Patent Protection, Technological Change and Wage Inequality

Transcription

1 Patent Protection, Technoogica Change and Wage Inequaity Shiyuan Pan y Taiong Li Heng-fu Zou x Abstract We deveop a directed-technoogica-change mode to address the issue of the optima patent system and investigate how the optima patent system in uences the direction of technoogica change and the inequaity of wage, where patents are categoried as ski- and abor-compementary. The major resuts are: (i) Finite patent breadth maximies the socia wefare eve; (ii) Optima patent breadth increases with the amount of skied (unskied) workers; (iii) Optima patent protection is ski-biased, because an increase in the amount of skied workers increases the dynamic bene- ts of the protection for ski-compementary patents via the economy of scae of ski-compementary technoogy; (iv) Ski-biased patent protection skews inventions towards skis, thus increasing wage inequaity. Keywords: Patent Breadth; Ski-Biased Patent Protection; Ski-Biased Technoogica Change; Wage Inequaity; Economic Growth JEL Cassi cation: O3; O34; J3 We thank Angus C. Chu, Menghong Yang, Zhongai Kou, Ziyin Zhuang and participants of the st Chinese Economics, Finance and Management Forum at Shenhen University for hepfu comments. A errors are ours. y Schoo of Economics and Center for Research of Private Economy, Zhejiang University, 38 Zheda Road, Hanghou 30027, P.R.China. E-mai Address: shiyuanpan@ju.edu.cn. Schoo of Economics & Management, Zhejiang Sci-Tech University, West District, Xiasha Higher Education Zone, Hanghou 3008, P.R. China. E-mai Address: m9845@63.com. x Centra University of Finance and Economics CEMA, 39 South Coege Road, Haidian District, Beijing 0008, P.R.China, Wuhan University IAS, Peking University and China Deveopment Bank. E-mai Address: ouhengfu@gmai.com.

2 Introduction One important poicy issue is how to design optima patent protection to promote innovation. Numerous papers examines optima patent protection in the framework of endogenous growth theory. This iterature, however, does not consider the direction of technoogica change. Indeed, severa modes have been deveoped to investigate directed technoogica change, in which technoogy is compementary to skied or unskied workers (Acemogu, 998, 2002a, 2003, 2007, Thoenig and Verdier, 2003). To the best of our knowedge, the iterature on directed technoogica change does not address the issue how to protect patents. We woud ike to combine the iterature on the optima patent system and the one on directed technoogica change in this paper. For this purpose, an endogenous growth mode is constructed to study the optima patent system in the directedtechnoogica-change economy and how the optima patent system impacts the direction of technoogica change and wage inequaity, in which patents are cassi ed as ski- and abor-compementary. The rst main resuts are: (i) Optima patent breadth is nite, and increases with the quantity of skied and unskied workers; (ii) Optima patent protection is ski-biased. The basic idea is that the more skied workers, the greater the economy of scae of ski-compementary technoogy, and thus the more the dynamic bene ts of protection for ski-compementary patent; (iii) Ski-biased patent protection induces technoogica change towards skis, thus increasing wage inequaity. Consequenty, an important poicy impication is that it is optima to have di erent patent protection for di erent type of innovation. For instance, it wi be bene cia to strengthen protection for abor-compementary patent in deveoping countries, whereas it wi be better o to increase protection for ski-compementary patent in deveoped countries. Consequenty, it may be time to consider whether or not patent rues that are neutra to technoogies in the rea word are the best. Ever since Schumpeter (942), we have known that it is necessary to provide inventors with some form of market power to give them incentives to invent in the rst pace. Indeed, two branches of iterature have been devoted to the topic of optima patent protection, namey, that of the patent system that best soves the trade-o between providing enough incentives to invent ex ante and minimiing dead-weight osses ex post. One branch of iterature has studied optima patent protection in a partia equiibrium mode (for exampe, Nordhaus, 969, Gibert and Shapiro, 990, Kemperer, 990, Gaini, 992). Another branch of iterature has ooked at the Scotchmer (2004) provides a comprehensive review on this iterature. 2

3 optima patent system within a framework of genera equiibrium (e.g., Judd, 985, Goh and Oiver, 2002, Kwan and Lai, 2003, Iwaisako and Futagami, 2003, Horii and Iwaisako, 2007, Futagami and Iwaisako, 2007, Chu, 200, Chu and Pan, 202). This paper foows and compements the second branch of iterature by taking account of the direction of technoogica change. This paper reates to Acemogu and Akcigit (2009), Mose (2009), Chu (20). These modes a concude that one-sie- t-a patent poicy is unikey to provide the appropriate incentives for innovation in every industry. 2 Our mode adds this iterature by anaying optima patent protection across industries in a framework of directed technoogica change. The reated iterature aso incudes Coi and Gai (2009) and Adams (2008). Coi and Gai (2009) emphasie that a strengthening of inteectua property rights wi ead to an increase in wage inequaity. Adams (2008) reports that strengthening inteectua property rights and openness are positivey correated with income inequaity in deveoping countries. The main di erence between our paper and Coi and Gai s is that the direction of technoogica change is considered, whie the one between our paper and Adam s is that the atter focus on income inequaity instead of wage inequaity. The paper proceeds as foows. In the next section, buiding on Acemogu (998, 2007, 2009) and the iterature on an optima patent system, we introduce the mode. In Section 3, we address the issue of optima patent protection and expore the e ect of patent protection on the direction of technoogica change and wage inequaity. Concusions are found in Section 4. 2 The Mode Consider an economy popuated with H skied workers and L unskied workers, who suppy one unit abor ineasticay. Representative consumers are with constant reative risk aversion (CRRA) preference. These consumers maximie intertemprora utiity 3 Z C (t) e t dt; () 0 where C(t) is consumption at time t, is the coe cient of reative risk aversion (or intertempora easticity of substitution) and is the subjective discount rate. We drop the time index as ong as this causes no confusion. 2 Burk and Lemey (2009) provide overwheming evidence that innovation works di erenty in di erent industries, and state that patent protection shoud be di erent across sectors. 3 In this paper, we assume 0 < <, because ony in the this circumstance, an increase protection for patent has the dynamic bene t and static cost as pointed out in Section 2. 3

4 The budget constraint of the consumer is: C + I + R Y = i h(y ) + (Y h ) ; (2) where I is investment, and R is tota R&D expenditure. The production function in (2) impies that output aggregate is de ned over a constant easticity of substitution (CES) aggregate of a abor-intensive good, Y, and a ski-intensive good, Y h. Parameter 2 [0; ) is the easticity of substitution between the two goods. When =, the two goods are perfect substitutes, and the function is inear. When =, the function is Cobb-Dougas. And when = 0, there is no substitution between the two goods, and the production function is Leontie. Foowing Acemogu (998, 2002, 2009), the abor-intensive good is produced from unskied workers and di erent types of abor-compementary machines or intermediates, whie the ski-intensive good is produced from skied workers and a set of di erentiated ski-compementary machines. The key assumption is that none of these machines are used by both types of workers. Speci cay, the production functions of the ski-intensive and the abor-intensive good are as foows 4 : and Y h = Z Ah k h (i) di (ZH) ; (3) 0 Y = Z A k (i) di L ; (4) 0 where 2 (0; ), A is the number of machines compementary to, k (i) is the quantity of machines of variety i together with workers of ski eve, = h or. Indexes h and denote skied and unskied workers, respectivey. Z > measures the reative productivity of skied workers. Consequenty, ZH represents the e ective amount of skied workers. The production functions in (3) and (4) exhibit constant returns to scae in input factors: the doube of abor and the quantity of a intermediate goods doubes output. However, the production possibiities set of the economy wi exhibit increasing returns to scae because technoogica knowedge, A, are endogenied. Technoogica progress takes the form of the increase in A over time. Using units of the na good, a rm can deveop a new variety of either type of machine. Therefore, the accumuation equation of technoogica knowedge is given by 5 4 The main resuts are not atered when we assume that Y h = R A 0 k (i) di L, where 6=. 5 Assumption A = X where h 6= wi not change our main resuts. R Ah 0 k h (i) di (ZH) and Y = 4

5 A = X ; (5) where X denotes tota output devoted to improving the technoogy compementary to = h or. Equation (5) impies that with a tota expenditure of X, there wi be X= new varieties invented. For convenience, we assume the margina cost for the production of any machine is constant and equa to one unit of the na good. A rm that invents a machine obtains the protection granted by patent. As do Goh and Oiver (2002), we assume that the ife of a patent is in nite and we focus on the issue of patent breadth. Foowing Gibert and Shapiro (990), Diwan and Rodrik (99), Iwaisako and Futagami (2003), we de ne patent breadth as the abiity of the patentee to raise the prices for the patented goods over the ifetime of patent. Strong protection granted by the patent increases the number of substitute products that infringe on the patent or raises the costs of imitation, thus aowing the patentee to raise prices. In particuar, the prices charged by rms producing the goods that embody inventions are given by (i) = = + ; = h or ; (6) where is the measure of patent breadth. The bigger the, the greater the patent breadth. Using some agebra, we know that the monopoy price maximiing the pro ts of patent products is. Therefore,, and patent breadth is in nite as equaity hods. Taking advantage of (3), (4) and (6), we obtain the quantity of machines compementary to skied and unskied workers: and k h (i) = [p h = ( + h )] = ZH; (7) k (i) = [p = ( + )] = L; (8) where p h and p are the prices of the ski-intensive and the abor-intensive good, respectivey. We normaie the price of consumption aggregate as one. Therefore, the monopoy pro ts of any intermediate good used by skied and unskied workers at time are: h () = h [p h = ( + h )] = ZH; (9) 5

6 and () = [p = ( + )] = L: (0) It is simpe to show that > 0 and = 0. This impies that the monopoy pro ts of a new variety of machines increase with patent breadth, and maximum extent of patent breadth makes the pro ts highest. Pugging (7) and (8) into (3) and (4) respectivey, we obtain Y h = [p h= ( + h )] ( )= A h ZH and Y = [p = ( + )] ( )= A L: () The market for the ski-intensive and the abor-intensive good is competitive, thus using (2) and (), we nd that the reative price of the two goods is p = p h p = + h + +( ) A h ZH A L +( ) : (2) This shows that when either the technoogy is highy ski-biased (high A h =A ) or the reative suppy of skied workers is great (high H=L), the reative suppy of the skiintensive good is arge and the reative price is ow. The reativey strong protection for the ski-compementary patent (high h = ) eads to a decrease in the reative demand for machines compementing skis, thus decining the reative suppy of the ski-intensive good and increasing the reative price. 3 Equiibrium We now characterie the equiibrium deveop a simpe mode to expore why the optima patent protection shoud be ski-biased and how the ski-biased patent protection a ects the degree of ski bias of technoogy and wage inequaity. 3. Optima Patent System Free-entry in the R&D business impies that the cost of invention,, shoud be equa to the present vaue of pro ts of any intermediate good, V. That is, = Z t e R t r(s)ds ()d = V ; (3) where r is the renta price of capita. It suggests that in the equiibrium the ow 6

7 pro ts from seing abor- and ski-compementary machines shoud be equa, i.e., h =. Therefore, inspection of (9) and (0) resuts in p h p = + h h ZH + L : (4) Intuitivey, the bigger the amount of skied workers, the arger the market for skicompementary machines, thus the ower the reative price of the ski-intensive good to ensure h =. Moreover, it is cear ) 0. This impies that the stronger the protection for the ski-compementary patent, the arger the pro ts of new invention compementing skis, thus the reative price of the ski-intensive good has to be ower to make the ow pro ts of abor- and ski- compementary machines equa. Combining (2) and (4), we obtain A h A = + h h + +( ) ( ) ZH : (5) L By (2), () and (2), we know that the easticity of substitution between skied and unskied workers is + ( ). Thus, (5) shows that the reative degree of ski bias of technoogy, A h =A, is determined by the reative factor suppy and the easticity of substitution between the two factors, skied and unskied workers. When >, i.e., when the two factors are gross substitutes, technoogica change is towards skied workers; whie <, i.e., when the two factors are gross compements, technoogica change is unskied-biased (ski-repacing). 6 Amost a estimates show an easticity of substitution between skied and unskied workers greater than, most ikey greater than.4, and perhaps as arge as 2, that is, + ( ) 2 (see, for exampe, Freeman, 986, Acemogu, 2002b). Hence, we take to be greater than in the rest of the paper. In addition, using some agebra, we nd ) +( ) ( > 0 as. Therefore, the degree of ski bias of technoogy, A h =A, rises with the breadth of the ski-compementary patent. Strong protection for the ski-compementary patent resuts in an increase in the pro ts seing ski-compementary machines, thus inducing ski-biased technoogica change. Inspection of the production function in (2) yieds p + p h = (6) 6 When the easticity of substitution is equa to, technoogica change is never biased towards skied or unskied workers. 7

8 Using (4) and (6), we nd the price indices to be " + h p h = + + # ( ) h ZH L " + h and p = + + # ( ) h ZH : L (7) Therefore, the price of the ski-intensive good is ower (and the price of the aborintensive good is higher) when the reative suppy of skied workers is arger or the reative protection for the ski-compementary patent is stronger. Maximiation of utiity function in (), subject to a standard budget constraint, yieds the usua formua for the growth rate of consumption: C C = (r ) ; (8) It says that either the intertempora easticity of substitution is bigger or the discount rate is smaer, the growth rate of consumption is higher. Combining (9), (0), (3), (7) and (8), we obtain the equiibrium economic growth rate: 7 g = >: 8 >< h( + h ) ( h ZH) ( ) + ( + ) ( )i ( ) ( L) 9 >= >; : 0, that is, strong patent protection increases the growth rate. This denotes the dynamic bene ts of the patent system. Intuitivey, great patent breadth raises the pro ts of invention, thus inducing resources devoted to inventing technoogy and a high growth rate. When there is a bigger amount of skied (unskied) workers, the market for machines compementing skied (unskied) workers is arger, hence the pro ts of invention are higher. Therefore, the growth rate increases with the quantity of skied and unskied @H > 0 > 0. Now et us brie y investigate the stabiity of the equiibrium. For given patent breadth, according to Acemogu and Ziibotti (200), we know that o the baanced growth path, there wi ony be one type of innovation. That is, if V h =V >, ony skicompementary innovation is taken pace, and if V h =V <, innovators ony undertake abor-compementary R&D. Ceary, when A h =A is ower than in (5), V h =V >, and 7 The growth rate expression suggests that the parameters must be assumed to be such that g 0. Otherwise, the constraint that A cannot be decreasing woud be vioated, and the free-entry condition for R&D woud not hod with equaity. Therefore, the measure of patent breadth, h and, wi not be ero. 8

9 vice versa when A h =A is too high. As a consequence, the transitiona dynamics of the economy are stabe. 8 h Since the utiity function is continuous on 2 0; i, there exists maximiing socia wefare. 9 Output growth maximiation, however, may not be equivaent to wefare maximiation. Therefore, in nite patent breadth, which maximies the output growth rate, is not optima. Intuitivey, when patent breadth is in nite, the bene of increasing patent protection is ero = 0), but the cost is bigger = than ero because of monopoy pricing. Hence, we state the foowing proposition Proposition The patent breadth that maximies socia wefare is nite, i.e., <. Proof. See the Appendix. Propositions shows that nite patent breadth maximies the socia wefare eve. This resut is simiar to the resuts of many existing studies. For instance, Gibert and Shapiro (990) have argued that narrow patent is optima because broad patent is costy for society in that it gives excessive monopoy power to the patent hoder. In recent decades, the reative quantity of skied workers has increased sharpy in most deveoped countries. To the best of our knowedge, there is no previous paper investigating the impact of an increased suppy of skis on the optima patent system. Proposition 2 The more the quantity of (unskied) skied workers, the greater the breadth of patent, that > 0 > 0. Moreover, if the e ective amount of skied workers is bigger (smaer) than the amount of unskied workers, the optima breadth of the ski-compementary patent is broader (narrower) than that of the aborcompementary patent, and if the e ective amount of skied workers is equa to the amount of unskied workers, the optima breadth of the two types of patents is same. That is, when ZH > L, h > ; whie ZH = L, h = ; when ZH < L, h <. 0 Proof. See the Appendix. 8 See Acemogu and Ziibotti (200) for a forma and detai proof. 9 Maybe is unique. However, the anaysis is substantiay compicated. Fortunatey, the foowing resuts are independent of whether is singeton or not. Indeed, when is not unique, it is reasonabe to choose that maximies the growth rate as the optima patent breadth. Moreover, it is easy to know h ; ) > 0 which satis es g h; = 0. Therefore, optima is interior soution, h ; ) = 0. When we take into account of the other di erences in the production of the ski-intensive and the abor-intensive good, even though ZH < L, h > may hod. For exampe, if the production technoogy of the ski-intensive good is assumed to be Y h = R Ah 0 k h (i) di (ZH) where >, then = ZH > L, h >. 9

10 The intuition is straightforward. The more the suppy of skied workers, the greater the economy of scae of knowedge compementing skied workers, thus the broader the breadth of the ski-compementary patent. That h > 0. Moreover, the reative price of the abor-intensive good rises with the amount of skied workers, thus the economy of scae of knowedge compementing unskied workers becomes arge. As a consequence, the protection for abor-compementary patent strengthens, > 0. By h > 0. As an important impication of this proposition, the patent poicy shoud be skibiased, that h = > 0. Precisey, it is easy to see that, in any ine segment of ZH = h > 0 h > 0, respectivey. This impies, at east in a neighborhood of the segment, there is a ski-biased patent protection, that is, for ZH > L, ZH 0 > L 0 and ZH0 L > ZH, h) (0 0 L, and for ZH < L, ZH 0 < L 0 ( 0 ) > h and ZH0 L < ZH, h) (0 0 L ( 0 ) < h. Hence, we woud ike to caim that a reative increasing of ski workers wi ead to a reativey stronger protection towards ski-compementary patent. That is, Coroary Optima patent protection is ski-biased, h = > 0, when ZH L. Over the decades, patent systems have been harmonied wordwide. During the 980s and eary 990s, patent protection in some deveoping countries proved inadequate. For instance, biotechnoogy, software and business methods were widey considered unpatentabe, and most importanty, procedures and resources for the enforcement of patent aws were often inadequate at protecting patent hoders under even the existing weak standards of patent protection (Maskus, 2000). A major turning point in the goba protection of inteectua property (patent protection) was the enforcement of an agreement on TRIPs in 995. A WTO member countries have signed on to TRIPs and are obigated to impement the most rudimentary rues for IPRs protection based on the U.S. and EU practices. Moreover, when additiona countries join the WTO, they must meet the minimum standard required by TRIPs. Therefore, the impementation of the agreement on TRIPs has signi canty increased harmoniation of IPRs protection wordwide. It is notabe that pressure from the United States and the European Union payed a critica roe in the internationa harmoniation of IPRs protection (e.g., Grossman and Lai, 2004). Widey pubicied American negotiations and threats in the 980s and 990s resuted in stronger IPRs egisation in South Korea, Argentina, Brai, Thaiand, Taiwan, and China. Since TRIPs ony sets out minimum standards with which countries must compy, it does not aim at compete harmoniation. 0

11 Simiary, European Union negotiations and assistance advanced IPRs protection in Egypt and Turkey (Maskus, 2000). Since there are ess skied workers in deveoping countries, one impication of Proposition 2 and Coroary is that the harmoniation of patent systems wordwide may decrease the socia wefare eve of many deveoping countries Wage Inequaity As vast quantities of empirica papers have documented over past decades, there has been an increase in wage inequaity in most deveoped and deveoping countries. 3 The standard expanation for this pattern is that technoogica change has been skibiased over this period. We argue here that ski-biased patent protection coud be another cause. Now et us rst address the probem of wage inequaity in deveoped countries. Inspection of () yieds the ratio of wages paid for skied workers to the ones paid for unskied workers w h w = ph p = + h + ( )= A h Z A : (20) It suggests that ski premia are greater when either the reative price of the skiintensive good is higher or the technoogy is more ski-biased. Equation (20) aso says that when the impact of ski-biased patent protection on the direction of technoogica change is not considered, great protection for ski-compementary patent reduces the demand for machines used by skied workers, due to the high price, and thus the margina product of skied workers (or wages) decines. Combining (2), (5) and (20), we get the ratio of wages paid for skied workers to the ones paid for unskied workers: ( ) ( ) ( ) w h + h h ZH = Z : (2) w + { L } { } e ect of ski-biased e ect of ski-biased patent protection technoogica change 2 Deardor (992), Grossman and Lai (2004) stress that whie the wefare of deveoped countries certainy rises with harmoniation of patent systems wordwide, that of deveoping countries may fa, and may we fa by more than the increase in the wefare of the deveoped countries. Maskus (2000) point out that in the short-term, goba patent protection decreases the wefare of deveoping countries, but it woud bene t a countries in the ong-run. 3 See Acemogu (2002b) and Godbegr and Pavcnik (2007) for detais.

12 When and h increase, + h + ( ) h ( ) becomes arger. Thus, we state Proposition 3 When ( ) > 0, ski-biased patent protection and technoogica change ead to an increase in ski premia in deveoped countries. Taking advantage of (5) and Proposition 2, we know that ski-biased patent protection encourages technoogica change towards skis, and therefore raising wage inequaity. A number of studies have documented that the reative suppy of skis have increased over the past decades (see Acemogu, 2002b, and references therein). As consequence, based on Proposition 2, we conject that patent protection is biased towards skis 4, and thereby enarging wage inequaity. Let us assume ZH N > L N in deveoped countries and ZH S < L S in deveoping countries. 5 By proposition 2, we know that in the autarkic economy S h < S. Therefore, (2) impies that ski premia in deveoping countries in autarky are wh S w S ZH S ( ) < Z : (22) L S Now if patent protection is harmonied a over the word and under pressures of the deveoped countries the deveoping countries are forced to enforce the patent protection, N h ; N, then by (2), we obtain 6 0 wh S w S 0 ZH S ( ) > Z (23) Comparing (22) and (23), we obtain the foowing proposition Proposition 4 Harmoniation of goba patent protection increases wage inequaity in deveoping countries. The United States and Europe have been dissatis ed with the situation of weak IPRs protection in many deveoping countries. Hence, they have tried to force 4 Ceary, most inventions of biotechnoogy, software and business methods are compementing to skis, athough some of them are used by unskied workers. Therefore, to some extent, the greater breadth of these patents demonstrates that patent protection is ski-biased. A famous exampe of broader caims in biotechnoogy is when Johns Hopkins University was assigned a patent that they caimed covered not ony the My-0 antibody but aso other antibodies that bind to CD34, athough that patent ony showed e ects with respect to the My-0 antibody (Bar-Shaom and Cook-Deegan, 2002). Owing to function caims in the ed of software and business methods, that is, patents caiming probems rather than soutions, any invention deveoped for that probem woud infringe the patent (Martine and Gueec, 2003). Therefore, the breadth of patent for software and business methods is substantiay broad. 5 As a matter of fact, the main resuts remain unchanged if N h N > S h S. 6 The main resuts sti appy to the case where the patent protection in the deveoping country is N h ; N, where <, <, N h > S h, N > S and N h > N. L S 2

13 deveoping countries to increase the protection for patents, in particuar for skicompementary patents. Intuitivey, stronger protection of ski-compementary patents forced by the U.S. and European countries induces ski-biased technoogica change, thus increasing ski premia in deveoping countries. 4 Concusion In this paper, a simpe mode has been constructed to expore the optima patent system in the directed-technoogica-change economy and the impact of the optima patent system on the direction of technoogica change and wage inequaity, in which patent is categoried as ski- and abor-compementary. We show that: (i) Optima patent breadth is nite, and rises with the quantity of skied workers; (ii) An increase in the amount of skied workers increases the protection for the skicompementary patent, i.e., optima patent protection is ski-biased; (iii) Ski-biased patent protection increases wage inequaity by encouraging ski-biased technoogica change. It must be stressed that we reied on the strong assumption of in nite patent ife to obtain unambiguous resuts. The obvious next issue on the research agenda is to check the robustness of our resuts to departures from the assumption. 3

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17 Appendix: Proof of Propositions Proof of Proposition : As in most of iterature, we assume the economy is in its BGP at time 0. Obviousy, (5) suggests that A h (0) = h( + ) ( + h ) h A (0); (A) where h = (+ h ) ( h ZH) ( ) and = (+ ) ( L) ( ). In order to investigate whether the optima patent protection is nite or not, we have to compute the utiity of a representative agent in the equiibrium. From (), the discounted sum of utiity coud be written as, for 0 < <, C(0) U( h ; ) = ( )[ ( )g] ( ) ; (A2) where C(0) = p h Y h (0) + p Y (0) A h (0)k h (0) A (0)k (0) ga h (0) ga (0) (A3) denotes the agent s consumption. Since the amount of initia tota capita, A h (0)k h (0)+ A (0)k (0), shoud be taken as given, thus From (9), the in nite patent breadth (A h (0)k h (0) + A (0)k = = maximies the growth rate, expicity, = 0: (A5) In fact, and @ = = = 0; (A6) = 0: (A7) 7

18 By () and (A), direct computation on (A3) gives C(0) = h( + h ) ( + ) p h ( ) X ZHA (0) + ( + ) p h LA (0) h ( + [A h (0)k h (0) + A (0)k (0)] g ) + A (0); ( + h ) h where we take A (0) be [ h (+ h ) j = h = 0 = As the partia derivative of U( h ; ) = Then, we deduce from (A4) (A7) and the fact ( + = g ) ( + h ) 2 A (0) h C(0) + ( = C(0) ( < 0: ; (A8) we concude from above < h 0. Simiary, < L 0. Therefore, = < hods, that is, the patent breadth maximiing socia wefare is nite. Proof of Proposition 2: The economy is assumed to be in its BGP. By (A3), when A (0) is xed, we consider the utiity U( ) as a function with parametric variabes H and L, and rewrite it as U( ; H; L). Let () be the nite optima patent breadth in the economy with H skied workers and L unskied workers. For the simpicity, denote () by. For xed A (0) and, C(0) increases as a function of. Otherwise, we can nd a su cient arge such that C(0; ) < 0, which is impossibe. Since the more the workers are, the higher the growth rate g is, we obtain, for >, C(0; ) ( )g() > C(0) ( )g : In C(0; : Therefore, we get from (A8) ( ; H; L) > 0: 8

19 It says that () >. Hence, we can caim here ZH > L > 0) L > ZH > 0) hod. If the caim is fase, we wi have H L < 0) L ZH < 0), and then create a contradiction to () >. From (A8), at, we C(0) g h and the foowing ong identity C(0) ( )g C(0) (A9) ( + )( + ( ) ( )( ) h ) ( + h )( + ( ) ( )( ) ) = ( + h) ( ) ( + ) h ( ( ) ( ( ) h)(zh) ( ) )L (g + ) +( ) ( h) : (A0) ( )g h ( + ) ( ) After a simpe agebra, we know that when H is enarged and is given, the right hand of the identity becomes greater than the eft hand of the identity. Consequenty, > 0. In addition, when ZH = L, H (g + ) +( ) ( h) = 2 (A) ( )g h ( + ) ( ) Equation (A) > 0 suggests that (g+)+( ) g is an increasing function of g. Therefore, when L increases and is given, the right hand of (A0) becomes smaer than the eft hand. It is foowed > 0. Using symmetry, H > > 0. In order to prove the rest of proposition 2, we now rst prove when ZH = @ZH > 0. Due to symmetry, when ZH = L, we have h =. Suppose that ZH = ( + ) ZH and L = ( &) ZH where! 0 +, &! 0 + and ( + ) ( ) +( &) ( ) = 2, and that patent protection remains unchanged. Then, the growth rate remains the same. Hence, when A (0) is unchanged, p Y (0) A (0) k (0) ga (0) = + + p Y (0) ga (0) remains invariant. Some cacuation yieds that p hy h (0) A h (0)k h (0) ga h (0) p Y (0) A (0)k (0) ga (0) = A h (0). Therefore, (A3) says that when A (0) ZH = ( + ) ZH and L = ( &) ZH, if =, then C (0) goes up. This means that in order to increase utiity, patent protection shoud be adjusted to increase the growth rate. Using (A), we know that when ( ) =,. Thus, some agebra impies that for ( 9

20 0, h (( + ) ZH; ( &) ZH) > h (ZH; ZH) and (( + ) ZH; ( &) ZH) < (ZH; ZH) hod, or h (( + ) ZH; ( &) ZH) < h (ZH; ZH) and (( + ) ZH; ( &) ZH) > (ZH; ZH) are satis ed. Suppose the atter two inequaities hod. Then, h (( + ) ZH; ( &) ZH)+ (( + ) ZH; ( &) ZH) > h (ZH; ZH) + (ZH; ZH) must be satis ed to increase the growth rate. when ZH = L, this inequaity cannot hod. Thus, inequaities h (( + ) ZH; ( &) ZH) > h (ZH; ZH) and (( + ) ZH; ( &) ZH) (ZH; ZH) shoud be satis ed. It is foowed that when ZH = L, > > 0. Suppose when ZH > L, h <. Then, there exists ZH g > L such that h =. By symmetry, the foowing two equations shoud be satis ed and ( + )( + ( ) ( )( ) h ) ( + h )( + ( ) ( )( ) ) = ( + h) ( ) ( + ) h ( ( ) ( ( ) h)( ZH) g ( ) )L (g + ) +( ) ( h) : (A2) ( )g h ( + ) ( ) = ( + )( + ( ) ( )( ) h ) ( + h )( + ( ) ( )( ) ) ( + h ) ( ) h ( ( ) ( ( ) h)l ( + ) )( ZH) g ( ) (g + ) +( ) ( h) : (A3) ( )g h ( + ) ( ) Ceary, when (A2) and (A3) cannot hod simutaneousy. Therefore, we get a contradiction to the case h = when ZH g > L. As a resut, when ZH > L, h > ; when ZH < L, h <. 20