# Industrial heterogeneity and international product cycles

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## Abstract

This paper proposes a model of quality ladder in the context of North–South trade to examine the emergence of product cycles in industries of different research and development (R&D) intensity levels. To acquire the dominant advantage, firms as a whole can strategically undertake either quality upgrades through R&D or cost saving through the channels of market penetration—foreign direct investment (FDI) or offshoring. In an infinite-horizon game, the uses of mixing moving-up and moving-out strategies in high-tech and medium-tech industries generate product cycles. Furthermore, in low-tech industries, FDI is a strongly dominant strategy for the industry leaders and followers. Under certain conditions, firms leapfrog over each other and product cycles thus emerge.

## Keywords

North–South trade Industrial heterogeneity Vertical innovation Penetration channels International product cycles## JEL Classification

C72 D21 F12 F23## 1 Introduction

With the continued globalization of the world economy over the last decade or so, the world has witnessed a phenomenal increase in the volume of trade penetration along with commitments to market openness. Foreign direct investment (FDI) and offshoring are common channels of penetration. Market penetration usually generates strategic competition and produces a cycle in the international production. The product cycle features prominently in trade between the Northern developed economies and the Southern newly-industrialized economies. In other words, a new product is first invented and manufactured in the advanced North. Once technology becomes routine and standardized, production is moved to the low-cost South through FDI. It follows that the South now manufactures and exports the product, which reverses the earlier trade pattern. It is well known that Vernon’s (1966) product-cycle hypothesis remains the dominant explanation of the North–South trade dynamics. In terms of newly-industrialized states, China, for instance, is a remarkable example of a country in the product cycle phase. With the entry of China into the World Trade Organization, China’s economy is becoming increasingly opened up to foreign investors. Gaulier et al. (2007) examined how China’s foreign trade is, to a large extent, the result of the investments of outsourcing foreign firms. They further showed that about half of China’s foreign trade has been fostered by the assembly and processing of imported inputs for re-exports since the early 1990s. This is served as the reason for China’s rapid diversification from textiles to electronics.^{1} In the use of revealed comparative advantage analysis, for instance, Savsin (2011) studied the trade patterns between Sweden and China.^{2} When Sweden shifts production lines to China, it directly and indirectly helps China develop relevant industries with the advanced technology. As a result, product cycles emerge.

The theoretical literature on North–South trade has explored the formation of product cycles starting with Krugman (1979). He found that the technological lag between countries gives rise to trade, with the North exporting new products and importing old products from the South. Higher Northern per capita income depends on the quasi-rents from the Northern monopoly of new products, so that the North must continually innovate not only to maintain its relative position but also to maintain its real income in absolute terms.^{3} On the technology improvements and product cycles, the frameworks are essentially developed as two alternatives.^{4} The first one views innovation as an increase in the number of product varieties. Jensen and Thursby (1987) showed that the existence of technology transfer lowers the Northern industry’s Research and Development (R&D) benefits. The innovation rate is then lower than the globally optimal rate. Grossman and Helpman (1991a) found that the size of the resource base and the productivity of resources in terms of learning activities are important for the determination of the steady-state growth rate. The larger the resource base of the South, the greater the productivity of its resources in learning the production processes that are originally developed in the North. Furthermore, the Southern market penetration for each product is either complete penetration or zero. Palokangas (2008) studied the growth effects of competition in a product-cycle model with innovation, imitation and non-diversifiable risk. He showed that only if the intensity of competition exceeds a critical level, its increase enhances growth. Gustafsson and Segerstrom (2011) examined that a strengthening of the protection on intellectual property rights (IPRs) in the South increases multinational firms’ R&D expenditures. It follows that the rate of technology transfer within multinational firms becomes faster, and long-run consumer welfare in both regions increases.^{5} Furthermore, Suzuki (2015) distinguished IPRs protection and IPRs enforcement by incorporating illegal imitation and seizure activity into a North–South quality ladder model. He focused on the Southern IPRs enforcement rather than IPRs protection, and showed that import prohibitions significantly affect IPRs enforcement policies in the South.

The second framework considers innovation as a quality improvement for the existing varieties of products. The use of the quality ladder framework is wider in the existing literature than that of the product variety framework. Flam and Helpman (1987) developed a static product cycle model and showed that if the North starts producing at a higher level of quality, then the South stops producing the lowest quality levels and takes over the production of some quality levels that were previously produced by the North. Grossman and Helpman (1991b) further developed a two-country model with endogenous innovation and imitation. They showed that the steady-state equilibrium is characterized by constant aggregate rates of innovation and imitation. They also revealed how these rates respond to changes in the sizes of the two regions and to policies in each region to promote learning. Glass (1997) demonstrated that a weakening of the protection of IPRs in the South results in an increase in the level of Southern market penetration. Furthermore, an increase in Southern market penetration implies a reduction in the rate of innovation. Segerstrom and Zolnierek (1999) showed that if the industry leader has R&D cost advantages, it is optimal for the government to subsidize all firms’ R&D expenditure, subsidize the leader’s production cost, and tax the profits of a leader in a new industry. Yang and Maskus (2001) examined the effects of stronger IPRs in the South on the incentives of firms in the North to innovate and to license advanced technologies.

Earlier studies consider product cycles to have resulted from the Northern innovation and the Southern imitation.^{6} When specifically considering other entry channels such as FDI and offshoring, Glass and Saggi (1998) developed a general equilibrium model wherein the quality of technology is transferred through FDI to the technology gap between countries. They showed that if the technology gap is small, successful imitation of low quality levels makes FDI with high quality levels possible. Furthermore, a subsidy to Southern imitation or a tax on low-quality FDI encourages high-quality FDI by reducing the technology gap. Lu (2007) introduced industrial heterogeneity in R&D productivity into the quality-ladder model of North–South trade to study how firms’ choices made between R&D and FDI vary across industries, and how such choices consequently determine the evolution of comparative advantage and trade. She found that trade reveals product-cycle dynamics in medium-tech industries but remains static in others. She further showed that the high-tech industries experience continued innovation in the North and that no product lines migrate to the South; further, the low-tech industries are not presented with the option of moving up and thus they stagnate.

Evidence shows us that product-cycle trade truly exists across almost all industries including high-, medium-, and low-tech levels; see, e.g., Feenstra and Rose (2000), Yeaple (2003), Zhu (2005) and Hallak (2006).^{7} Although Lu (2007) theoretically showed that product-cycle dynamics in medium-tech industries, none of the relevant literature have theoretically explored how product cycles emerge in industries in high- or low-tech levels. The contribution of this paper is to utilize rigorous theoretical reasons in support of the analyses of firms’ behavior and the formation of product cycles. To exhibit the formation of product cycles in high-, medium- and low-tech industries, we extend Lu (2007)’s model to envisage a model of quality ladder in North–South trade with two countries: the innovating North and non-innovating South. Industrial heterogeneity is based on R&D intensity and labor is the only factor input in both countries. Different from the literature, in view of the channels of market penetration, we separate the production line by two parts: (i) production of intermediate goods; and (ii) production/assembly of final goods. To seek the dominant advantage, a Northern firm has two options: (i) quality advantage through R&D investment^{8} and (ii) cost advantage through FDI or offshoring. In each period, there is always a two-stage, one-shot game. At the first stage, all firms in heterogeneous industries simultaneously decide on one strategy from catch-up R&D, FDI, and offshoring, and then compete via price at the second stage. Incorporating the strategic form game, we comprehend that for the producers of state-of-the-art products and the second-to-top, the cost difference is the key force for the Nash equilibria.

Our model is motivated to capture the contemporary scenario of industrial growth experienced in response to the strategic moves of Northern firms. The moves generate strategic competitions and produce cycles in the international production. We show that the cost difference plays the key role for firms to undertake the strategic choices between FDI or outsourcing, and thereby raises the competition level. Intuitively, if a firm moves production lines to the South, it is reasonable to conclude that it will save more labor costs than the scenario in which it merely produces in the North. Furthermore, we apply the quality-adjusted price to reveal the consumer perceived price on a quality of product.^{9} In equilibrium, it is always the highest available quality providing the lowest quality-adjusted price. Thereby, the version with the lowest quality-adjusted price will capture the world demand. This offers firms incentives on R&D investments. We explore in detail that the uses of mixing moving-up and moving-out strategies would result in product cycles in industries of any level in an infinite-horizon game. In high-tech industries, continuous investment in R&D is a strategy for a dominant quality leader. To save the production cost, the leading firm can consider a mix of moving-up and moving-out strategies. By doing this, if it can earn the same expected profits as a firm that engages in R&D at all times, then the uses of mixing strategies generate the emergence of product cycles. In medium-tech industries, if FDI is a strongly dominant strategy for both quality leaders and followers, the production of leading-edge product agglomerates in the South. Furthermore, if a follower specifically chooses R&D, R&D then weakly dominates FDI and offshoring for the leader. It follows that the leader in some cases will mix strategies and then product cycles emerge. In low-tech industries, FDI is a strongly dominant strategy for both leaders and followers. Under certain conditions, product cycles will emerge.

The remainder of this paper is as follows. Section 2 depicts model frameworks. Section 3 analyzes firms’ behavior and dominant strategies through the use of a strategic form game. Section 4 illustrates that Northern firms’ strategies would induce product cycles in industries at any level. Section 5 characterizes the concluding remarks.

## 2 Model frameworks

### 2.1 Overview

We envisage a model of North–South trade with endogenous, vertical innovations and technology transfers. In both the North and South, labor is the only factor of production and is immobile across countries. The Northern and Southern labor endowments are constant. In 1817, David Ricardo laid out the theory of comparative advantage, which showed that all nations could benefit from free trade even if a nation was less efficient than its trading partners at producing all kinds of goods.

Let the wage in the South be \(w_{S}=1\) and the wage in the North be \(w_{N}>1\).^{10} Suppose that South has a Ricardian advantage in upstream industries for intermediate production, that is, \(\gamma \ge 1\) units of labor are required to produce a unit of intermediate goods in the North while only one unit of labor is needed to produce a unit of intermediate in the South. Specifically, we assume that each intermediate good is variety-specific: once produced for a certain production/assembly line, it has no alternative use. As for the production of final goods, labor in North and South supposedly share identical productivity. We further assume that initial Northern firms are vertically integrated on production process and initial Southern firms produce only intermediate goods. Thus, the production of final goods in the South requires Northern firms’ successful FDI in downstream divisions. It turns out that Northern firms become multinational entrepreneurs and can start to produce final goods in the South. A transplanted firm hires Southern workers, retains full control over production, and remits all profits back to the North. Supposing that there is no trade friction, the lower the production costs in the South, the greater the incentive for Northern firms to engage in FDI.

Moreover, Northern firms can benefit from offshoring their intermediate divisions to the South due to its relatively lower production costs. Offshoring is associated with additional costs besides production costs. If produced in the South, intermediate goods must be shipped back to the North, which incurs iceberg trade costs \(\tau >1\) for one shipped unit to reach the destination. Contracts between the upstream (intermediate plant) and downstream (assembly plant) divisions are complete when both are located in the North or in the South. However, contracts are incomplete when the upstream division is offshored to the South and the production of final goods is left in the North. This case then generates a hold-up problem: if the upstream division has supplied its specific inputs, it has to reach an arrangement with the downstream division on how to share the joint surpluses (revenues) from final sales. In the bargaining stage, we denote the bargaining weights of upstream division and of downstream division by \(\beta \) and \(1-\beta \), respectively.

The product space is defined uniformly on a unit square indexed by \((z,y)\in [0,1]\times [0,1]\), where *z* denotes a continuum of industries, and *y* denotes a continuum of varieties. Let \(j\in \{0,1,2,\ldots ,J\}\) be an index of different versions along the quality ladder, where *J*(*z*, *y*) denotes the leading-edge version of product (*z*, *y*). The quality of version *j* of product (*z*, *y*) is denoted by \(q(z,y,j)=\lambda (z,y)^{j}\), where \(\lambda (z,y)=\lambda (z)\), \(\lambda (0)=1\), and \(\lambda ^{\prime }(z)>0\), that is, the industries are ranked such that the quality jump of each innovation is increasing in *z*. Furthermore, each version *j* is produced by only one firm, and the technologies used to produce version zero (i.e. \(j=0\)) of all products always exist in the economy (see, also, Grossman and Helpman 1991a, b; Lu 2007; Gustafsson and Segerstrom 2010, 2011). Since R&D (innovation) usually requires specific knowledge and techniques, we then assume that all R&D is undertaken in the North. A success in R&D (innovation) moves a product one step up its quality ladder and creates quality improvements. Each innovation is rewarded with a worldwide perpetual patent, which grants the inventor an exclusive right over the blueprint that he or she invents. Production of any patented good takes place in the North, and each product can potentially be improved by an infinite number of times. To distinguish skilled from unskilled labors in the North, we let \(w_{R}\) be the wage paid to a skilled worker if a firm engages in the process of R&D, where \(w_{R}>w_{N}>1\).

*via*FDI/offshoring for raising its cost advantages. Recall that R&D processes happen only in the North, and the South has a Ricardian advantage in upstream divisions of intermediate goods. For readers’ easy understanding, we summarize a Northern firm’s three possible strategies in Table 1, and further analyze them in detail in Sect. 2.3.

Summary of a Northern firm’s three strategies: R&D, FDI and offshoring

R&D | FDI | Offshoring |
---|---|---|

1. Since R&D (innovation) usually requires specific knowledge and techniques, R&D process occurs in the North | 1. A North firm undertakes FDI and produces in the South | 1. A North firm moves upstream division to the south but still produces final goods in the North |

2. A sucess in R&D (innovation) creates quality improvements and moves a product supplied by a vertical integrated Northern firm one step up its quality ladder | 2. A transplanted firm hires Southern workers, retains full control over production, and remits all profits back to the North | 2. When produced in the South, intermediate goods must be shipped back to the North, and the firm incurs iceberg trade costs \(\tau > 1\) per shipped unit |

3. The contract between the upstream (intermediate plant) and downstream (assembly plant) divisions is complete | 3. The contract between the upstream and downstream divisions is complete | 3. The contract between the upstream and downstream divisions is incomplete. \(\beta \) and \(1-\beta \) are, respectively, the bargaining powers of the upstream and downstream producers |

### 2.2 Preferences

*u*(

*t*) is the utility flow at time

*t*. Obviously, the utility function is assumed to be instantaneously Cobb–Douglas and intertemporally CES with the unit elasticity of intertemporal substitution. The instantaneous preference is composed with a Cobb–Douglas utility function over the product space and a linear utility function over different versions within each product line. It follows that:

*x*(

*z*,

*y*,

*j*,

*t*) denotes the demand for version

*j*of product (

*z*,

*y*).

We assume that the consumer allocates an equal expenditure share to every product (*z*, *y*). Given prices *p*(*z*, *y*, *j*, *t*), the level of total spending is \(E(t)=\int _{0}^{1}\int _{0}^{1}[\sum \nolimits _{j}p(z,y,j,t)x_{jt} (z,y,j,t)]dydz\). Within the same product line, different versions are perfect substitutes, and only the version that provides the lowest quality-adjusted price, \(\frac{p(z,y,j,t)}{q(z,y,t)}\), is purchased. In equilibrium, it is always the highest available quality provided with the lowest quality-adjusted price.

*I*(0) is the present value of income streams,

*E*(

*t*) is the total expenditure level at time

*t*, and

*R*(

*t*) is the cumulative interest rate. Specifically, \(R(t)=\int _{0}^{t}r(s)ds\), where

*r*(

*t*) is the instantaneous interest rate. Using multi-stage budgeting to solve their utility maximization problem, households first allocate their income flow between savings and expenditures. This consequently yields a time path of

*E*(

*t*) that follows the Euler equation of a standard Ramsey problem, that is,

*r*(

*t*), exceeds the discount rate, \(\rho \).

In both countries, each individual member of a household lives forever and is endowed with one unit of labor, which is inelastically supplied. To attain their incomes, households offer labor services in exchange for wage payments. As we have mentioned in Sect. 2.1, for different needs in either industry, there are three types of labor in the North: skilled labor working for R&D process if needed, unskilled labor working for the production of intermediate goods, and unskilled labor working for the production of final goods. There are two types of labor in the South: unskilled labor working for intermediates, and unskilled labor working in the production of final goods if needed.

*t*, the representative consumer’s income in either the North or the South equals her expenditures. Note that households in both countries share an identical utility, and only the version with the lowest quality-adjusted price is purchased. We then obtain the function of world demand,

*X*(

*z*,

*y*,

*j*,

*t*), as:

### 2.3 Market strategies and game structure

Dixit and Stiglitz (1977) asserted that, under monopolistic competition, a commodity should be produced if the costs can be covered by the sum of revenues and a properly defined measure of the consumer’s surplus. The optimum amount is then found by equating the demand price and the marginal cost. If the increments to quality are common to all products and a firm earns a profit stream \(\pi (\widetilde{t})\) for \(\widetilde{t}\geqq t\), Grossman and Helpman (1991b) showed that profits accrue only to firms that are the lowest cost producers of state-of-the-art products. The stream of profits for such a producer continues until another firm succeeds in catching up with this producers of state-of-the-art products through copying the product or improving upon it. If it happens, then the profit of the initial leading firm falls to zero, and the leapfrog emerges.^{11}

Accordingly, in each period *t*, within each product line, firms compete as price-setting oligopolists. The version/firm with the lowest quality-adjusted price captures the entire world demand of the product if no other firms at the same period succeed in either (1) acquiring cost advantages *via* FDI or offshoring or (2) catching up with the producers of state-of-the-art products *via* R&D investments.

#### 2.3.1 FDI

^{12}

*dt*has some probability of success in transferring production technology to the South. Let \(a_{F}^{*}(z,y,j)\) denote the unit Southern labor requirement per unit of FDI intensity for the version

*j*of product (

*z*,

*y*). Let \(V_{F}\) be the expected present value of the profits earned by the dominant firm. Each FDI firm acts to maximize \((V_{F}-a_{F}^{*})\iota _{F}dt\). A long-term equilibrium with positive and finite FDI investments shows that \(V_{F} =a_{F}^{*}\). If \(V_{F}<a_{F}^{*}\), then no FDI will be undertaken, that is, \(\iota _{F}(z,y)=0\).

#### 2.3.2 Offshoring

*z*,

*y*):

*p*(

*z*,

*y*,

*j*,

*t*)

*X*(

*z*,

*y*,

*j*,

*t*). According to the bargaining weights, each of divisions decides the optimal share distribution

*à la*Nash’s axiomatic bargaining: \(\underset{\alpha }{\max }\) \([\alpha pX]^{\beta }[(1-\alpha )pX]^{1-\beta }=\underset{\alpha }{\max }\) \(\beta \ln \alpha +(1-\beta )\ln (1-\alpha )+\ln [pX]\). It follows that \(\alpha =\beta \) at optimum, and then a share \((1-\beta )\) of the profit goes to the downstream division while a share \(\beta \) goes to the upstream division. Notice that we assume that one unit of intermediate goods can be used for the production of one unit of final goods in a one-to-one fashion. Due to iceberg costs, \(\tau >1\), we deduce that the production of one unit of final output will require \(\tau \) units of intermediate inputs (see, also, Naghavi and Ottaviano 2009). We then derive the downstream operating profit

*j*of product (

*z*,

*y*) and \(a_{O} ^{*}<a_{F}^{*}\). Let \(V_{O}\) be the expected present value of the profits earned by a dominant firm. Each offshoring firm acts to maximize \((V_{O}-a_{O}^{*})\iota _{O}dt\), where \(\iota _{O}dt\) is the probability of success in achieving offshoring at intensity \(\iota _{O}\) for duration

*dt*. A long-term equilibrium with positive and finite offshoring investments shows that \(V_{O}=a_{O}^{*}\). If \(V_{O}<a_{O}^{*}\), then no offshoring will be undertaken, that is, \(\iota _{O}(z,y)=0\).

#### 2.3.3 R&D and quality upgrades

We refer to the firm holding the leading-edge blueprint as the quality leader in an industry (denoted by *J*), and to the firms holding obsolete blueprints as the followers, denoted by \(j_{F}\in \{0,1,2,\ldots ,J-1\}\). A firm with a worldwide perpetual patent enjoys an exclusive right over the blueprint that it invents. Specifically, we assume that intertemporal knowledge spillovers are quite strong. R&D adoption by firms follows Poisson processes with success rates depending on the current R&D or so-called knowledge accumulation as described by Jones (1995).^{13} For a given product space (*z*, *y*), the probability of improvements *m* in an interval of length *t* is \(f(m,t)=\frac{(\iota t)^{m}e^{-\iota t}}{m!}\), where \(\iota \) denotes the aggregate intensity of R&D. A successful quality-improvement raises the version one step up. Eq. (2) then becomes \(\int _{0}^{1}\int _{0} ^{1}\ln q(z,y,t)dydz=\sum _{m=0}^{\infty }f(m,t)\ln \lambda ^{m}=\iota t\ln \lambda \), where \(\iota t\) is the expected number of improvements in an interval of length *t*. An innovator engaging in R&D aims to invent the blueprint of the next version at intensity \(\iota \) for a length of time *dt*. It follows that the firm has a probability of success, \(\iota dt\). Furthermore, the patent reward earned by the new leader is \(V_{R}\) and the cost paid for research is \(w_{R}a_{R}\), where \(a_{R}\) is the number of skilled labors working for R&D. Recall that \(w_{R}>w_{N}>1\). Hence, the expected present value of the profit in R&D is \((V_{R}-w_{R}a_{R})\iota dt\). Free entry in the R&D races generates an equilibrium with positive and finite R&D investments. It shows that \(V_{R}=w_{R}a_{R}\); if \(V_{R}<w_{R}a_{R}\), then no R&D will proceed, that is, \(\iota (z,y)=0\).

^{14}

^{15}It implies that the cost of the production of patented goods, \(w_{N}(1+\gamma )X\), is definitely higher than that of the production without R&D, \(2w_{N}X\). Hence, R&D expenditure is the most expensive investment, and only the North can deal with R&D by our assumption. As mentioned in the Sect. 2.1, since R&D (innovation) usually requires specific knowledge and techniques, it costs firms more to generate new patents than to transfer technology.

Comparing the cost functions in three cases, we then obtain the following remark:

### Remark 1

\(w_{N}(1+\gamma )>1+w_{N}\tau >2\) for \(\exists \) \(w_{N}>1,\) \(\tau >1,\) \(\gamma \ge 1\).

The labor in the South is more efficient in the production of intermediate goods than the labor in the North. The Northern and Southern labors have identical productivity in terms of final goods. It is reasonable to conclude that if a Northern firm moves all production lines as FDI to the South, it will intuitively reduce more labor costs than the scenario in which it just offshores its upstream division to the South.

From the inequality, \(w_{N}(1+\gamma )>1+w_{N}\tau \), we can obtain that \(w_{N}>\frac{1}{1+\gamma -\tau }\). Together with \(w_{N}>1\), we then obtain the second remark:

### Remark 2

If \(w_{N}>\frac{1}{1+\gamma -\tau }>1\), then \(\tau >\gamma \) while if \(w_{N}>1>\frac{1}{1+\gamma -\tau }\), then \(\gamma >\tau \).

#### 2.3.4 Scenarios and game structure

*t*, there is always a two-stage, one-shot game. At the first stage of the game, all heterogeneous firms simultaneously decide on one strategy from the pool of catch-up R&D, FDI and offshoring, and then compete

*via*price at the second stage. The pure Nash equilibria exist while we compare outcomes in each period

*t*. In an infinite-horizon game, we consider the following possible scenarios which may occur at

*t*:

- (i)
Quality leaders in the North would decide on whether to engage in R&D for a two-step quality advantage over their nearest rivals, or to offshore the intermediate division for cost advantages. During the same

*t*, Northern quality followers including producers of the second-to-top try to beat and then replace the producers of state-of-the-art products. If the followers’ strategies fail, then Northern producers of state-of-the-art products win the market. It follows that all quality followers become potential competitors. - (ii)
If a quality-follower succeeds in its strategies as the case in (i) and wins the market, then the producers of state-of-the-art products and other quality followers become potential competitors.

- (iii)
Consider Northern producers of the second-to-top have cost advantages due to FDI or offshoring. Northern producers of state-of-the-art products will decide on their market strategies in order to win the market back. If such the strategies fail, then Northern producers of the second-to-top keep occupying the market.

- (iv)
If Northern producers of state-of-the-art products succeed in their strategies as mentioned in (iii), then they seize back the market. Thereby, all low-quality producers including producers of the second-to-top become potential competitors.

## 3 Final goods market and Bertrand pricing game

As we detailed in Sect. 2.3, in each time *t*, within each product line, firms compete as price-setting oligopolists. The firm (version) that offers the lowest quality-adjusted price will capture the entire world demand of the product if no other firms at the same *t* succeed in catching up with the highest-quality leader *via* R&D investments or in acquiring cost advantages through FDI or offshoring. In the following, Sects. 3.1 3.2 and 3.3 will examine three strategies in sequence, and Sect. 3.4 will illustrate the short-run equilibria in a strategic form game.

### 3.1 R&D and quality upgrades

In the absence of FDI and offshoring, every version involves R&D in the North, and each faces the same marginal cost, \(w_{N}(1+\gamma )\). Along with the success in R&D, the initial state-of-the-art producer *J* still attains its superior quality and sets its price a little less than \(\lambda w_{N}(1+\gamma )\) in order to maintain its quality-adjusted price slightly lower than \(\frac{p}{q}=\frac{\lambda w_{N}(1+\gamma )}{\lambda ^{J+1}} =\frac{w_{N}(1+\gamma )}{\lambda ^{J}}\). The value of \(\frac{w_{N}(1+\gamma )}{\lambda ^{J}}\) is the minimum quality-adjusted price that the initial quality-followers \(j_{F}\in \{0,1,2,\ldots ,J-1\}\) can provide. Therefore, *J* defeats \(j_{F}\) and then seizes the market as a monopolist with a positive profit. As Etro (2004) showed, the persistence of monopoly is based on a precommitment of the incumbent monopolist to invest in R&D, and the incumbent has more incentive to invest than any outsider.

Substituting \(p=\lambda w_{N}(1+\gamma )\) into Eq. (9), we can derive that the instantaneous profit is \(\pi _{N}=(1-\frac{1}{\lambda })E^{W}\), which is increasing in *z* because of \(\lambda ^{\prime }(z)>0\). In such a case, \(j_{F}\) becomes a potential rival.

### 3.2 Offshoring

In this subsection, all firms in the North can undertake either R&D or offshoring to win the market. We then have two possible situations to discuss. One is that all firms in the North offshore their upstream divisions to the South. The alternative is that *J* stays in the North and continues engaging in R&D while \(j_{F}\) \((\le J-1)\) offshores the upstream division to the South for the the purpose of cost reduction.

In the first case, all firms engage in offshoring and face the same production costs. Similar to the case shown in Sect. 3.1, *J* still attains its superior quality and sets its price a little less than \(\lambda (1+w_{N}\tau )\) for its quality-adjusted price slightly lower than \(\frac{p}{q}=\frac{\lambda (1+w_{N}\tau )}{\lambda ^{J}}=\frac{(1+w_{N}\tau )}{\lambda ^{J-1}}\), which is the minimum value that the quality-follower \(j_{F}\) can provide. In this case, *J* defeats \(j_{F}\) and earns the instantaneous profit \(\pi _{O}=(1-\frac{1}{\lambda })E^{W}\). Also, \(\pi _{O}\) is increasing in *z*.

Turning to the second case, among the Northern firms facing the same production cost, *J*, possessing the leading-edge technology, is the most competitive firm. Likewise, among the offshoring firms facing the same production cost, \(j_{F}\), for example, is the most competitive firm. If *J* is defeated by \(j_{F}\), so are other Northern offshoring firms. On the other hand, if *J* defeats \(j_{F}\), *J* must defeat all offshoring firms whose quality is relatively low.

The success in R&D allows *J* to acquire a two-step quality advantage compared to the quality followers, and then *J* becomes the \(J+1\) version. The pricing strategy for the initial *J* is \(p(J)=\max \{\lambda ^{(J+1)-(J-1)}(1+w_{N}\tau ),\) \(w_{N}(1+\gamma )\}=\max \{\lambda ^{2}(1+w_{N} \tau ),\) \(w_{N}(1+\gamma )\}\), where the first term is the minimum quality-adjusted price that \(j_{F}\) may provide, and the second term is the minimum marginal cost. The pricing strategy for the nearest rival \((J-1)\) is \(p(j_{F})=\max \{\frac{w_{N}(1+\gamma )}{\lambda ^{2}},\) \(1+w_{N}\tau \}\), where the first term is the minimum quality-adjusted price that *J* can set, and similarly, the second term is the minimum marginal cost faced by the offshoring firms. If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>1+w_{N}\tau \), \(j_{F}\) gains cost advantages through offshoring and defeats *J*; otherwise, *J* dominates the market due to its superior quality. Consequently, if \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>1+w_{N}\tau \), dominant offshoring firm \(j_{F}\)’s instantaneous profit is derived as \(\pi _{O}=[1-\phi ]E^{W}\), where \(\phi =\frac{\lambda ^{2}(1+w_{N}\tau )}{w_{N}(1+\gamma )}\). If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<1+w_{N}\tau \), dominant R&D firm *J*’s instantaneous profit is derived as \(\pi _{N}=[1-\frac{1}{\phi }]E^{W}\).

Comparing profits between R&D and offshoring strategies, we characterize the industry leader *J*’s strategies against \(j_{F}\)’s offshoring in the following lemma:

### Lemma 1

The strategy for industry leader *J* in response to the nearest follower \((J-1)\)’s cost-saving offshoring is to undertake (i) offshoring if \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<1+w_{N}\tau <\frac{w_{N}(1+\gamma )}{\lambda }\) or (ii) R&D if \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<\frac{w_{N}(1+\gamma )}{\lambda }<1+w_{N}\tau \).

### 3.3 FDI

Besides R&D and offshoring, in this subsection, FDI is placed as the third option for firms’ consideration. We then have three further scenarios for discussions. First, all firms in the North directly invest in the South. Second, *J* stays in the North and continues engaging in R&D while \(j_{F}\) \((\le J-1)\) undertakes FDI in the South for the cost advantage. The third case is that *J* offshores the upstream division to the South while \(j_{F}\) \((\le J-1)\) undertakes FDI in the South. Specifically, when \(j_{F}\) engages in FDI in the South, \(J-1\) becomes the superior quality firm in the South. In this subsection, we define \(J^{*}\) \((\equiv J-1)\) as being the version of the FDI leader in cases 2 and 3 above. In other words, \(J^{*}\) in such cases is the highest version that has been successfully transferred to the South.

Concerning the first case, all firms undertake FDI and face the same production cost. As we have studied in Sect. 3.2, *J* still attains its superior quality and sets its price a little less than \(2\lambda \) in order to keep its quality-adjusted price slightly lower than \(\frac{p}{q}=\frac{2\lambda }{\lambda ^{J}}=\frac{2}{\lambda ^{J-1}}\). Again, \(\frac{2}{\lambda ^{J-1}}\) is the minimum quality-adjusted price that the quality-followers can provide. In this case, *J* defeats \(j_{F}\), and then earns the instantaneous profit, \(\pi _{F}=(1-\frac{1}{\lambda })E^{W}\), which is increasing in *z*.

The second and third cases are analogous to the above second case in Sect. 3.2. If *J* is defeated by \(J^{*}\), so are other FDI firms; on the other hand, if *J* defeats \(J^{*}\), *J* has to defeat all FDI firms whose quality is relatively lower than that of *J*. With a successful R&D, *J* acquires a two-step quality advantage over all the followers and becomes the \(J+1\) version. The pricing strategy for the initial *J* is \(p(J)=\max \{2\lambda ^{2},w_{N}(1+\gamma )\}\). The pricing strategy for \(J^{*}\) is \(p(J^{*})=\max \{\frac{w_{N}(1+\gamma )}{\lambda ^{2}},2\}\), where the first term is the minimum quality-adjusted price that *J* can set, and similarly, the second term is the least marginal cost for FDI firms. If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>2\), \(J^{*}\) gains cost advantages through FDI and defeats *J*; otherwise, *J* dominates the market due to the superior quality. Accordingly, if \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>2\), dominant FDI firm \(J^{*}\)’s instantaneous profit is derived as \(\pi _{F}=[1-\frac{2\lambda ^{2}}{w_{N}(1+\gamma )}]E^{W}\). If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<2\), dominant R&D firm *J*’s instantaneous profit is derived as \(\pi _{N}=[1-\frac{w_{N}(1+\gamma )}{2\lambda ^{2}}]E^{W}\).

Furthermore, if *J* chooses to offshore the upstream division to the South rather than R&D or FDI, then the pricing strategies of *J* and \(J^{*}\), \(p(J)=\max \{2\lambda ,1+w_{N}\tau \}\) and \(p(J^{*})=\max \{\frac{1+w_{N}\tau }{\lambda },2\}\), will sufficiently determine the outcomes in the pricing game. If \(\frac{1+w_{N}\tau }{\lambda }>2\), \(J^{*}\) defeats *J*; otherwise, *J* dominates the market due to the successful offshoring. Therefore, if \(\frac{1+w_{N}\tau }{\lambda }>2\), dominant FDI firm \(J^{*}\)’s instantaneous profit is derived as \(\pi _{F}=[1-\frac{2\lambda }{1+w_{N}\tau }]E^{W}\). If \(\frac{1+w_{N}\tau }{\lambda }<2\), dominant offshoring firm *J*’s instantaneous profit is derived as \(\pi _{O}=[1-\frac{1+w_{N}\tau }{2\lambda }]E^{W}\).

*J*if \(J^{*}\) chooses FDI as its strategy in Table 2:

*J*’s payoffs given \(J^{*}\)’s FDI

Payoffs of | ||
---|---|---|

R&D | Offshoring | FDI |

If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>2, \pi _{N}(J)=0\) | If \(\frac{1+w_{N}\tau }{\lambda }>2\), \(\pi _{o}(J)=0\) | \(\pi _{F}(J)=\left[ 1-\frac{1}{\lambda }\right] E^{W}\) |

If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<2,\quad \pi _{N}(J)=\left[ 1-\frac{w_{N}(1+\gamma )}{2\lambda ^{2}}\right] E^{W}\) | If \(\frac{1+w_{N}\tau }{\lambda }<2,\quad \pi _{o}(J)=\left[ 1-\frac{1+w_{N}\tau }{2\lambda }\right] E^{W}\) |

In other words, in the case where \(J^{*}\) undertakes FDI in the South, we can show that (i) if \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>2\) and *J* chooses to engage in R&D, then \(J^{*}\) can earn \([1-\frac{2\lambda ^{2}}{w_{N}(1+\gamma )}]E^{W}\) while *J* earns zero profit; (ii) if \(\frac{1+w_{N}\tau }{\lambda }>2\) and *J* undertakes offshoring, then \(J^{*}\) can earn \([1-\frac{2\lambda }{1+w_{N}\tau }]E^{W}\) while *J* earns zero profit.

Supposing \(J^{*}\) engages in FDI, we obtain pure optimal strategies for *J* in the following lemma:

### Lemma 2

In the case where followers undertake FDI, the best strategy for quality leader *J* is (i) FDI if \(\lambda >1\), \(\frac{w_{N}(1+\gamma )}{1+w_{N}\tau }>1\) and \(\frac{w_{N}(1+\gamma )}{\lambda }>2\); (ii) R&D if \(\lambda>\frac{w_{N}(1+\gamma )}{1+w_{N}\tau }>1\) and \(2>\frac{w_{N}(1+\gamma )}{\lambda }\); (iii) offshoring if \(\frac{w_{N} (1+\gamma )}{1+w_{N}\tau }>\lambda >1\) and \(2>\frac{w_{N}(1+\gamma )}{\lambda }\).

### 3.4 Strategic form game and the equilibria

*J*and \(J-1\) for the optimum in the single-period

*t*. The following table illustrates all the payoffs we obtained in Sect. 3.

\(3\times 3\) strategic form game

( | ||||
---|---|---|---|---|

R&D | Offshoring | FDI | ||

( | R&D | \(\pi _{N}=\left[ 1-\frac{1}{\lambda }\right] E^{W} \hbox { and } \pi _{N}(j)=0\) | \(\pi _{o}(J)=\left[ 1-\frac{1}{\lambda }\right] E^{W} \hbox { and } \pi _{N}(j)=0\) | \(\pi _{F} (J)=\left[ 1-\frac{1}{\lambda }\right] E^{W} \hbox { and } \pi _{N}(j)=0\) |

Offshorting | If \(\frac{W_{N}(1+\gamma )}{\lambda ^{2}}>1+w_{N}\tau ,\quad \pi _{N}(J)=0 \hbox { and } \pi _{o}(j)=[1-\phi ]E^{W}\) | \(\pi _{o}(J)=\left[ 1-\frac{1}{\lambda }\right] E^{W}\hbox { and } \pi _{o}(j)=0\) | \(\pi _{F}(J)=\left[ 1-\frac{1}{\lambda }\right] E^{W} \hbox { and } \pi _{o}(j)=0\) | |

If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<1+w_{N}\tau ,\pi _{N}(J)=\left[ 1-\frac{1}{\phi }\right] E^{W}\hbox { and } \pi _o(j)=0\) | ||||

FDI | If \(\frac{W_{N}(1+\gamma )}{\lambda ^{2}}>2,\quad \pi _{N}(J)=0\hbox { and } \pi _{F}(j)=\left[ 1-\frac{2\lambda ^{2}}{w_{N}(1+\gamma )}\right] E^{W}\) | If \(\frac{1+w_{N}\tau }{\lambda }>2,\quad \pi _{o}(J)=0\hbox { and }\pi _{F}(j)=\left[ 1-\frac{2\lambda ^{2}}{w_{N}(1+\gamma )}\right] E^{W}\) | \(\pi _{F}(J)=\left[ 1-\frac{1}{\lambda }\right] E^{W}\hbox { and }\pi _{F}(j)=0\) | |

If \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}<2, \pi _{N}(j)=\left[ 1-\frac{w_{N}(1+\gamma )}{2\lambda ^{2}}\right] E^{W}\hbox { and } \pi _{F}(j)=0\) | If \(\frac{1+w_{N}\tau }{\lambda }<2,\quad \pi _{o}(J)=\left[ 1-\frac{1+w_{N}\tau }{2\lambda }\right] E^{W}\hbox { and }\pi _{F}(j)=0\) |

Notice that in Table 3, \(j\in \{j_{F},J^{*}\}\), where \(j_{F}\le J-1\) and \(J^{*}\le J-1\); as we have defined, \(j_{F}\) denotes a follower engaging in R&D or offshoring while \(J^{*}\) denotes a follower directly investing in the South. Together with Lemmas 1 and 2, we evidently show that if \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau \), FDI is the weakly dominant strategy for *J*. Hence, *J* chooses FDI no matter how \(j_{F}\) behaves. If \(1+w_{N}\tau>2>\frac{w_{N}(1+\gamma )}{\lambda }\), R&D is the best strategy for *J*.

Furthermore, since FDI is the only strategy for \(j_{F}\) if \(\lambda \) is small enough such that \(\frac{w_{N}(1+\gamma )}{\lambda ^{2}}>2\), *J* will undertake FDI and then achieve an equilibrium. If \(1+w_{N}\tau>\frac{w_{N}(1+\gamma )}{\lambda }>2>\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\), three strategies are weakly dominant for \(j_{F}\). On the one hand, *J* must engage in R&D when \(j_{F}\) chooses either R&D or offshoring. On the other hand, if \(j_{F}\) chooses FDI, *J* then engages in FDI. Therefore, we can conclude that if \(1+w_{N}\tau>\frac{w_{N}(1+\gamma )}{\lambda }>2\), (FDI, FDI), (R&D, Offshoring), and (R&D, R&D) are the Nash equilibria for players (*J*, *j*).

In the present Bertrand game, the version/firm with the lowest quality-adjusted price will capture the entire world demand. Intuitively, the cost difference will play the key role for firms to undertake the optimal strategies. As we have discussed above, if a Northern firm moves totally/partially production lines to the South, it is reasonable to conclude that it will reduce more production costs than the scenario in which it produces in the North.

We thus characterize the findings in the following proposition:

### Proposition 1

In time *t*, (i) if \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau \), FDI is the weakly dominant strategy for *J*; (ii) if \(1+w_{N}\tau>2>\frac{w_{N}(1+\gamma )}{\lambda }\), R&D is the dominant strategy for *J*; (iii) if \(1+w_{N}\tau>\frac{w_{N}(1+\gamma )}{\lambda }>2\), (FDI, FDI), (R&D, Offshoring), and (R&D, R&D) will be the Nash equilibria.

## 4 The formation of product cycles in an infinite-horizon game

\(\lambda (z,y)=\lambda (z)\), \(\lambda (0)=1\), and \(\lambda ^{\prime }(z)>0\). Since \(\lambda \) is monotonically increasing in *z*, a high *z* induces a high \(\lambda \). It means that high-tech industries produce high-quality final products and low-tech industries produce low-quality final products. We set a \(\overline{z}\), which makes \(\frac{w_{N}(1+\gamma )}{\lambda }\) less than 2. With a decrease in *z*, the value of \(\frac{w_{N}(1+\gamma )}{\lambda }\) increases. Thereby, there exists a \(\underline{z}\) such that \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau \). In general, any *z* in the range \((\underline{z},\overline{z})\) will result in \(1+w_{N}\tau>\frac{w_{N}(1+\gamma )}{\lambda }>2\).

### 4.1 High-tech industries

For any high-tech product \(z_{H}\in [\overline{z},1]\), we have that \(1+w_{N}\tau>2>\frac{w_{N}(1+\gamma )}{\lambda (z_{H})}\). Proposition 1 reveals that the leading firm \(J(z_{H})\) will engage in R&D at every *t* against followers by setting price \(p(J;z_{H})=\lambda w_{N}(1+\gamma )\). The premium charged reflects consumers’ willingness to pay for a higher-quality version than what the nearest follower can produce. Subsequently, \(J(z_{H})\) sells \(X(J;z_{H})=\frac{E^{W}}{\lambda w_{N} (1+\gamma )}\) with a positive instantaneous profit, \(\pi _{N}=(1-\frac{1}{\lambda })E^{W}\). Although \(j_{F}(z_{H})\) may try either moving up for the quality improvements or moving out for the cost advantages, \(j_{F}(z_{H})\) is not able to defeat the leader’s dominant position as long as \(J(z_{H})\) succeeds in R&D. Therefore, without the occurrence of imitation, in high-tech industries, the production of leading-edge products agglomerates in the North. This is equivalent to Krugman’s (1979) argument: the North has an incentive for continuous innovation in order to maintain its dominant position.

*dt*. We let \(\pi _{N}dt\) be the flow of profits and \(\iota dt\) be a probability of success in R&D. Then the expected present value of the profits earned by the dominant firm in a representative industry \(z_{H}\) is

*z*.

Lu (2007) showed that productions of leading-edge products merely agglomerate in the North since all firms in high-tech industries only consider R&D as their strategy. However, her arguments can not sufficiently explain why high-tech firms like Apple corporate, for instance, move their production lines to East Asia or to emerging countries. To study the practices such as Apple did, we address a possible scenario showing that a high-tech firm can apply a strategy combination involving both R&D and FDI (or offshoring). In other words, the leader \(J(z_{H})\) can undertake a moving-out strategy after every successful R&D phase and still maintains market dominance.

Without considering any transformation and transportation costs, we show a case of strategy combination: \(J(z_{H})\) engages in R&D at \(t_{1}\), then undertakes FDI/offshoring at \(t_{2}\), then engages in R&D at \(t_{3}\), then undertakes FDI/offshoring at \(t_{4}\), and so on. At even *t* such as \(t_{2},t_{4},t_{6},t_{8},\ldots \), *J* still dominates the market even if a follower succeeds in achieving a one-step quality upgrade and \((J-1)\) upgrades to be *J*.^{16} Since initial leader *J* gains a cost advantage through FDI/offshoring, it can set a much lower quality-adjusted price than initial follower \((J-1)\) can. If so, *J*’s expected present value of profits can be derived as \(V_{R2}(z_{H})\iota _{2}dt+V_{F}(z_{H})\iota _{F}dt\) (if *J* undertakes FDI) or \(V_{R2}(z_{H} )\iota _{2}dt+V_{O}(z_{H})\iota _{O}dt\) (if *J* undertakes offshoring), where \(V_{R2}\) and \(\iota _{2}\) denote the expected present value of profits and the R&D intensity in the case wherein the industry leader invests in R&D at odd *t*.

*J*’s \(\iota _{2}(z_{H})\) and \(\iota _{F}(z_{H})\):

^{17}Mixing two strategies,

*J*still achieves the same expected profits as the case where

*J*only undertakes R&D. In such the mixing strategy, a product cycle emerges. The analogous results can be obtained in the offshoring case. This intuitively explains why high-tech industries are willing to move their production lines outside of the North. Noticeably, moving-out is one of strategies in the pool.

*J*still cannot cease seeking quality-upgrading R&D in all periods; otherwise, it will be displaced by the nearest rival at some point as we have discussed in the Sect. 3.4.

^{18}Furthermore, if initial follower \((J-1)\) insists on conducting R&D at all times, then initial leader

*J*must engage in R&D in order to guarantee its lower quality-adjusted price in the market. Otherwise,

*J*may lose the market at some point.

We characterize the findings in the following proposition:

### Proposition 2

Based on \(\frac{a_{R}}{a_{F}^{*}}>1\), if high-tech industry leader \(J(z_{H})\) in an infinite-horizon game can achieve the same expected present value of profits in use of a mix of moving-up and moving-out strategies as that in the case where \(J(z_{H})\) always engages in R&D, then product cycle emerges even no imitation is considered.

### 4.2 Medium-tech industries

*J*or \(j_{F}\). (FDI, FDI) is therefore the Nash equilibrium. If so, for such industries, production of leading-edge products agglomerates in the South. The expected present value of profits earned by dominant firm

*J*in this specific industry \(z_{M}\) is

*J*, R&D strongly dominates offshoring and FDI. In this case,

*J*monopolizes the market with a low quality-adjusted price. The expected present value of profits earned by dominant firm

*J*in a representative industry \(z_{M}\) is shown as

*J*, R&D weakly dominates FDI and offshoring. For this case, we already have shown two similar discussions in the section of high-tech industries. First, no matter what \(j_{F}(z_{M})\) conducts either moving-up or moving-out strategies, it cannot defeat the leader’s dominant position as long as \(J(z_{M})\) succeeds in R&D. Therefore, in such medium-tech industries, the production of leading-edge products agglomerates in the North. Equations (18) and (19) thus express the expected present value of profits earned by dominant firm

*J*in the industry \(z_{M}\) and the R&D intensity of the dominant leader, respectively.

For *J* in the present case, R&D weakly dominates FDI and offshoring. In use of strategy combination, we consider a situation wherein \(J(z_{M})\) engages in R&D at odd *t*, then undertakes FDI/offshoring at even *t*, which is similar to the case studied in the section of high-tech industries. Accordingly, we can solve the expected present value of profits of the initial *J* as \(V_{R2}(z_{M})\iota _{2}dt+V_{F}(z_{M})\iota _{F}dt\) (if *J* undertakes FDI) or \(V_{R2}(z_{M})\iota _{2}dt+V_{O}(z_{M})\iota _{O}dt\) (if *J* undertakes offshoring). Recall that \(V_{R2}\) and \(\iota _{2}\) denote the expected present value of profits and the R&D intensity in the case wherein the industry leader invents R&D in odd *t*. Similar to Eqs. (12) and (13), we hereby take into account the case of FDI for example. Then the expected present values obtained in cases of FDI and R&D are, \(V_{F}(J;z_{M})\iota _{F}dt\) and \(V_{R2}(J;z_{M})\iota _{2}dt\), respectively.

While \(j_{F}\) engages in R&D, \(\pi _{N}(J)=\pi _{F}(J)=(1-\frac{1}{\lambda (z_{M})})E^{W}\). It follows that \(V_{R1}(z_{M})\iota _{1} dt=V_{R2}(z_{M})\iota _{2}dt+V_{F}(z_{M})\iota _{F}dt\). Therefore, \(\iota _{F}(z_{M})=\frac{(1+\rho )}{\left( 2+\rho \right) \rho a_{F}^{*}} [(1-\frac{1}{\lambda (z_{M})})E^{W}]<\iota _{F1}(z_{M1})\) and \(\iota _{2} (z_{M})=\frac{(1+\rho )^{2}}{\left( 2+\rho \right) \rho w_{R}a_{R}} [(1-\frac{1}{\lambda (z_{M})})E^{W}]<\iota _{1}(z_{M})\).

In case of (FDI, FDI), FDI is the only strategy for firms, that is, they intuitively invest in FDI with high intensity compared to the firms in case of (R&D, R&D). In case of (R&D, R&D), we know that R&D is a weakly dominant strategy for the quality leader *J*. When *J* mixes R&D and FDI strategies, firms will distribute their resources on both such that \(\iota _{F} (z_{M})<\iota _{F1}(z_{M1})\) and \(\iota _{2}(z_{M})<\iota _{1}(z_{M})\). Similarly, the strategy combination results in product cycles. As the case in high-tech industries, the initial leaders cannot renounce R&D processes at all times; otherwise, they would be taken over by their nearest rivals. In the similar manner, (R&D, Offshoring) incurs product cycles, as does (R&D, R&D).

### Proposition 3

In medium-tech industries, (FDI, FDI), (R&D, Offshoring) and (R&D, R&D) are the Nash equilibria. Among the equilibria, (R&D, Offshoring) and (R&D, R&D) in an infinite-horizon game may result in product cycles.

### 4.3 Low-tech industries

For any low-tech product \(z_{L}\in [0,\underline{z}]\), we have \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau \), and then FDI is the weakly dominant strategy for *J*. Hence, *J* chooses FDI in the first stage *t* and produces in the South no matter what strategy \(j_{F}\) chooses. If \(j_{F}\) undertakes offshoring or FDI, *J* still holds the advantage with the low quality-adjusted price and keeps occupying the market. Thus, \(\pi _{F}(J)=(1-\frac{1}{\lambda })E^{W}\). If \(j_{F}\) engages in R&D and acquires any success in its patent, \((J-1)\) then moves its quality up to *J*. In this case, the initial *J* still monopolizes the market because of the cost advantage. However, if \(j_{F}\) continues succeeding in R&D in the second stage, the initial \((J-1)\) will become \((J+1)\).

We further consider \(2>\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\) such that \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau>2>\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\). In detail, the new \((J+1)\) in the third stage moves outside to seek lower production costs no matter what strategy the initial *J* chooses. It can be seen that the new \((J+1)\) defeats the initial *J* and earns the profit: \(\pi _{N}(J+1)=[1-\frac{w_{N}(1+\gamma )}{2\lambda ^{2}}]E^{W}\). In other words, the new \((J+1)\) monopolizes the market and forces *J* to carry out new R&D in the North. Recall that only the North has skilled workers for the process of R&D. In the fourth stage, the initial *J* upgrades its quality as the new \((J+2)\) and then recaptures the market back from \((J+1)\). Subsequently, firms leapfrog over each other; if the game repeats,^{19} then product cycles always exist. This explains why some low-tech industries have product cycles but not all of them have the cycles.

Moreover, in the case of \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau >\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\) \(>2\), even if the new \((J+1)\) owns the leading-edge version in the third period, the initial *J* still monopolizes the market because of its significantly large cost advantage. Notice that, in the case of \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau \), FDI is always the weakly dominant strategy for the dominant firm.

At present, we have two critical scenarios: (i) if \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau >\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\) \(>2\), the initial *J* monopolizes the market because of cost advantages; (ii) if \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau>2>\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\), product cycles emerge.

*z*.

*J*, the expected present values in cases of FDI and R&D are, respectively,

*J*are derived as

Moreover, in the case of \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau>2>\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\), we obtain that \(\pi _{F}>\pi _{N}\). The expected present value of the initial *J*’s profits is \((1+\rho )^{4}\pi _{F}+(1+\rho )\pi _{N}\). It is definitely larger than the expected present value of the initial \(j_{F}\)’s profits, \((1+\rho )^{2}\pi _{F}+(1+\rho )^{3}\pi _{N}\). Specifically, if \(\pi _{N}=\left( 1+\rho \right) \pi _{F}\), we can argue that the expected present values of the initial *J* and \(j_{F}\) are the same.

We then characterize the findings in the following proposition:

### Proposition 4

In low-tech industries, if \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N}\tau >\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\) \(>2\), (FDI, FDI) is a Nash equilibrium and no product cycles exist. If \(\frac{w_{N}(1+\gamma )}{\lambda }>1+w_{N} \tau>2>\frac{w_{N}(1+\gamma )}{\lambda ^{2}}\), then \(\iota (j_{F};z_{L} )>\iota (J;z_{L})\) and \(\iota _{F}(j_{F};z_{L})<\iota _{F}(J;z_{L})\), and product cycles may occur.

## 5 Concluding remarks

In real world scenarios, product cycles occur frequently across almost all industries. To dominate a market, a firm can take advantage of either quality upgrades through R&D or cost saving through the channels of market penetration. These strategic options have been commonly observed in all stages of economic development. Due to differences in the intensity of investments and firms’ dominant strategies, an industry in a developed country may move to a developing country and later move back.

In this paper, we envisage a model of a quality ladder in North–South trade to examine the formation of product cycles in high-, medium-, and low-tech industries. Incorporating the strategic form game, we identify the optimal short-run strategies for the producers of state-of-the-art products and of the second-to-top products. We show that cost differentials can give rise to the Nash equilibria and then product cycles in any industry. In an infinite-horizon game, we further show in detail that (i) in high-tech industries, the investment in R&D at all times is a dominant strategy for the quality leader firm. If the leading firm mixes the moving-up and moving-out strategies, product cycles occur. (ii) In medium-tech industries, if FDI is a strongly dominant strategy for both quality leaders and followers, then the production of leading-edge products agglomerates in the South. If a follower chooses to offshore its upstream division, then R&D for the leader strongly dominates FDI and offshoring; if a follower chooses R&D, then R&D for the leader weakly dominates FDI and offshoring. If the leader mixes the moving-up and moving-out strategies, product cycles occur in some cases. (iii) In low-tech industries, FDI is a strongly dominant strategy for both quality leading and following firms. Under certain conditions, product cycles emerge.

## Footnotes

- 1.
- 2.
According to “Statistics Sweden” (http://www.scb.se/en), China is now the 10th-largest trading partner of Sweden with 3.1% share of its total exports of goods. Compared to other nine major trading partners with Sweden, China is the one whose gross domestic product (at purchasing power parity) per capita is less than 15,000 international dollars (Int$). See, World Bank 2015 data source: http://data.worldbank.org/indicator/.

- 3.
Krugman’s work has been extended by Dollar (1986) and Jensen and Thursby (1986). Dollar maintained Krugman’s assumption of an exogenous rate of product innovation, but related the rate of technology transfer to the North–South terms of trade, albeit in an entirely

*ad hoc*manner. Jensen and Thursby assumed that all innovation is carried out by a single entrepreneur in the North, and that the allocation of resources in the South is performed by a social planner. - 4.
In addition to the variety-based model and the quality ladder model, other frameworks can be used to analyze product cycles. For instance, Zhu (2004) extended the Dornbusch–Fischer–Samuelson (DFS) model with Northern product innovation and product-cycle-driven technology transfers. Antràs (2005) incorporated the model of incomplete contracts into the product cycle issue.

- 5.
In contrast to the strong scale effects in the earlier literature, economic growth in their model is characterized by weak scale effects. See, also, Gustafsson and Segerstrom (2010).

- 6.
- 7.
Specifically, Lu’s (2007, p. 326) Figure 1 in her introduction utilized the measurement and data of product-cycle trade from Zhu (2005), and then illustrated the product-cycle trade intensity and the R&D intensity across industries. The figure evidently showed that product cycle emerges in industries in almost all levels. Notice that, in her figure, industries are arranged along the axis according to their R&D intensities with the most R&D-intensive industry located at the far right.

- 8.
As the second trand we have shown above, R&D in this paper is to upgrade quality for the existing varieties of products.

- 9.
- 10.
Although the wages might be endogenously influenced by the results of strategic choices between R&D and FDI/outsourcing, we assume that the wage rates are exogenously fixed as traditional trade theories for solving outcomes in the present fairly complicated model.

- 11.
This is equivalent to the Schumpeterian “process of creative destruction.” In each period, a firm enjoys its temporary monopoly power until a more inventive challenger appears. See, e.g., Reinganum (1985).

- 12.
In Lu (2007), she considered only the production of final goods. Thereby, a multinational firm’s instantaneous profit in her model is \(\pi _{F}=pX-X\) with the wage rate in the South being 1. See, also, Suzuki (2015). In the present paper, we specifically separate the production line by two divisions respectively on intermediate and final goods. It follows that a multinational firm’s cost in its instantaneous profit function is 2

*X*. Notice that each intermediate good is variety-specific by assumptions. - 13.
- 14.
- 15.
This cost can be seen as a training cost equivalently. To undertake R&D, a firm requires skilled labors, and then pays for upgrading workers’ skills.

- 16.
We simply assume that the follower after the first period

*t*learns and uses the same development path as the leader did, that is, move up, then move out, then move up, then move out, and so on. However, if the follower insists on undertaking R&D, then there is no chance for the leader to seek possible cost advantages*via*FDI or offshoring. Otherwise, the leader may lose the market at some point. - 17.
- 18.
For instance, if \(\frac{1+w_{N}\tau }{\lambda }>2\) and

*J*always undertakes offshoring and never engages in R&D, then*J*earns zero profit at the end. - 19.
In low-tech industries, a dominant firm considers FDI as its strategy until it no longer earns profit, and then invests in R&D. The game mentions the rotation of the undertaking of FDI and R&D.

## Notes

### Acknowledgements

We are indebted to the editor-in-chief, two referees, Haitao Mao, Wenshou Yan, Xiaopeng Yin and Qi Zhang for constructive comments and helpful suggestions. Also, we are grateful for many useful comments and discussions by conference participants at 2016 Asia Meeting of Econometric Society in Japan and 2017 Annual Meeting of Chinese Society of International Trade.

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