# Outsourcing Innovation in Product Cycles

Authors: Hsiao-Lei Chu
Journal: ISRN Economics
 (2012)

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## Abstract

In this quality-ladder product-cycles model, a southern firm can undertake innovation by collaborating with a northern firm through R&D outsourcing. Generally, I find that the initial steady-state scale of R&D outsourcing and the fraction of innovative tasks undertaken by southern labor through R&D outsourcing critically affect the results of comparative statics. Particularly, the friendly policy to promote R&D outsourcing may be beneficial for both of the North and the South only if the scale of R&D outsourcing is small.


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## Body

## 1. Introduction

The Vernon-type product-cycles model generally consists of two countries, North and South. The North is distinguished from the South by its superior ability of innovation and higher wage rate. In addition, a production process in the model generally consists of two stages, R&D and manufacturing. Given the key character that the latest innovation through R&D replaces outdated products, various patterns of production process have been discussed in this Vernon-type product-cycles framework. Particularly, outsourcing part of the production process, which is consistent with the observation of global fragmentation in production and trading semifinished products [1, 2], has been investigated. (In addition, FDI (e.g., [3–5]) and licensing (e.g., [6]) have also been discussed in the product-cycles framework.)  Glass and Saggi [7] divide the manufacturing stage into an advanced stage and a basic stage, where the basic stage can be outsourced to the South. Their framework is extended by Glass [8] with imitation, and by Sayek and Sener [9] with heterogenous labor. Sener and Zhao [10] push out the outsourcing frontier further outward by considering that the North is able to outsource domestically developed innovation directly to the South without manufacturing the innovation at home in advance. Although they all recognize the importance of international outsourcing to product cycles, no R&D outsourcing is discussed. (Lai et al. [11] model outsourcing innovation in a principal-agent framework. They mainly focus on optimal contracting. Innovation in the South actually has been discussed in product-cycles models by Chui et al. [12] and Chu [13]; however, neither of the two articles considers outsourcing.)If the South is also capable of undertaking innovation, it might stimulate the North to outsource at least a part of R&D stage to the South because of the lower wage rate in the South. In fact, some reports do point out that international outsourcing of innovation is rising. For example, Technology Forecasters Inc. reports that USA companies have increased outsourcing innovation overseas from less than US $30 billion in 2000 to over US $60 billion in 2004 [14]; BusinessWeek (March 21, 2005) tells that “Today, the likes of Dell, Motorola and Philips are buying complete designs of some digital devices from Asian developers, tweaking them to their own specifications, and slapping on their own brand names….About 30% of digital cameras are produced by ODMs, 65% of MP3 players, and roughly 70% of personal digital assistants.” Based on a survey of over 300 executives worldwide, Economist Intelligence Unit [15] concludes that the proportion of respondents with at least some of their R&D function overseas is 65% in 2007 and the percentage is expected to be higher over time. These observations of outsourcing innovation are consistent with the upward trends of granted patents in the developing countries (which may be classified as southern countries in the product-cycles framework) such as China, India, South Korea, and Taiwan as showed in Appendix A. (A report from Markets, Patents & Alliances, LLC [16], shows that particularly in the computer and telecommunication industries, many Taiwanese ODM firms, like Hon Hai Precision Industry (also known as Foxconn Electronics) and Taiwan Semiconductor Manufacturing Corporation (TSMC), experience a huge growth in patent activities since 1995.These ODMs design and assemble products for companies with brand names. Multinational corporations also offshore R&D. For example, Nortel Networks and GECIS in India, HP in Taiwan have hired local scientists to work on R&D.)Motivated by the rising of international outsourcing of innovation, this paper incorporates southern innovation into a product-cycles framework. Specifically, it is assumed that a southern firm can undertake innovation by collaborating with a northern firm through R&D outsourcing. I intend to investigate how the steady-state relative wage, the aggregate rate of innovation, and the measure of northern firms which outsource R&D (dubbed the scale of R&D outsourcing hereafter) are determined in this model. Particularly, I investigate the impacts of R&D subsidies, of an improvement on southern R&D ability and of the sizes of regional labor force on the steady-state equilibrium. In this model I leave imitation out of the picture, which can be justified by a sufficiently high cost of a violation of the intellectual property right faced by a country once it integrates into the global trade system.Similar to Glass and Saggi [7] and Sener and Zhao [10], this paper also finds that an R&D friendly policy in the South raises southern relative wage; however, its impacts on the aggregate rate of innovation and on the scale of outsourcing (though the paper refers to R&D outsourcing and their articles refer to manufacturing outsourcing) are uncertain in this paper. Glass and Saggi [7] consider manufacturing outsourcing and find that subsidies to adapting technologies for southern production lead to more international outsourcing and the subsidies also raise the aggregate rate of innovation and the southern relative wage. Sener and Zhao [10] emphasize the phenomenon of simultaneous innovation and outsourcing by northern firms. They show that if the South subsidizes on northern firms which directly transfer technologies to the South or if the South subsidizes on local manufacturing, all of the aggregate rate of innovation, the scale of outsourcing, and the southern relative wage should rise.In this paper, the South controls two variables,ss and sn, for R&D friendly policies. ss denotes the rate of R&D subsidy on the southern firms and sn denotes the rate of R&D subsidy on the northern firms that outsource R&D to the South. An increase in ss or an increase in sn raises southern relative wage and the size of this effect depends on the fraction of innovative tasks undertaken by the southern labor employed in R&D outsourcing. An improvement on southern R&D ability, an increase in southern labor force, a higher ss and a higher sn generate the same qualitative effect on the scale of R&D outsourcing and on the aggregate rate of innovation. The qualitative effect can be positive or negative, and it will be positive if the initial steady-state scale of R&D outsourcing is not too large (will define “too large” later). If the development of R&D outsourcing raises the southern relative wage but reduces the aggregate rate of innovation then R&D outsourcing casts shadow on the welfare of the North and might even reduce the welfare of the South, which is in contrast with the conclusion of Glass and Saggi [7]. They argue that international (manufacturing) outsourcing can potentially create gains sufficient to offset the decline in the northern wage because the rate of innovation is raised by outsourcing. In addition, in this paper a larger northern labor force might reduce the aggregate rate of innovation; however, it always speeds up the global rate of innovation in the literature because the northern labor is the sole input for innovation. Generally, I find that the initial steady-state scale of R&D outsourcing and the fraction of innovative tasks undertaken by the southern labor through R&D outsourcing are two variables which critically affect the results of comparative statics. These two variables can emerge because of taking international R&D outsourcing into consideration.In the rest of the paper, Section2 sets up the model, Section 3 derives the steady-state equilibrium and comparative statics, and Section 4 concludes the paper.


## 2. The Model

The world consists of two countries: North and South denoted byN and S, respectively. Agents worldwide consume goods indexed by ω,  ω∈[0,1]. The quality of good ω can be improved by innovation and one improvement raises the quality by λ times. Let qj(ω)=λj, λ>1, qj(ω) is the quality of good ω that experiences j times of improvement on quality. Labor is the only input in production and the wage rate in the North, wn, is strictly greater than the wage rate in the South, ws. For any good ω, firms which possess the technologies to produce various qualities of good ω compete with each other, leading to only one firm that offers the lowest quality-adjusted price in the industry ω. Only northern firms are capable of inventing new quality independently and any southern firm can undertake innovation only if it collaborates with a northern firm through R&D outsourcing.


### 2.1. Types of Firms and Technology

The production process in a northern industryω includes an R&D stage to innovate the state-of-the-art quality of good ω and a manufacturing stage to produce good ω featuring this top quality. A northern firm becomes the incumbent firm of an industry ω once its new invention is launched in the market. At the same time, the technology utilized by the precedent of this incumbent firm, that is, the technology for the second-to-the-top quality, becomes common knowledge to all firms in the world. Each northern firm has the option to outsource the manufacturing stage and/or the R&D stage to the South. An incumbent northern firm is type-jr (r indicates northern innovation) if its precedent outsources the manufacturing stage only and is type-jo (o indicates outsourcing innovation) if its precedent outsources both of the R&D and the manufacturing stages, where j=r if this incumbent northern firm undertakes the R&D stage at home and outsources the manufacturing stage to the South and j=o if it outsources both of the R&D and manufacturing stages. A southern firm that undertakes innovative activity in the R&D stage is called type-s. Define Jn≡{jk,j=r,oandk=r,o}.The success of an R&D activity exhibits a continuous Poisson process. In the R&D stage, fori=rr,ro, an innovation with intensity ιi for a time interval dt requires anιidt units of northern labor at a cost of wnanιidt. On the other hand, for i=or,oo, an innovation with intensity ιi for a time interval dt requires asιidt units of southern laborand aoιidt units of northern labor for a time interval dt. A type-s firm incurs an R&D cost of wsasιidt and receives psιidt from its northern outsourcer which also spends wnaoιidt on northern labor to impose its desired intensity of R&D in collaborating with the southern innovation. Any innovation with intensity ιi for a time interval dt leads to a success with probability ιidt. We assume that an<ao+as since the North is superior in innovation and assume that an>ao so that outsourcing is worthwhile in the equilibrium. The ratio of (an-ao)/an represents the fraction of R&D undertaken by southern labor. We, therefore, define S-share and N-share in R&D as
(1)S-share≡an-aoan,N-share≡aoan.
In an industry ω, an outsourcing connection between an incumbent northern firm and its southern R&D partner ceases when another northern firm launches a better quality in the market. (In reality, firms change their outsourcing partners flexibly. Of course, a long-term contractual relationship between the trading parties can be considered in a more complex framework.)North and South have the same manufacturing technology which requires one unit of labor to produce one unit of output. To emphasize that outsourcing the manufacturing stage is relatively less complex and to simplify the analysis, it is assumed that outsourcing the manufacturing stage incurs no outsourcing cost which implies that all manufacturing stages are operated in the South givenwn>ws.


### 2.2. Preferences

All agents have identical preferences and each consumer maximizes his utilityU,
(2)U=∫0∞‍e-ρt∫01‍log[∑jqj(ω)xjt(ω)]dωdt,ω∈[0,1],qj(ω)=λj,λ>1,
where ρ is the discount rate; t indexes time; ω indexes goods; qj(ω) and xjt(ω), respectively, are the quality and the consumption of good ω that experiences j times of improvement on quality. Let E(t) denote the expenditure at time t and E(t)=∫01‍[∑j‍pjt(ω)xjt(ω)]dω, where pjt(ω) is the price of good ω with quality λj. A consumer allocates an equal expenditure share to every good ω that offers the lowest quality adjusted price pjt(ω)/qj(ω). Let pt(ω) represent the price of the top-quality good ω. The indirect utility function UI is derived as
(3)UI=∫0∞‍e-ρt{logE(t)-∫01‍log[pt(ω)qt(ω)]dj}dt.
In the equilibrium, we also have
(4)E•/E=r-ρ,
where r is the instantaneous interest rate.


### 2.3. No-Arbitrage Conditions and Free Entry

Letπi denote the instantaneous profit of a type-i firm and vi be the value of a type-i firm, i∈Jn∪{s}. For a type-jk firm, jk∈Jn, over a time interval dt, its stockholders receive πjk(t) as dividend payments and suffer a loss of vjk(t) with probability (ιoj+ιrj)dt. That is, the stockholders of this type-jk firm have a probability of [1-(ιoj+ιrj)dt] to capture a capital gain (or loss) of vjk•dt. We, therefore, define the no-arbitrage condition to be that the expected rate of return from holding the stock of the type-jk firm equals the risk-free interest rate, that is, πjk(t)dt+[1-(ιoj+ιrj)dt]vjk•dt-(ιoj+ιrj)vjk(t)dt=r(t)vjk(t)dt. As dt→0, the no-arbitrage condition can be written as
(5)vjk=πjkr+ιoj+ιrj-(vjk•/vjk),
where
(6)πjk=Eλws(λws-ws),jk∈Jn,λws is the unit price, ws is the marginal cost, and E/λws is the amount of products sold. Because the quality of a good increases by λ times through one successful innovation and the competing price is ws, an incumbent firm charges the limit price of λws for its products.A type-s firm receives ps from its northern outsourcer as the payment for its R&D service, spends wsas on employment for R&D, receives a subsidy of sswsas (recall that ss denotes the rate of R&D subsidy on the southern firms), and generates a profit of πs so that we can write
(7)πs=ps-wsas(1-ss).
Since no type-s firm is involved with any manufacturing task, the no-arbitrage condition for a type-s firm is πs(t)dt=r(t)vs(t)dt or
(8)vs≤πsr.Each firm exante chooses its type and R&D intensity to maximize the expected value of the R&D. Free entry in the markets implies that the expected value does not exceed the cost, with equality holding if the R&D intensity is strictly positive. Recall thatsn is the rate of subsidy, representing a friendly policy from the South to encourage R&D outsourcing. We can, therefore, write
(9)vi≤wnao(1-sn)+pswithequalityforιi>0,i=or,oo,vi≤wnanwithequalityforιi>0,i=rr,ro,vs≤wsas(1-ss)withequalityforιi>0forsomei,i∈{or,oo}.


### 2.4. Measures of Firms and Labor Markets

Letni be the measure of type-i firms, for i∈Jn. Since the consumption goods are of measure 1, we get
(10)∑i∈Jnni=1.
In the steady-state equilibrium, the measure of type-i firms is a constant for all i in Jn. Thus, we have the following equations describing the instantaneous exit and entry of firms among types:
(11)njk(ιrj+ιoj)=ιjk(nkr+nko),forj=r,o,k=r,o,
where the LHS of each equation represents the measure of firms which exit from the type and the RHS gives the measure of entry.Among the labor force of the SouthLs,  E/λws units of labor are allocated in the manufacturing stage, and as∑k=r,oιoknok units of labor work on innovation, that is,
(12)Ls=as∑k=r,oιoknok+Eλws.
In the northern labor market, domestic R&D activities require an∑k=r,oιrknrk units of labor and R&D outsourcing require ao∑k=r,oιoknok units of labor. Therefore, we can present the aggregate labor supply n North, Ln, by
(13)Ln=an∑k=r,oιrknrk+ao∑k=r,oιoknok.


## 2.1. Types of Firms and Technology

The production process in a northern industryω includes an R&D stage to innovate the state-of-the-art quality of good ω and a manufacturing stage to produce good ω featuring this top quality. A northern firm becomes the incumbent firm of an industry ω once its new invention is launched in the market. At the same time, the technology utilized by the precedent of this incumbent firm, that is, the technology for the second-to-the-top quality, becomes common knowledge to all firms in the world. Each northern firm has the option to outsource the manufacturing stage and/or the R&D stage to the South. An incumbent northern firm is type-jr (r indicates northern innovation) if its precedent outsources the manufacturing stage only and is type-jo (o indicates outsourcing innovation) if its precedent outsources both of the R&D and the manufacturing stages, where j=r if this incumbent northern firm undertakes the R&D stage at home and outsources the manufacturing stage to the South and j=o if it outsources both of the R&D and manufacturing stages. A southern firm that undertakes innovative activity in the R&D stage is called type-s. Define Jn≡{jk,j=r,oandk=r,o}.The success of an R&D activity exhibits a continuous Poisson process. In the R&D stage, fori=rr,ro, an innovation with intensity ιi for a time interval dt requires anιidt units of northern labor at a cost of wnanιidt. On the other hand, for i=or,oo, an innovation with intensity ιi for a time interval dt requires asιidt units of southern laborand aoιidt units of northern labor for a time interval dt. A type-s firm incurs an R&D cost of wsasιidt and receives psιidt from its northern outsourcer which also spends wnaoιidt on northern labor to impose its desired intensity of R&D in collaborating with the southern innovation. Any innovation with intensity ιi for a time interval dt leads to a success with probability ιidt. We assume that an<ao+as since the North is superior in innovation and assume that an>ao so that outsourcing is worthwhile in the equilibrium. The ratio of (an-ao)/an represents the fraction of R&D undertaken by southern labor. We, therefore, define S-share and N-share in R&D as
(1)S-share≡an-aoan,N-share≡aoan.
In an industry ω, an outsourcing connection between an incumbent northern firm and its southern R&D partner ceases when another northern firm launches a better quality in the market. (In reality, firms change their outsourcing partners flexibly. Of course, a long-term contractual relationship between the trading parties can be considered in a more complex framework.)North and South have the same manufacturing technology which requires one unit of labor to produce one unit of output. To emphasize that outsourcing the manufacturing stage is relatively less complex and to simplify the analysis, it is assumed that outsourcing the manufacturing stage incurs no outsourcing cost which implies that all manufacturing stages are operated in the South givenwn>ws.


## 2.2. Preferences

All agents have identical preferences and each consumer maximizes his utilityU,
(2)U=∫0∞‍e-ρt∫01‍log[∑jqj(ω)xjt(ω)]dωdt,ω∈[0,1],qj(ω)=λj,λ>1,
where ρ is the discount rate; t indexes time; ω indexes goods; qj(ω) and xjt(ω), respectively, are the quality and the consumption of good ω that experiences j times of improvement on quality. Let E(t) denote the expenditure at time t and E(t)=∫01‍[∑j‍pjt(ω)xjt(ω)]dω, where pjt(ω) is the price of good ω with quality λj. A consumer allocates an equal expenditure share to every good ω that offers the lowest quality adjusted price pjt(ω)/qj(ω). Let pt(ω) represent the price of the top-quality good ω. The indirect utility function UI is derived as
(3)UI=∫0∞‍e-ρt{logE(t)-∫01‍log[pt(ω)qt(ω)]dj}dt.
In the equilibrium, we also have
(4)E•/E=r-ρ,
where r is the instantaneous interest rate.


## 2.3. No-Arbitrage Conditions and Free Entry

Letπi denote the instantaneous profit of a type-i firm and vi be the value of a type-i firm, i∈Jn∪{s}. For a type-jk firm, jk∈Jn, over a time interval dt, its stockholders receive πjk(t) as dividend payments and suffer a loss of vjk(t) with probability (ιoj+ιrj)dt. That is, the stockholders of this type-jk firm have a probability of [1-(ιoj+ιrj)dt] to capture a capital gain (or loss) of vjk•dt. We, therefore, define the no-arbitrage condition to be that the expected rate of return from holding the stock of the type-jk firm equals the risk-free interest rate, that is, πjk(t)dt+[1-(ιoj+ιrj)dt]vjk•dt-(ιoj+ιrj)vjk(t)dt=r(t)vjk(t)dt. As dt→0, the no-arbitrage condition can be written as
(5)vjk=πjkr+ιoj+ιrj-(vjk•/vjk),
where
(6)πjk=Eλws(λws-ws),jk∈Jn,λws is the unit price, ws is the marginal cost, and E/λws is the amount of products sold. Because the quality of a good increases by λ times through one successful innovation and the competing price is ws, an incumbent firm charges the limit price of λws for its products.A type-s firm receives ps from its northern outsourcer as the payment for its R&D service, spends wsas on employment for R&D, receives a subsidy of sswsas (recall that ss denotes the rate of R&D subsidy on the southern firms), and generates a profit of πs so that we can write
(7)πs=ps-wsas(1-ss).
Since no type-s firm is involved with any manufacturing task, the no-arbitrage condition for a type-s firm is πs(t)dt=r(t)vs(t)dt or
(8)vs≤πsr.Each firm exante chooses its type and R&D intensity to maximize the expected value of the R&D. Free entry in the markets implies that the expected value does not exceed the cost, with equality holding if the R&D intensity is strictly positive. Recall thatsn is the rate of subsidy, representing a friendly policy from the South to encourage R&D outsourcing. We can, therefore, write
(9)vi≤wnao(1-sn)+pswithequalityforιi>0,i=or,oo,vi≤wnanwithequalityforιi>0,i=rr,ro,vs≤wsas(1-ss)withequalityforιi>0forsomei,i∈{or,oo}.


## 2.4. Measures of Firms and Labor Markets

Letni be the measure of type-i firms, for i∈Jn. Since the consumption goods are of measure 1, we get
(10)∑i∈Jnni=1.
In the steady-state equilibrium, the measure of type-i firms is a constant for all i in Jn. Thus, we have the following equations describing the instantaneous exit and entry of firms among types:
(11)njk(ιrj+ιoj)=ιjk(nkr+nko),forj=r,o,k=r,o,
where the LHS of each equation represents the measure of firms which exit from the type and the RHS gives the measure of entry.Among the labor force of the SouthLs,  E/λws units of labor are allocated in the manufacturing stage, and as∑k=r,oιoknok units of labor work on innovation, that is,
(12)Ls=as∑k=r,oιoknok+Eλws.
In the northern labor market, domestic R&D activities require an∑k=r,oιrknrk units of labor and R&D outsourcing require ao∑k=r,oιoknok units of labor. Therefore, we can present the aggregate labor supply n North, Ln, by
(13)Ln=an∑k=r,oιrknrk+ao∑k=r,oιoknok.


## 3. The Steady-State Equilibrium

The steady state equilibrium consists of a sequence of variables(r,E/wn,ps/ws,ws/wn,ni,ιi,i∈Jn) which satisfies (4) to (13). We can simplify (4) to (13) by taking the symmetries among firms into consideration. The northern firms of the same type face the same exante conditions of production and the same prospects of profitability; therefore, these forward-looking firms should make the same decisions on their R&D intensity and should have the same expected revenue and cost in production. Thus, we define xj by
(14)xj≡xjr=xjo=xjsforx=ι,v,π,j=r,o.
Define η as the aggregate rate of innovation,
(15)η≡ιono+ιrnr.
Equations (4) to (15) are reduced to the following three equations as shown in Appendix B to determine the steady-state values of ιo, no, and η:
(16)asιono+ϕλe(ιono+ρ)-Ls=0,anη+ιono(ao-an)-Ln=0,η-2ιono+2ιo-ιono=0,
where
(17)ϕ≡(1+ρ)as(1-ss)anan-ao(1-sn).
Once ιo,  no,  and  η are determined, we can get
(18)Ewn=λλ-1an(ιo+ιr+ρ),njk=ιjιk(ιr+ιo)2,forj=r,o,k=r,o.
In addition, by (4) to (15), we derive
(19)r=ρ,psws=(1+ρ)(1-ss)as,wswn=an-ao(1-sn)(1+ρ)as(1-ss).Proposition 1.
Givenws/wn<1 and sn=ss=0 in the steady-state equilibrium, we have d(ws/wn)/dsn=ao/(1+ρ)as and d(ws/wn)/dss=(an-ao)/(1+ρ)as.Proposition1 states that a marginal increase in the rate of subsidy on R&D in the South raises the southern relative wage. Particularly, subsidizing southern firms generates a larger (smaller) effect than subsidizing northern outsourcing firms if S-share is larger (smaller) than N-share.Total differentiation of (16) evaluating at sn=ss=0 yields
(20)A[dηdιodno]=[-(ιono+Φas)das+Φdss+Φaoan-aodsn+dLsdLn0],
where
(21)A=[0asno+ϕλe1noasιo-ϕλeιono1noanno(ao-an)ιo(ao-an)1-2no+2-1no-2ιo+ιono1no],Φ≡ϕ(ιo/no+ρ)λe.Proposition 2.
In the steady-state equilibrium, we havedx/dLs=Φ-1dx/dss=(aoΦ/(an-ao))-1dx/dsn=-(ιo2/(ιr+ιo)+Φ/as)-1dx/das, for x=η,ιo,no.Proposition2 derived from (20) tells that an increase in Ls, in ss, in sn and a decrease in as generate the same qualitative impact on η,ιoand no. If S-share is larger (smaller) than N-share, sn generates a larger (smaller) effect than ss.Propositions1 and 2 imply some welfare consequences of R&D friendly policy. For example, suppose that the government of the South chooses to raise one of sn and ss around the steady state and that S-share is greater than 1/2. If the impacts mentioned in Proposition 2 are negative, raising ss is better than raising sn, since the former increases the southern relative wage more and reduces the aggregate rate of innovation less. On the other hand, if the impacts are positive then it is ambiguous whether raising ss is better than raising snbecause raising ss increases the southern relative wage more but it increases the aggregate rate of innovation less. In addition, if the impacts in Proposition 2 are negative, an increase in R&D subsidy and/or an increase in labor productivity in R&D aimed to encourage R&D outsourcing will get the contrary results—a smaller ιoor a smaller no.To figure out the sign of the impact,|A| is computed according to (20) and
(22)|A|=2anasιono[no-1+(an+aoan-1no)(1+ρ)an(λ-1)(an-ao)]≡2anasιonoX.AppendixC shows that we can define no*, 0<no*<1 such that |A|=0 when no=no*. Since dX/dno>0, we get that if no>no* then |A|>0 and if no<no* then |A|<0. That the initial steady-state scale of R&D outsourcing, that is, no, is not too large means no<no*. Particularly, by (22), we have |A|<0if no≤an/(an+ao) and we also get that 1/2<an/(an+ao)<no*. In Appendix D, some results of comparative statics are derived and the results with unambiguous signs are summarized in Proposition 3 and Figure 1.Figure 1
Results of Comparative Statics.Proposition 3.
In the steady-state equilibrium with|A|<0, we get dη/dLs>0 and dno/dLn<0 for no∈(0,no*);  dη/dLs>0, dno/dLn<0, and dιo/dLs>0 for no∈(0,an/(an+ao)]; dη/dLs>0, dno/dLn<0, dιo/dLs>0,  dη/dLn>0, and dno/dLs>0 for no∈(0,1/2]. In the steady-state equilibrium with |A|>0, we get dη/dLs<0, dη/dLn>0, and dno/dLn>0 for no∈(no*,1).Proposition3 indicates that the comparative statics mentioned in Proposition 2 are with negative signs if no is greater than no*. Thus, many policy implications depend on if the scale of R&D outsourcing is large or is small.In contrast with the general result in the literature (e.g., [7, 10, 17]) that an increase in the labor force for R&D raises the aggregate rate of innovation, Proposition 3 tells that an increase in the labor force in the North generates an indeterminate effect on the aggregate rate of innovation if no∈(1/2,no*) and that an increase in the labor force in the South reduces the aggregate rate of innovation if no>no*. That a larger Ls corresponds to a smaller η when no>no* is mainly because Ls are employed in two ways, R&D and manufacturing. As no increases, the southern labor for manufacturing becomes more scarce. When no is sufficiently large, a marginal increase in Ls will be allocated to manufacturing and will result in a smaller ratio of employment in R&D to employment in manufacturing and, therefore, a reduction in η.Thoughno is implied by (16), unfortunately it is not easy to tell what values of the parameters facilitate us to derive an analytical solution to no or how no is affected by each of the parameters. Nonetheless, Figure 1, together with Propositions 1 and 2, offers a guideline for policymakers once the steady-state equilibrium is realized and no is known.


## 4. Concluding Remarks

This model incorporates southern innovation into product cycles and is structured as simple as possible while it still keeps the key character that the latest innovation through R&D replaces outdated products. It is shown that there are certain boundary conditions of the scale of R&D outsourcing which affect policy implications. Particularly, according to Propositions1 and 2, when the scale of R&D outsourcing is large, the aggregate rate of innovation is negatively related with the rate of R&D subsidy. Therefore, a friendly policy to promote R&D outsourcing (given that all manufacturing tasks are outsourced from the North to the South) may be beneficial for both of the North and the South only if the scale of R&D outsourcing is small. This finding is in contrast to the result of Glass and Saggi [7] that the rate of innovation is raised by outsourcing (they only consider manufacturing outsourcing); therefore, international outsourcing can potentially create gains sufficient to offset the decline in the northern wage and to benefit the North as well as the South.In general, I find that the results of comparative statics are affected by the sign of|A| and by the scale of R&D outsourcing as shown in Figure 1, which indicates that considering R&D in the South through outsourcing is important. These contraries of comparative statics under different environments may offer a guideline for making policies to affect the relative wage and the rate of innovation.



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*Source: 702974-2012-09-26.xml*