n f oArticle history:Received3November2010Received in revised form9September2011Accepted30December2011Available online14January2012JEL classification:O31O34O4Keywords:Intellectual property rightsInnovationLab equipment modelPatent protectionGrowtha b s t r a c tWe present an analytical solution to the lab-equipment R&D growth model with anexogenous rate of imitation and apply it to study the optimal level of intellectualproperty rights(IPR)protection.This has already been studied in Kwan and Lai(2003);however,a mistake in writing out the dynamics of the problem has contaminated thatanalysis.For the whole parameter space considered there,the conclusion is no longer tostrengthen IPR protection partially,but fully(a result which we prove analytically forthe logarithmic utility function).The usual tradeoff persists,though,for differentchoices of parameters.&2012Elsevier B.V.All rights reserved.1.IntroductionIt is well known that the issue of protection of intellectual property rights(IPR)presents governments with a tradeoff. On the one hand,by lessening the public good character of ideas and being beneficial to entrepreneurship,IPR protection alleviates possible underinvestment problems that arise in the knowledge market and may,thereby,foster economic growth and welfare.1On the other hand,by favoring a less competitive economic environment,it might also bring about a short-term welfare reduction.Since Tandon(1982),Judd(1985)and Gilbert and Shapiro(1990)prescriptions of an infinite-length patent policy, several studies using diverse models and methods have shown that,on the contrary,optimal patent length could be expected to befinite.Some examples are Horowitz and Lai(1996),Iwaisako and Futagami(2003),Kole´da(2004),and, more recently,Horii and Iwaisako(2007),Futagami and Iwaisako(2007),Furukawa(2007),Bessen and Maskin(2009),Acs and Sanders(forthcoming)and Chen and Iyigun(2011).Here we shall pursue a simple solution to the IPR tradeoff by working within the‘‘lab-equipment’’general equilibrium model of endogenous growth of Rivera-Batiz and Romer(1991),in which the R&D sector usesfinal good as input for the production of blueprints.Here,R&D is responsible for‘‘horizontal’’innovation(in product variety),in contrast to SchumpeterianContents lists available at SciVerse ScienceDirectjournal homepage:/locate/jedcJournal of Economic Dynamics&Control0165-1889/$-see front matter&2012Elsevier B.V.All rights reserved.doi:10.1016/j.jedc.2011.12.005n Corresponding author.Praia de Botafogo190,s.1100,Rio de Janeiro,RJ22250-900,Brazil.Tel.:þ552137995832;fax:þ552125538821.E-mail addresses:rubens.cysne@fgv.br(R.P.Cysne),davidturchick@(D.Turchick).1This effect has been empirically evaluated in Gould and Gruben(1996),Park and Ginarte(1997),Schneider(2005)and Falvey et al.(2006).Journal of Economic Dynamics&Control36(2012)851–861‘‘vertical’’innovation (in product quality).Added to the model is an exogenous imitation rate a la Krugman (1979)(also used by Barro and Sala-i-Martin,2004,Chapter 6.2;and by Gancia and Zilibotti,2005,Section 2.3),associated with the prevailing level of IPR protection.The first effect of the tradeoff above,related to long-term growth,will derive essentially from the usual Euler equation.By giving the model a global analytical solution (unlike any other analyses of this model of which we are aware in the literature),it becomes possible to precisely gauge the second effect,the one which emerges from the instantaneous change in the consumption level.In this way all the dynamic welfare gains/losses related to a change in IPR protection policy can be taken into account.Kwan and Lai (2003)uses this same model in order to analyze optimal patent protection.However,that analysis contains a mistake in the writing out of the dynamics of the problem which ends up contaminating the results.2Iwaisako and Futagami (2003)also addresses this issue in a model equivalent to the one presented here,although using a fixed-length patent system,rather than an imitation rate of invented varieties.However,they limit their welfare analysis to steady states.The present paper considers the whole dynamic path.This paper shows,in contrast with Kwan and Lai (2003),that for the whole set of parameter vectors considered there,the optimal policy is always that of providing full (infinite-length)protection of IPR —and not an incomplete one.This is to say that,when restricted to the parameter values used by Kwan and Lai,the IPR protection tradeoff has in fact a corner solution,and the model’s policy implication is that governments should pursue an infinite-length patent policy.In particular,if utility is logarithmic,we prove that this conclusion is valid irrespective of any other parameter values.We also provide examples based on a parameter set other than the one worked out by Kwan and Lai (2003).In such cases,their insight that the present model is able to generate incomplete optimal patent protection happens to be correct.The IPR protection tradeoff may then have an interior solution,meaning that the optimal policy may be that of inaction (in very specific situations),loosening,or partial tightening of patent protection.The structure of the paper is as follows.Section 2presents the model and covers the growth effect of the IPR protection tradeoff.Section 3gives the closed-form solution for its dynamic path and evaluates the current-consumption effect.Section 4builds on the foundations established in the two previous sections to consider the optimal IPR protection problem,first analytically (through Proposition 3)and then numerically.Section 5concludes.2.The modelFollowing Kwan and Lai (2003)or Gancia and Zilibotti (2005,Section 2.3),the representative household seeks to maximizeU ¼Z þ1e Àr t u ðc t Þdt ,ð1Þwhere u is the CRRA functionu ðc Þ¼c 1Ày À11Ày if y a 1,log c if y ¼1(subject to the budget constraint _b¼w þrb Àc ,where consumption c is in terms of the final good,r is the rate of return on assets held b (which equals b 0at time 0and must be nonnegative for sufficiently advanced time),and w is the wage rate paid by final-good firms.This problem implies the standard Euler equationg :¼^c ¼r Àry ,ð2Þas well as the transversality condition r 4g (throughout the paper,‘‘^’’will stand for growth rate).Firm i 2I produces final goods according to the homogeneous production function Y i ¼L 1Àa i R A 0x a i ,j dj ,where L i is thelabor input,x i ,j is the quantity of index-j intermediate good being used as input,to which an elasticity of a 2ð0;1Þcorresponds,and A is the measure of existing intermediate goods.In order to maximize profit Y i ÀwL i ÀR A 0p j x i ,j dj ,the demand of firm i (who is a taker of prices p j in the intermediate goods market)satisfies x i ,j ¼L i ða =p j Þ1=ð1Àa Þ.Let x j :¼P i 2I x i ,j ¼L ða =p j Þ1=ð1Àa Þ,where L :¼P i 2I L i .It is convenient to write the production function in an aggregate form,noting that x i ,j =L i ¼x j =L :Y :¼X i 2I Y i ¼X i 2I L 1Àa i Z A 0L i x j a dj ¼X i 2IL i Z A 0x j a dj ¼L 1Àa Z A 0x a j dj :ð3ÞThe version of the lab-equipment model we use here prescribes that the representative of the R&D sector who invented intermediate good j will produce it using a unit of the final good.3Thus,it chooses p j seeking to maximize ðp j À1Þx j within2Edwin i and Yum K.Kwan have been invited by the editor to submit their views,but have declined to do so.3The original lab-equipment model (see Rivera-Batiz and Romer,1991)assumes that firms produce intermediate goods using capital.In this paper,we borrow this terminology (lab equipment)from Rivera-Batiz and Romer (1991)but assume,as in Barro and Sala-i-Martin (2004),that the production of intermediate goods demands final good,rather than capital.R.P.Cysne,D.Turchick /Journal of Economic Dynamics &Control 36(2012)851–861852its monopolistically competitive environment.The resulting price is p j ¼a À1.If IPR protection regarding this good is lost somehow,the inventor is no longer a price maker and will have to accept for p j his marginal cost,1.Following Krugman (1979),an exogenous imitation rate of m 40,underlying the patent-breaking process,is assumed:_A c ¼m ðA ÀA c Þ,ð4Þwhere both A ð0Þand A c ð0Þare given.Here,½0,A c is the set indexing intermediates whose patents have been broken (or secrets have been discovered),while ðA c ,A are those still being protected.The rate m is thus inversely related to the level of enforcement of IPR protection laws.Since x j ¼L ða =p j Þ1=ð1Àa Þ,the demand for intermediates can be written as x j ¼x c ¼L a 1=ð1Àa Þif j 2½0,A c ,x m ¼L a 2=ð1Àa Þif j 2ðA c ,A :(Finally,entry into the R&D sector is conditioned to the possibility of being rewarded with monopoly rents to the extent that IPR is being protected.The relevant rate of return to the potential innovator is the interest rate adjusted,through a no-arbitrage argument,for imitation risk:r þm .4He must check if the present value of the returns from discovering intermediate good j pays at least the cost of invention,which is assumed to be a constant,b 40.Assuming there is free entry into the R&D sector of the economy,the ‘‘pays at least’’in the previous sentence becomes in equilibrium ‘‘pays exactly’’,and,at a given time s ,b ¼R þ1s e Àðr þm Þðt Às Þðp j À1Þx m dt ¼ða À1À1ÞL a 2=ð1Àa Þ=ðr þm Þ,whence at all times it is necessary to haver þm ¼Lb ða À1À1Þa 2=ð1Àa Þ:ð5ÞBy plugging (5)into the Euler equation (2),one obtainsg ¼L b ða À1À1Þa 2=ð1Àa ÞÀm Àry :ð6ÞWe see that in order to guarantee a nonnegative growth rate g ,the upper bound ðL =b Þða À1À1Þa 2=ð1Àa ÞÀr should be imposed on m .Also,d g =dm ¼À1=y o 0.This is to say that a tightening of IPR protection (by means of lowering m )brings about a faster growth of consumption on the whole equilibrium path.Let C ¼Lc represent aggregate consumption and X ¼R A 0x j dj total intermediate good production,so that the resource constraint for the economy is Y ÀX ¼C þb _A(total product,net of resources used in the production of intermediates,is used for either consumption or innovation).Using (3),this can be rewrittenb _A ¼L 1Àa ðAc x a c þðA ÀA c Þx a mÞÀA c x c ÀðA ÀA c Þx m ÀC ð7Þ¼L A c a a =ð1Àa ÞÀa 1=ð1Àa Þ þðA ÀA c Þa 2a =ð1Àa ÞÀa 2=ð1Àa Þ h i ÀC :ð8ÞFollowing Kwan and Lai (2003),consider two new variables,a scaled version of consumption,h :¼C =ðb A Þ,and the fraction of intermediate goods that have already been imitated,g :¼A c =A .Then _h ¼ðg À^A Þh and _g ¼m ð1Àg ÞÀg ^A (using (4)).Dividing both sides of (8)by b A yields ^A¼ðL =b Þ½g ða a =ð1Àa ÞÀa 1=ð1Àa ÞÞþð1Àg Þða 2a =ð1Àa ÞÀa 2=ð1Àa ÞÞ Àh .We thus get the model’s equilibria full dynamics:_g ¼m þðk 2Àm Þg þk 1g 2þgh ,_h ¼ðg þk 2Þh þk 1gh þh 2,(ð9Þwhere k 1:¼ð1þa À1Àa À1=ð1Àa ÞÞðr þm Þand k 2:¼Àð1þa À1Þðr þm Þ.5Not only k 2but also k 1must be negative,since k 1Z 0would imply 1þa Z a Àa =ð1Àa Þ,or a a ð1þa Þ1Àa Z 1¼aa þð1Àa Þð1þa Þ,contradicting the arithmetic-mean/geometric-mean inequality.The economically relevant steady state is ðg ,h Þ¼m m þg ,Àk 1m m þgÀg Àk 2 ,a saddle point (the degenerate case m þg ¼0,i.e.,m ¼g ¼0,would lead to an infinity of steady states,and is dismissed).In order to guarantee that h 40,the minimal condition to be imposed on the parameters (see Appendix A )is r þðL =b Þð1Àa Þa 2a =ð1Àa Þðy þa ðy À1ÞÞ40.6This implies g þk 2will always be negative,irrespective of the m chosen.Simply taking the coefficient of relative risk aversion y Z 0:5will do the job,for instance.Now,since ^h -0on the equilibrium path toward the steady state,we have ^A -g .Dividing both sides of Y ¼L ðA c a a =ð1Àa ÞþðA ÀA c Þa 2a =ð1Àa ÞÞby A shows that Y =A -L ða a =ð1Àa Þg þa 2a =ð1Àa Þ1Àg ðÞÞ,a constant,so that d Y =A -0,and also^Y -g .And since X ¼L ðA c a 1=ð1Àa ÞþðA ÀA c Þa 2=ð1Àa ÞÞ,one obtains X =A -L ða 1=ð1Àa Þg þa 2=ð1Àa Þð1Àg ÞÞ,a constant,so that also4For this argument in detail,the reader is referred to Gancia and Zilibotti (2005,ft.11).5It may be noted that this dynamical system is not equivalent to expression (8)in Kwan and Lai (2003),where the left-hand sides should read ^g and ^hinstead.This mistake was carried on throughout that paper.6By ‘minimal’it should be understood a condition not depending on a specific m .R.P.Cysne,D.Turchick /Journal of Economic Dynamics &Control 36(2012)851–861853^X-g .Therefore,the long-run growth rate of the gross domestic product Y ÀX is also g (and GDP per capita as well,since L is constant).Since we have already seen that d g =dm ¼À1=y o 0,we get a simple confirmation of one of the effects mentioned in the introduction.Proposition 1.Stronger (weaker )IPR protection brings faster (slower )long-run growth of the economy.This effect is more important the less risk-averse (or the more intertemporally elastic )society is .Also,as shown in Appendix A ,one has dg =dm 40and =dm 40.Thus two other long-term effects of strengthening the enforcement of IPR protection laws are a lower fraction of intermediates which have been imitated g and a lower consumption/assets ratio h .Welfare effects will be addressed in full in Section 4.3.Analytical solutionAt this point,one possibility is to find a linear approximation or a numerical solution to the path ðg ,h Þapproaching ðg ,h Þ,as attempted in Kwan and Lai (2003).However,it may be noted that Eq.(9)can be written in the form of a matricial differential equation of the Riccati type,asd dt g h !¼m 0 !þÀm 00g "#g h !þg h!k 2½ þg h !k 11ÂÃg h !,so its exact solution can also be sought for.7By imposing the terminal condition lim t -þ1ðg ðt Þ,h ðt ÞÞ¼ðg ,h Þ,one obtains the following expression for the stable saddle path:g ðt Þh ðt Þ"#¼z 0 !þ11Àg 0Àg g 0Àze Àðg þk 2þk 1z Þt g Àz h !,ð10Þwhere g 0is the given g ð0Þ(since both A ð0Þand A c ð0Þare given),andz ¼m Àk 2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðm Àk 2Þ2À4k 1m q 2k 1ð11Þis the negative root of k 1x 2þðk 2Àm Þx þm ¼0.Checkings are done in Appendix B .Note that the stable solution (10)is simply a straight line in the phase diagram.8Since h ðt Þ=¼ðg ðt ÞÀz Þ=ðg Àz Þ,the method employed to solve (9)also rewards us with an exact,explicit and elementary solution for h 0(¼h (0)):h 0¼g 0Àz g Àz ð12ÞEq.(12)makes it possible to compute the exact magnitude of the shift in h (and therefore in consumption,since the A variable cannot adjust immediately)once a strengthening or loosening of IPR protection takes place.This allows for the assessment of the welfare effect regarding the instantaneous response of consumption in face of a once-and-for-all change in IPR protection policy.Assume that government mandates for such a change.For example,a tightening of patent protection,so that m is lowered to m 0.Then,although (5)implies that k 1and k 2remain unaltered,it also forces the movement in r to instantaneously counterbalance the movement in m ,with no transitional dynamics whatsoever,so that r 0¼r Àðm 0Àm Þ,and the rate of growth of consumption in (2)immediately becomes g 0¼g Àðm 0Àm Þ=y .If the economy was initially at the steady state ðg ,h Þ,we would then haveg 00¼g ¼mm þg ð13Þand h initially jumping from h toh 00¼g 00Àz 0g 0Àz00,ð14Þwhere the new steady state ðg 0,0Þand the auxiliary expression z 0have the same expressions as before,but with m and greplaced by m 0and g 0.7For this purpose one may follow the strategy developed in Levin (1959)of first solving a related linear system of three first-order ordinary differential equations,and then applying Theorem 2there in order to get the solution paths of the original Riccati system.8In case this fact is somehow seen in advance (and assuming one is interested solely in the stable solution),an alternative course of action would be to first use the method of undetermined coefficients (in this case,D 1and D 2in h ¼D 1g þD 2)to transform the system of differential equations (9)into a single differential equation.R.P.Cysne,D.Turchick /Journal of Economic Dynamics &Control 36(2012)851–861854Since cutting m reduces both g and h,as noticed in the previous section,it is a priori ambiguous from(12)if this jumpin h,from h to h00,is up or downward.In other words,it is unclear what is to happen with current consumption,althoughthe expected effect may be that of a decrease.That this expectation is always correct is formally proved in:Proposition2.Stronger(weaker)IPR protection has an immediate negative(positive)effect on consumption.Again,this effect is more important the less risk-averse(or the more intertemporally elastic)society is.Proof.In Appendix C it is shown thatdh0¼yÀ1yg0zÀ1þk1ÀðmþgÞg0z!d zð15Þand d z=dm¼ð1ÀzÞ=ðk1ðz zÞÞo0,where z is the other zero of k1x2þðk2ÀmÞxþm.Since this parabola is concave and at x¼0takes on a positive(or zero if m¼0)value,while at x¼1it takes on the negative value k1þk2,we have z2½0;1Þ. From(15),it is immediate that,if y r1,then dh0=dm40,as wished.And sinced d y dh0dm¼1y2g0zÀ1þrÀry2g0z2d zdmo0,where(2)was used in the calculation of d g=d y,in order to conclude that dh0=dm40irrespective of y we just have to show that lim y-þ1dh0=dm Z0.Now,lim y-þ1dh0¼g0zÀ1þk1Àmg0z!d z,which is just an affine function of g0,whence it suffices to show that lim y-þ1,g0¼0dh0=dm Z0and lim y-þ1,g¼1dh0=dm Z0.And indeed,limy-þ1,g0¼0dh0¼À1þ1ÀzÀz¼1ÀzÀz40and,using Girard’s relation z¼m=ðk1zÞ,limy-þ1,g0¼1dh0dm¼1À1þk1Àmz!1k11Àzmk1zÀ¼1ÀzÀ1Àz¼0:Thus dh0=dm408y40,and current consumption drops with a decrease in m.We have also shown that this effect is larger the lower is y—that is,dðdh0=dmÞ=d y o0.&In words,the larger the intertemporal elasticity of substitution(1=y),the more people are willing to sacrifice their current consumption today(given the establishment of a less competitive environment)in the name of more consumption from some point on in the future(since g04g).Propositions1and2together confirm that the tradeoff mentioned in the introduction is a sensible one,and that it can be easily explained in terms of the lab-equipment model.However,as we show in the next section,this does not mean that we should always expect it to have an interior solution.4.The optimal level of IPR protectionStill in the situation described in the previous section,of an instantaneous exogenous change in IPR protection from m to m0,given the tradeoff represented by Propositions1and2,the government should want to choose the optimal m0.It may then substitute b Að0Þh00e g0t=L for c in the representative agent’s utility function,yielding the total intertemporal utility level associated to m0,accumulated throughout the new equilibrium path:U nðm0Þ¼1yb Að0ÞL1Àyh01Àyr y g0À1r!if y a1,1rlog b Að0ÞLþlog h0þg0rh iif y¼1:8>><>>:ð16ÞThus it solves the problem offinding m n¼m02½0,ðL=bÞðaÀ1À1Þa2=ð1ÀaÞÀr that maximizes9Hðm0,g00,yÞ:¼r uðh0Þþg0rÀð1ÀyÞg0¼rrÀð1ÀyÞg0h01Ày1ÀyÀ11Àyif y a1,log h0þg0r if y¼1:8><>:ð17Þ9The focus on these variables as arguments of H will be made clear in the proof of Proposition3.R.P.Cysne,D.Turchick/Journal of Economic Dynamics&Control36(2012)851–861855This is equivalent to the maximization problem stated in expressions (24)and (25)in Kwan and Lai (2003).Note from (14)that h 00depends upon g 00and from (13)that g 00depends upon m .This makes the optimal choice of patent protection m n depend upon its initial level m .Economically,this happens because the initial rate of imitation is related to the initial steady-state fraction of imitated goods,which influences the rate of new imitations from Eq.(4).The math involved here is a bit more intricate than that of the previous section.We shall analyze the y ¼1case mathematically,then treat the y a 1case with the aid of a computer.An observation that will be important in the proof of the next proposition is that k 1=k 2o 1.In fact,otherwise,one would have1r k 1k 2¼a À1=ð1Àa ÞÀð1þa À1Þ1þa À1¼a À1=ð1Àa Þ1þa À1À1‘2ð1þa Þr a Àa =ð1Àa Þ‘2ð1þa Þð2ð1þa ÞÞÀa r a Àa‘2ð1þa Þr ð2ð1þa À1ÞÞa ‘ð1þa Þ1=a r 21À1=a ð1þa À1Þ:Now,Bernoulli’s inequality applied to both left-and right-hand sides of the previous inequality gives ð1þa Þ1=a 41þð1=a Þa ¼2and 21À1=a ð1þa À1Þo 21À1=a ð1þ1Þ1=a ¼2,a contradiction.Proposition 3.If y ¼1,then ,given any m Z 0,m n ¼0.That is ,optimal IPR protection is simply absolute IPR protection ,regardless of its initial level .Proof.As shown in Appendix D ,H is strictly concave in its first variable,so that m n is either 0or a possibly positive zero of D 1H .In order to show that the latter will never be the case,we prove that D 1H ð0,g 00,1Þo 0.In fact,from (C.1)and (D.2),one hasD 1H ð0,g 00,1Þ¼À1r À100k 1Àg 0g 00z !1Àz 0k 2þ2k 10¼À1r þ1ðg 0þk 2Þ1þk 1k 2g 00 k 1Àg 0g 00Àk 2k 10B @1C A 1þk 2k 1k 2k 2¼À1r À1ðg 0þk 2Þ1þk 1k 2g 00 k 11Àg 0k 1k 22g 00 !1k 1þ1k 2 ¼À1r þ1þk 1k 2g 0þk 2g 0k 1k 2g 00Àk 2k 1g 00þk 2,where we have used the trivial facts that,for m 0¼0,z 0¼Àk 2=k 1and h 00¼Àðg 0þk 2Þð1þðk 1=k 2Þg 00Þ,where g 0þk 2o 0from the assumption on the parameters made in Section 2and explained in Appendix A .SinceD 2D 1H ð0,g 00,1Þ¼1þk 1k 2g k 2g 0k 1k 2ðk 1g 00þk 2ÞÀg 0k 1k 2g 00Àk 2 k 1k 1g 00þk 2ÀÁ¼1þk 1k 2g k 2g 0k 1þk 1k 2k 1g 00þk 2ÀÁ¼k 11þk 1k 2ð1g 00þ2Þo 0,we may focus on the g 00¼0(i.e.,m ¼0)case.Thus we compute (with the aid of (2),which under the present hypothesis gives r þg 0¼r 0)sgn ðD 1H ð0;0,1ÞÞ¼sgn À1r À1þk 1k 2g 0þk 2 !¼sgn g 0þk 2þr 1þk 1k 2 ¼sgn r þg 0À1þa À1ÀÁr 0þr k 1k 2¼sgn Àa À1r 0þr k 1k 2,and since k 1=k 2o 1,Àa À1r 0þrk 1=k 2o Àa À1r 0þr r ð1Àa À1Þr 0o 0,whence D 1H ð0,g 00,1Þo D 1H ð0;0,1Þo 0,and m n ¼0necessarily.&Proposition 3shows that it would be misleading to try to solve the first-order condition D 1H ðm n ,g 00,1Þ¼0if y ¼1,since the optimum is not interior.A possible reason for the specificity of the logarithmic utility function behind Proposition 3may be that,in this model (as also happens in the Ramsey–Cass–Koopman model with a CRRA utility function),y ¼1implies a propensity to consume out of wealth constant with respect to r (and here,through (5),with respect to m ).Remark 1.One could also consider the social planner’s problem,of choosing c ,x c and x m in order to maximize (1)subject to the resource constraints (7)and (4).This leads to the product growth rate g sp ¼ððL =b Þðða À1À1Þa 1=ð1Àa ÞÞÀr Þ=y ,clearly larger than the decentralized one,(6),irrespective of the chosen value of m .Thus the social planner’s solution cannot be implemented by a choice of IPR policy within our framework.In other words,the distortions caused by the lack ofR.P.Cysne,D.Turchick /Journal of Economic Dynamics &Control 36(2012)851–861856competitiveness (a lower production of intermediate goods and a private rate of return lower than the social rate of return implied by the planner’s analysis)cannot be compensated by a specific choice of m only.10Numerical implementation of the optimization program described in (17)with a set of parameter values containing that used by Kwan and Lai (2003,Section 3)(r ¼0.065;g ¼0:016as in King et al.,1988;a À12½1:25,2:5 ;y 2½1;4 as in Stokey,1995;and T 2½1;100 )still yields the corner solution m n ¼0obtained in Proposition 3.11The model considered here,therefore,suggests a policy of forever patent protection with more ease than it had been suspected at first.Interior solutions do occur,however,for different parameter choices.In such cases,the optimal policy may be one of partially tightening (0o m n o m )or even loosening (m n 4m )IPR protection (besides the unlikely but possible m n ¼m case).Under these circumstances,the first-order condition D 1H ðm n ,g 00,y Þ¼0(equivalent to Eq.(26)in Kwan and Lai,2003),coupled with the present calculation of dh 0=dm in expression (15),is useful.Figs.1–3,which use g ¼0:016and a À1¼1:6,give an insight to the shape of the optimal policy regions involved in this problem.A blank region means that the given r ,y and T yield inconsistent parameters,such as a nonpositive r (¼r Ày g ),or that the necessary transversality condition does not hold (r r g ).A light gray region symbolizes the recommendation of governmental weakening of the patent protection policy (m n 4m ),a dark gray region,that of partial strengthening (0o m n o m ),while a black region,that of total strengthening (m n ¼0).E.g.,the upper left graph in Fig.2can simply be seen as a graphical representation of Proposition 3.These figures illustrate three facts.First,a larger interest rate brings a greater possibility that the optimal solution to the patent policy tradeoff is to favor more the present,by weakening IPR protection.This can be easily visualized in any one of the graphs in Figs.2and 3.Indeed,by departing from any point in the valid (nonblank)region and moving upward,regions never become darker —only lighter (whenever there is a change in color tone).Given the parameters used in the simulation,one can see that a higher y (meaning less desire to spread consumption intertemporally)favors the present against the future.This can be seen in any one of the graphs in Figs.1and 2.By starting at any valid point in the figures and moving up (graphs of Fig.1)or to the right (graphs of Fig.3),regions can only become lighter.Finally,it is more likely for a weakening in patent protection to be optimizing if the initial level of protection is a high one already,as can be seen by moving right in Figs.1and 2.It may also be noted that the borders between weakening (light gray)regions and strengthening (dark gray)regions in these figures correspond to parameter values for which the optimal level of enforcement or protection of patent policy happens to exactly match its current level (m n ¼m ).For instance,consider the parameters r ¼0.15,g ¼0:016,a À1¼1:6and y ¼6(more in line with those of a developing economy).The lower left graph in Fig.2shows that in this case,the corresponding value of T would be approximately 30.4years (m %0:001586).Finally,usage of this numerical implementation with the same parameter values as in Kwan and Lai (2003)enables us to recalculate Tables 1–3presented in their paper.12Direct comparison of our and their tables shows thequantitativeFig.1.Optimal policy regions,given r .10However,as is suggested in Barro and Sala-i-Martin (2004,p.309),the social planner’s solution can be implemented if our framework is extended with the possibility that government subsidizes,via lump-sum taxation,(a)the purchase of intermediate goods in the ðA c ,A interval (at the rate 1Àa ),and (b)inventions,by implementing a new b 0so that ðL =b Þðða À1À1Þa 1=ð1Àa ÞÞ¼ðL =b 0Þðða À1À1Þa 1=ð1Àa ÞÞÀm .11Here we are using the identification m ¼r =ðe rT À1Þ,equivalent to Eq.(30)in Kwan and Lai (2003).12We have not recalculated line 4of those tables (welfare percentual gain)both because we do not have the value of A ð0Þused there and because relative difference is not an appropriate measure of welfare gain when the base value is not necessarily positive,as is the case here.R.P.Cysne,D.Turchick /Journal of Economic Dynamics &Control 36(2012)851–861857。