economic growth revisited

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1 Intellectual property rihts protection and endoenous economic rowth revisited Rubens Penha Cysne y, David Turchick z November 00 Abstract An analytical solution to the lab-equipment rowth model (Rivera-Batiz and Romer, 99) with an exoenous imitation rate is presented and applied to study the policy tradeo between weaker levels of intellectual property rihts (IPR) protection yieldin more consumption today, and stroner levels yieldin more rowth tomorrow. This has already been studied in Kwan and Lai (003); however, a mistake in writin out the dynamics of the problem has contaminated that analysis. For the whole parameter space considered there, the conclusion is no loner to strenthen IPR protection partially, but fully. The tradeo persists, thouh, for di erent choices of parameters. Keywords: Intellectual property rihts; Innovation; Lab-equipment model; Patent protection; Growth. JEL: O3; O34; O4. y Correspondin author. Rubens Penha Cysne is a Professor at the Graduate School of Economics, Getulio Varas Foundation (EPGE/FGV). Address: Praia de Botafoo 90 s. 00 Rio de Janeiro, RJ Brazil. Telephone number: +55 () Fax: +55 () URL: address: rubens.cysne@fv.br. z David Turchick is a researcher at the Getulio Varas Foundation (FGV).

2 Introduction It is well known that the issue of protection of intellectual property rihts (IPR) presents overnments with a tradeo. On the one hand, by lessenin the public-ood character of ideas and bein bene cial to entrepreneurship, IPR protection alleviates possible underinvestment problems that arise in the knowlede market and may, thereby, foster economic rowth and welfare. On the other hand, by favorin a less competitive economic environment, it miht also brin about a short-term welfare reduction. We work within the "lab equipment" eneral-equilibrium model of endoenous rowth of Rivera-Batiz and Romer (99), in which the R&D sector uses nal ood as input for the production of blueprints. Here, R&D is responsible for "horizontal" innovation (in product variety), in contrast to Schumpeterian "vertical" innovation (in product quality). Added to the model is an exoenous imitation rate (as in Barro and Sala-i-Martin, 995, ch. 6, or Gancia and Zilibotti, 005, sec..3), associated with the prevailin level of IPR protection. The rst e ect above, related to lon-term rowth, derives essentially from the usual Euler equation, as will be seen in Section. By ivin the model a complete analytical solution (unlike any other analyses of this model of which we are aware in the literature), it becomes possible to precisely aue the second e ect, the one which emeres from the instantaneous chane in the consumption level (Section 3). In this way all the welfare ains/losses related to a chane in IPR-protection policy can be taken into account. Kwan and Lai (003) uses this same model in order to analyze optimal patent protection. However, that analysis contains a mistake in the writin out of the dynamics of the problem which ends up contaminatin the results. This paper shows, in contrast with Kwan and Lai (003), that for the whole set of parameter vectors considered there, the optimal policy is always that of providin full This e ect has been empirically evaluated in Gould and Gruben (996), Park and Ginarte (997), Schneider (005) and Falvey et al. (006).

3 protection of IPR. This is to say that, when restricted to the parameter values used by Kwan and Lai, the IPR-protection tradeo has in fact a corner solution, in which case overnments should always pursue an imitation rate equal to zero. Examples based on other parameter values are provided, thouh, in order to show that this tradeo may have interior solutions as well. In these cases, the optimal policy may be that of a partial tihtenin or a loosenin of IPR protection (or inaction). The structure of the paper is as follows. Section presents the model and covers the rowth e ect of the IPR-protection tradeo. Section 3 ives the closed-form solution for its balanced rowth path and evaluates the current-consumption e ect. Section 4 builds on the foundations established in the two previous sections to consider the optimal IPR-protection problem. Section 5 concludes. The model The representative household seeks to maximize U = Z + 0 e t u (c t ) dt, where u is the CRRA function 8 >< c, if 6= u (c) = >: lo c, if =, subject to the resource constraint _b = w + rb c, where consumption c is in terms of the nal ood, r is the rate of return on assets held b (which equal b 0 at time 0 and must be nonneative for su ciently advanced time), and w is 3

4 the wae rate paid by nal-ood rms. Implicit here are time subscripts, the fact that individuals inelastically supply one unit of labor to these rms, and the normalization of the price in the nal oods market to. This proram implies the standard Euler equation := ^c = r, () as well as the transversality condition r > (throuhout the paper, ^ will stand for rowth rate). Firm i I produces nal oods followin the Ethier (98) production function Y i = L i Z A 0 x i;j dj, () where L i is labor input, x i;j is the quantity of index-j intermediate ood bein used as input, to which an elasticity of (0; ) corresponds, and A is the measure of existin intermediate oods. In order to maximize pro t Y i wl i R A 0 p jx i;j dj, the demand of the rm (who is a taker of prices p j in the intermediate oods market) satis es x i;j = L i p j. (3) Let x j := P ii x i;j = L (=p j ), where L := PiI L i. It is convenient to write () in an areate form, notin that x i;j =L i = x j =L: Y := X ii Y i = X ii L i Z A 0 L i x j L X Z A dj = L i ii 0 xj L Z A dj = L x j dj (4) 0 The lab equipment model prescribes that the representative of the R&D sector who invented intermediate ood j will produce it usin a unit of the nal ood. Thus, it chooses p j seekin to maximize (p j ) x j. The resultin price is p j =. If IPR protection reardin this ood is lost somehow, the inventor is no loner a price maker and will have 4

5 to accept for p j his marinal cost,. Followin Kruman (979), an exoenous imitation rate of m > 0, underlyin the patent-breakin process, is assumed: _ A c = m (A A c ), (5) where both A (0) and A c (0) are iven. Here, [0; A c ] is the set indexin intermediates whose patents have been broken (or secrets have been discovered), while (A c ; A] are those still bein protected. The rate m is inversely related to the level of enforcement of IPR protection laws. So the demand for intermediates can be written as 8 >< x j = >: x c = L, if j [0; Ac ] x m = L, if j (Ac ; A]. (6) Finally, entry into the R&D sector is conditioned to the possibility of bein rewarded with monopoly rents to the extent that IPR is bein protected. The relevant rate of return to the potential innovator is the interest rate adjusted, throuh a no-arbitrae arument, for imitation risk: r + m. He must check if the present value of the returns from discoverin intermediate ood j pays at least the cost of invention, which is assumed to be a constant, > 0. Assumin 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 iven time s, = Z + s e (r+m)(t s) (p j ) x m dt = r + m L, whence at all times it is necessary to have r + m = L. (7) For this arument in detail, the reader is referred to Gancia and Zilibotti (005, ft. ). 5

6 From (7) and (), we see that in order to uarantee a nonneative rowth rate, the upper bound (L=) ( ) should be imposed on m. By pluin (7) into the Euler equation () one obtains d=dm = = < 0. This is to say that a tihtenin of IPR protection (by means of lowerin m) brins about a faster rowth of consumption on the whole equilibrium path. Let C = Lc represent areate consumption and X = R A 0 x jdj total intermediate-ood production, so that the resource constraint for the economy is Y X = C + _ A. (8) That is, total product, net of resources used in the production of intermediates, is used for either consumption or innovation. Pluin (6) into (4) ives Y = L (A c x c + (A A c ) x m), whence (8) can be rewritten A _ = L (A c x c + (A A c ) x m) A c x c (A A c ) x m C = L A c + (A A c ) C. (9) Followin Kwan and Lai (003), consider two new variables, a scaled version of consumption, h := C= (A), and the fraction of intermediate oods that have already been imitated, := A c =A. Then h _ = ^C ^A h and _ = m ( ) ^A (usin (5)). In order to nd ^A, one rst plus (6) into (4), obtainin Y = L Ac + (A Ac ). (0) Dividin both sides of (9) by A yields ^A = L + ( ) h. 6

7 We thus et the model s equilibria full dynamics: 8 >< >: _ = m + ( m) + + h, () _h = ( + ) h + h + h where := + (r + m) and := ( + ) (r + m). 3 Not only but also must be neative, since 0 would imply +, or ( + ) = + ( ) ( + ), contradictin the arithmetic mean-eometric mean inequality. The economically relevant steady state is ; h m = m + ; m m +, a saddle point (the deenerate case m + = 0, i.e. m = = 0, would lead to an in nity of steady states, and is dismissed). In order to uarantee that h > 0, the minimal condition to be imposed on the parameters (see Appendix A) is + (L=) ( ) ( + ( )) > 0. 4 This implies + will always be neative, irrespective of the m chosen. Simply takin the coe cient of relative risk aversion 0:5 will do the job, for instance. Now, since ^h! 0 on the equilibrium path toward the steady state, we have ^A!. Dividin both sides of (4) by A shows that Y=A! L + ( ), a constant, so that d Y=A! 0, and also ^Y!. And since X = A c x c + (A A c ) x m, one obtains X=A! x c + x m ( ), a constant, so that also ^X!. Therefore, the lon-run rowth rate of the ross domestic product Y X is also (and GDP per capita as well, since L is constant). Since we have already seen that d=dm = = < 0, we et a simple 3 It may be noted that this dynamical system is not equivalent to expression 8 in Kwan and Lai (003), where the left-hand sides should read ^ and ^h instead. This mistake was carried on throuhout that paper. 4 By minimal it should be understood a condition not dependin on a speci c m. 7

8 con rmation of one of the e ects mentioned in the introduction: Proposition Stroner (weaker) IPR protection brins faster (slower) lon-run rowth of the economy. This e ect is more important the less risk-averse (or the more intertemporally-elastic) society is. Also, as shown in Appendix A, one has d=dm > 0 and dh=dm > 0. Thus two other lon-term e ects of strenthenin the enforcement of IPR protection laws are a lower fraction of intermediates which have been imitated and a lower consumption/assets ratio h. Welfare e ects will be addressed in full in Section 4. 3 Analytical solution At this point, one possibility is to nd a linear approximation or a numerical solution to the path (; h) approachin ; h, as attempted in Kwan and Lai (003). However, it may be noted that equation () can be written in the form of a matricial di erential equation of the Riccati type, as d dt 6 4 h = 6 4 m m h h [ ] h h 3 7 5, its exact solution can also be souht for. 5 By imposin the terminal condition lim t!+ ( (t) ; h (t)) = ; h, one obtains the followin expression for the stable saddle 5 Lemma in Jódar and Navarro (989) can be used to enerate all of its solution paths. The fact that a b 5A = 4 eat (b= (a c)) (e at e ct ) 5 should come in handy to the reader interested in 0 c 0 e ct followin this math. 8

9 path: 6 4 (t) h (t) = e (+ + )t 6 4 h 3 7 5, () where 0 is the iven (0) (since both A (0) and A c (0) are iven), and = m + q (m ) 4 m (3) is the neative root of x + ( m) x + m = 0. Checkins are done in Appendix B. Note that the stable solution () is simply a straiht line in the phase diaram. 6 Since h (t) =h = ( (t) ) = ( ), the method employed to solve () also rewards us with an exact, explicit and elementary solution for h 0 (= h (0)): h 0 = 0 h. (4) Equation (4) makes it possible to compute the exact manitude of the shift in h (and therefore in consumption, since the A variable cannot adjust immediately) once a strenthenin or loosenin of IPR protection takes place. This allows for the assessment of the welfare e ect reardin the instantaneous response of consumption in face of a once-and-for-all chane in IPR-protection policy. Assume that overnment mandates for such a chane. For example, a tihtenin of patent protection, so that m is lowered to m 0. Then, althouh (7) implies that and remain unaltered, it also forces the movement in r to instantaneously counterbalance the movement in m, with no transitional dynamics whatsoever, so that r 0 = r (m 0 m), and 6 In case this fact is somehow seen in advance (and assumin one is interested solely in the stable solution), an alternative course of action would be to rst use the method of undetermined coe cients (in this case, D and D in h = D + D ) to transform the system of di erential equations () into a sinle di erential equation. 9

10 the rate of rowth of consumption in () immediately becomes 0 = (m 0 m) =. If the economy were initially at the steady state ; h, we then have 0 0 = = m m + (5) and h initially jumpin from h to where the new steady state h 0 0 = h0, (6) 0 ; h 0 and the auxiliary expression 0 have the same expressions as before, but with m and replaced by m 0 and 0. Since cuttin m reduces both and h, as noticed in the previous section, it is a priori ambiuous from (4) if this jump in h, from h to h 0 0, is up- or downward. In other words, it is unclear what is to happen with current consumption, althouh the expected e ect may be that of a decrease. That this expectation is always correct is formally proved in Proposition Stroner (weaker) IPR protection has an immediate neative (positive) e ect on consumption. Aain, this e ect is more important the less risk-averse (or the more intertemporally-elastic) society is. Proof. In Appendix C it is shown that dh 0 dm = 0 (m + ) 0 d + dm (7) and d=dm = ( ) = < 0, where is the other zero of x + ( m) x + m. Since this parabola is concave and at x = 0 takes on a positive (or zero if m = 0) value, while at x = it takes on the neative value +, we have [0; ). 0

11 From (7), it is immediate that, if, then dh 0 =dm > 0, as wished. And since d dh 0 d dm = 0 + r 0 d dm < 0, where () was used in the calculation of d=d, in order to conclude that dh 0 =dm > 0 irrespective of we just have to show that lim!+ dh 0 =dm 0. Now, dh 0 lim!+ dm = 0 + m 0 d dm, which is just an a ne function of 0, whence it su ces to show that lim!+;0 =0 dh 0 =dm 0 and lim!+;0 = dh 0 =dm 0. And indeed, dh 0 lim!+; 0 =0 dm = + = > 0 and, usin Girard s relation = m= ( ), dh 0 lim!+; 0 = dm = + m m = = 0. Thus dh 0 =dm > 0; 8 > 0, and current consumption drops with a decrease in m. We have also shown that this e ect is larer the lower is that is, d (dh 0 =dm) =d < 0. In words, the larer the intertemporal elasticity of substitution (=), the more people are willin to sacri ce their current consumption today (iven the establishment of a less competitive environment) in the name of more consumption from some point on in the future (since 0 > ). Propositions and toether con rm that the tradeo 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.

12 4 The optimal level of IPR protection Still in the situation described in the previous section, of an instantaneous exoenous chane in IPR protection from m to m 0, iven the tradeo represented by Propositions and, the overnment should want to choose the optimal m 0. It may then substitute A (0) h 0 0e 0t =L for c in the representative aent s utility function, yieldin the total intertemporal utility level associated to m 0, accumulated throuhout the new equilibrium path: 8 >< U (m 0 ) = >: A(0) L lo A(0) L Thus it solves the problem of ndin m = m 0 0 h0 ( ) 0 + lo h , if 6=, if = i h0; (L=) ( ). (8) that maximizes 7 8 H (m 0 ; 0; 0 ) := u (h0 0) + >< 0 = ( ) 0 >: h 0 ( ) 0 0, if 6= lo h , if =. (9) This is equivalent to the maximization problem stated in expressions 4 and 5 in Kwan and Lai (003). Note from (5) that h 0 0 depends upon 0 0 and from (6) that 0 0 depends upon m. This makes the optimal choice of patent protection m 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 oods, which in uences the rate of new imitations from equation (5). The math involved here is a bit more intricate than that of the previous section. We shall analyze the = case mathematically, then treat the 6= case with the aid of a computer. An observation that will be important in the proof of the next proposition is that 7 The focus on these variables as aruments of H will be made clear in the proof of Proposition 3.

13 = <. In fact, otherwise, one would have = ( + ) = + + ) ( + ) ) ( + ) ( ( + )) ) ( + ) + ) ( + ) +. Now, Bernoulli s inequality applied to both left- and riht-hand sides of the previous inequality ives ( + ) > + (=) = and ( + ) < ( + ) =, a contradiction. Proposition 3 If =, then, 8m 0, m = 0. That is, optimal IPR protection is simply absolute IPR protection, reardless of its initial level. Proof. As shown in Appendix D, H is strictly concave in its rst variable, so that m is either 0 or a possibly positive zero of D H. In order to show that the latter will never be the case, we prove that D H (0; 0; 0 ) < 0. In fact, from (A.3) and (A.), one has D H (0; 0; 0 ) = = = = h ( 0 + ) ( 0 + ) , C A + + where we have used the trivial facts that, for m 0 = 0, 0 = = and h 0 0 = ( 0 + ) ( + ( = ) 0 0), where 0 + < 0 from the assumption on the parameters 3

14 made in Section and explained in Appendix A. Since D D H (0; 0; 0 ) = + 0 ( ) 0 + ( ) = ( ) = + ( ) < 0, we may focus on the 0 0 = 0 (i.e., m = 0) case. Thus we compute (with the aid of (), which under the present hypothesis ives + 0 = r 0 ) sn (D H (0; 0; )) = sn = sn + = sn r = sn r 0 +, and since = <, r 0 + = < r 0 + < r 0 < 0, whence D H (0; 0 0; ) < D H (0; 0; ) < 0, and m = 0 necessarily. Proposition 3 shows that it would be misleadin to try to solve the rst-order condition D H (m ; 0; 0 ) = 0 if =. This would also be true if were close enouh to. In fact, since H is C, H would still be concave in its rst variable (lim! D D H (m; 0 ; ) = D D H (m; 0 ; ) < 0), and since lim! D H (0; 0; 0 ) = D H (0; 0; 0 ) < 0, the proof of Proposition 3 would still apply. Numerical implementation of the optimization proram described in (9) with a set of parameter values containin that used by Kwan and Lai (003, sec. 3) (r = 0:065, = 0:06 as in Kin et al. (988), [:5; :5], [; 4] as in Stokey (995), and 4

15 T [; 00]) still yields the corner solution m = 0 obtained in Proposition 3. 8 The model considered here, therefore, suests a policy of forever patent protection with more ease than it had been suspected at rst. Interior solutions do occur, however, for di erent parameter choices. In such cases, the optimal policy may be one of partially tihtenin (0 < m < m) or even loosenin (m > m) IPR protection (besides the unlikely but possible m = m case). Under these circumstances, the rst-order condition D H (m ; 0; 0 ) = 0 (equivalent to equation 6 in Kwan and Lai, 003), coupled with the present calculation of dh 0 =dm in expression (7), is useful. Fiures, and 3, which use = 0:06 and = :6, ive an insiht to the shape of the optimal policy reions involved in this problem. A blank reion means that the iven r, and T yield inconsistent parameters, such as a nonpositive (= r ), or that the necessary transversality condition does not hold (r ). A liht ray reion symbolizes the recommendation of overnmental weakenin of the patent-protection policy (m > m), a dark ray reion, that of partial strenthenin (0 < m < m), while a black reion, that of total strenthenin (m = 0). E.., the upper left raph in Fiure can simply be seen as a raphical representation of Proposition 3. [INSERT FIGURE ABOUT HERE] [INSERT FIGURE ABOUT HERE] [INSERT FIGURE 3 ABOUT HERE] These ures illustrate three facts in areement with the usual intuition reardin the IPR-policy tradeo. First, a larer interest rate brins a reater possibility that the 8 Here we are usin the identi cation m = r= e rt, equivalent to equation 30 in Kwan and Lai (003). 5

16 optimal solution to the patent-policy tradeo is to favor more the present, by weakenin IPR protection. This can be easily visualized in any one of the raphs in Fiures and 3. Indeed, by departin from any point in the valid (non-blank) reion and movin upward, reions never become darker only lihter (whenever there is a chane in color tone). The same happens with, meanin that a hiher (lower elasticity of substitution =) favors the present aainst the future. This can be seen in any one of the raphs in Fiures and 3. By startin at any valid point in the ures and movin up (raphs of Fiure ) or to the riht (raphs of Fiure 3), reions can only become lihter. Finally, it is more likely for a weakenin in patent protection to be optimizin if the initial level of protection is a hih one already, as can be seen by movin riht in Fiures and. It may also be noted that the borders between weakenin (liht-ray) reions and strenthenin (dark-ray) reions in these ures correspond to parameter values for which the optimal level of enforcement or protection of patent policy happens to exactly match its current level (m = m). For instance, consider the parameters r = 0:5, = 0:06, = :6 and = 6 (more in line with those of a developin economy). The lower left raph in Fiure shows that in this case, the correspondin value of T would be approximately 30:4 years (m 0:00586). Finally, usae of this numerical implementation enables one to recalculate tables, and 3 presented in Kwan and Lai (except for line 4 in those tables), usin the same parameter values as there. 9 [INSERT TABLE ABOUT HERE] [INSERT TABLE ABOUT HERE] [INSERT TABLE 3 ABOUT HERE] 9 We do not recalculate line 4 because we do not have the value of A(0) used there. 6

17 5 Conclusion The lab-equipment/horizontal-innovation model of endoenous rowth of Rivera-Batiz and Romer (99), coupled with an exoenous rate of imitation a la Kruman (979), enerates a dynamical system of the Riccati type that can be solved analytically. This solution allows for an illustration of the IPR-protection tradeo between lon-run rowth on one side, and short-run consumption on the other. This is shown in Propositions and. Our results show that, when =, the optimal IPR protection is always a full one (Proposition 3). The same applies, followin our numerical analysis, for the whole set of parameter values used in Kwan and Lai (003), a fact which establishes a clear contrast between our results and theirs. In all such cases, the underlyin problem eneratin the tradeo to which we have just referred has a corner solution. For other parameter values, as we show, the tradeo persists. Finally, a series of twelve raphs allows the reader to visualize many di erent reions in which protection should be strenthened, as well as weakened. The reader should be cautioned to the fact that all calculations developed here, as well as in Kwan and Lai, are based on an ad hoc assumption about how imitation develops over time (equation 5). One possible extension of this work would be tryin to ure out how robust the results are to chanes in this speci cation (for instance, as suested in Helpman, 993, ft. 5). 7

18 Acknowledement The authors are thankful to Victor Duarte, Reinan Ribeiro and André Okamoto Untem for their assistance. The usual disclaimer applies. 8

19 Appendix A We know that d dm = d r+m m dm = d m dm =, where the second equality is an application of (7). Then d dm = m + m (m + ) = m + (m + ) > 0 and, since h = and both and don t vary with m (because of (7)), we et dh dm = d dm d dm = m + (m + ) > 0. Therefore h is minimum with m = 0, in which case h = = r + + r = + + r = + L ( ) ( + ( ) ), where the last equality used (7). Appendix B We rst check for the validity of the limits lim t!+ (t) = and lim t!+ h (t) = h. Given (), we see that the limits follow immediately from inequality + + > 0, which holds due to Girard s relation: + + = = + + (m ) = m +, 9

20 where is the other zero of x + ( m) x + m, as mentioned in the proof of Proposition. As remarked there, 0, so that m + m + > 0, as wished. We now check that () indeed solves (). First an auxiliary calculation, based on the fact that h = h= ( ) ( ): h = = ( + + ) h h h h, + h = h + h since h + = h = + +. h Therefore _h (t) = 0 0 h e (+ + )t = ( + + ) h (t) = ( + + ) h (t) = ( + + ) h (t) ( + + ) 0 0 e (+ + )t 0 0 e 0 (+ + )t e (+ + )t e (+ + )t! h (t) = ( + ) h (t) + (t) h (t) + h (t). h Before movin on to the motion equation for, a second auxiliary calculation, which 0

21 departs from the rst one in this appendix: h = ( + + ) = m ( m) = m + = m + m ( ) = = (m + ) (m + ) m, where the third equality follows from the quadratic de nin, and the fourth from the expression for. Since = + ( ) =h h, we then et _ = h _h = h h _ = ( ) ( h) = ( h) (m + ) m = m + ( m) + + h. Appendix C We start with a di erent but equivalent expression for h 0. From (4), h 0 = h= ( ) ( 0 = ). Now note that ( ) (m + ) = m + ( m ) + = m + ( ) + ( m) + = h, where the identity + ( m) + m = 0 was used. Therefore 0 h 0 = (m + ),

22 and dh 0 dm = = = 0 0 d 0 dm (m + ) 0 (m + ) 0 d dm (m + ) 0 d dm. d dm The derivative d=dm comes from implicit di erentiation of + ( m) + m = 0: d dm = m + = m + = < 0, (A.) where Girard s relation + = ( m) = was employed. Appendix D Here we deal with the = case. Let H : R + [0; ] f! R be such that H (m; 0 ; ) = lo h 0 (m; 0 ) + (m) = (the second line entry in (9)), where h 0 is iven in (4), and comes from pluin (7) into (). The derivatives dh 0 =dm and d=dm have already been evaluated, in Appendices C and A respectively. In order to obtain the rst two derivatives of H, the derivatives of will also be needed. The rst one has been found in the previous appendix. Similarly, d dm =,

23 so that the second derivative of is d dm = = = d dm ( ) d dm d dm ( ) ( ) 3 = ( ) 3 > 0, (A.) since <, as explained in the proof of Proposition. So D H (m; 0 ; ) = (m + ) 0 d h 0 dm (A.3) and D D H (m; 0 ; ) = h 0 dh 0 (m + ) 0 dm + h 0 (m + ) 0 3 d dm d dm + h 0 (m + ) 0 d dm < 0, from Proposition, (A.) and (A.). The arbitrae relation (7) is also fundamental in showin that d (m + ) =dm = 0 in this case, since m + = r + m, a constant. Therefore H is a (strictly) concave function in its rst variable. 3

24 References [] Barro, R.J., Sala-i-Martin, X., 995. Economic Growth. McGraw-Hill, New York. [] Ethier, W.J., 98. National and international returns to scale in the modern theory of international trade. American Economic Review 7, [3] Falvey, R., Foster, N., Greenaway, D., 006. Intellectual property rihts and economic rowth. Review of Development Economics 0, [4] Gancia, G., Zilibotti, F., 005. Horizontal innovation in the theory of rowth and development, in: Ahion, P., Durlauf, S.N. (Eds.), Handbook of Economic Growth, Vol. A. Elsevier, Amsterdam, pp [5] Gould, D.M., Gruben, W.C., 996. The role of intellectual property rihts in economic rowth. Journal of Development Economics 48, [6] Helpman, E., 993. Innovation, imitation, and intellectual property rihts. Econometrica 6, [7] Jódar, L., Navarro, E.A., 989. On the eneralized Riccati matrix di erential equation. Exact, approximate solutions and error estimate. Applications of Mathematics 34, [8] Kin, R., Plosser, C., Rebelo, S., 988. Production, rowth and business cycles: I. The basic neoclassical model. Journal of Monetary Economics, [9] Kruman, P., 979. A model of innovation, technoloy transfer, and the world distribution of income. Journal of Political Economy 87, [0] Kwan, Y.K., Lai, E.L.-C., 003. Intellectual property rihts protection and endoenous economic rowth. Journal of Economic Dynamics & Control 7,

25 [] Park, W.G., Ginarte, J.C., 997. Intellectual property rihts and economic rowth. Contemporary Economic Policy 5, 5-6. [] Rivera-Batiz, L., Romer, P., 99. Economic interation and endoenous rowth. Quarterly Journal of Economics 06, [3] Schneider, P.H., 005. International trade, economic rowth and intellectual property rihts: A panel data study of developed and developin countries. Journal of Development Economics 78, [4] Stokey, N., 995. R&D and economic rowth. Review of Economic Studies 6,

26 Fiures Fiure. Optimal policy reions, iven r. 6

27 Fiure. Optimal policy reions, iven. 7

28 Fiure 3. Optimal policy reions, iven T. 8

29 Tables Table = :6, r = 0:065, = 0:06, = :5 Current IPR protection level: m (T ) 0:03 (7) 0:070 (0) 0:693 (5). Deviation from opt. IPR prot.: lo (m=m ) Optimal rowth rate: :89% 4:44% 8:37% 3. Consumption level shift: h h =h 4:0% 9:36% 33:7% Table = :5, r = 0:065, = 0:06, = :5 Current IPR protection level: m (T ) 0:03 (7) 0:070 (0) 0:693 (5). Deviation from opt. IPR prot.: lo (m=m ) Optimal rowth rate: :89% 4:44% 8:37% 3. Consumption level shift: h h =h :37% 5:9% 9:4% Table 3 = :5, r = 0:065, = 0:06, = :5 Current IPR protection level: m (T ) 0:03 (7) 0:070 (0) 0:693 (5). Deviation from opt. IPR prot.: lo (m=m ) Optimal rowth rate: :89% 4:44% 8:37% 3. Consumption level shift: h h =h 5:49% 30:97% 35:05% 9