## Abstract

Technological leadership has shifted at various times from one country to another. We propose a mechanism that explains this perpetual cycle of technological leapfrogging in a two-country model including the dynamic optimization of an infinitely lived consumer. In the model, each country accumulates knowledge stock over time because of domestic innovation and spillovers from foreign innovation. We show that if the international knowledge spillovers are reasonably efficient, technological leadership may shift first from one country to another, and then alternate between countries along an equilibrium path.

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## Notes

See also Ohyama and Jones (1995), Motta et al. (1997), Brezis and Tsiddon (1998), van de Klundert and Smulders (2001), and Desmet (2002). The present paper essentially differs from those analyses in its focus on perpetual cycles of leapfrogging, thus complementing these works by clarifying the intrinsically cyclical nature of national technological leadership.

As argued by Brezis (1995), foreign capital plays a role in industrialization and development processes. We may also accept that international capital flows, as well as imports, are important channels for international knowledge spillovers, as discussed in the literature (Grossman and Helpman 1991; Feenstra 1996). See Branstetter (2006) for recent empirical evidence.

A time lag can be seen in history, in which the technologically leading country is not necessarily the most active foreign direct investor country in the same period. A formal investigation of such a time lag is left for future work.

See Athreye and Godley (2009), Giovannetti (2013), and Petrakos et al. (2005) for more recent research. In the political economy literature, Krusell and Rios-Rull (1996) provide an endogenous explanation for a long cycle of stagnation and growth, similar to perpetual leapfrogging in the present paper, by focusing on vested interests in determining policies. See also Aghion et al. (2001) for perpetual leapfrogging at the firm level.

Another line of research in this area examines growth cycles in a perfectly competitive setting (Drugeon 2013). In a different context, a number of papers investigate growth cycles in an overlapping generations model; recent contributions include, for instance, Kitagawa and Shibata (2005), Kaas and Zink (2007), Barnett and Bhattacharya (2008), Yang (2010), Barnett et al. (2013) and Iwaisako and Tanaka (2014). From a broader viewpoint, the market quality theory (Yano 2006, 2009) is also related, which stresses the importance of more drastic industrial revolution cycles through market quality fluctuations. However, leapfrogging does not exist in these studies.

See the discussion at the end of Sect. 3.3 on the use of a discrete-time model. See also footnote 15.

See, for instance, Nishimura et al. (2014), who show a destabilization effect of international trade in a discrete-time model of dynamic general equilibrium.

As is well known, the index is defined as \(P(t)=\left( \int _{j\in \{\varLambda ^{A}(t)\cup \varLambda ^{B}(t)\}}p(j,t)^{1-(1/\theta )}dj\right) ^{\frac{1}{1-(1/\theta )}}\).

As Romer (1990) explains, this simplified setting is only a convenience since “(w)hether the owner of the patent manufactures the good itself or licenses others to do so, it can extract the same monopoly profit.” One potential oversimplifying factor here is the lack of explicit and costly adoption of innovation, which should be a limiting factor for the analysis in a broader context. While a complete analysis on costly innovation adoption is beyond the scope of this paper, we can incorporate a process of costly adoption of innovation without essentially changing the results by using a very simple setting; see Furukawa (2014).

Following the literature (Romer 1990; Matsuyama 1999), we consider a deterministic innovation process for the sake of simplicity, although without any qualitative change in our results, we can consider a simple stochastic innovation process in which success probability for a firm to innovate a new good is endogenous and increases with the firm’s R&D investment (see Furukawa (2014)). However, if we assume that the time for each innovation to be completed was not fixed at one period but was stochastic, the analysis becomes intractable. We leave the question of how such stochastic timing of innovation impacts leapfrogging for future research.

In line with the literature on international trade and growth (Lai 1998), we do not distinguish between the various forms of production transfer, including fully and partly owned subsidiaries and licensing.

Here, we simply consider that efficiency in manufacturing normalizes across countries. We can extend this simple setting by allowing for country-specific manufacturing efficiency and endogenous technological progress. In such an extended model, we can easily verify that the comparative advantage between R&D and manufacturing (rather than the absolute advantage in R&D) plays an important role in perpetual leapfrogging, although there is no fundamental change in the results and their implications for perpetual leapfrogging.

This assumption implies that all patents start and expire at the same time, although in reality patents overlap. We may deal with this undesirable property by interpreting the length of a period as very long (e.g., 40 years) and dividing each period into subperiods (e.g., two 20-year periods), although we need a continuous-time model to completely fix this problem. In the present paper, we view leapfrogging as a discrete-time phenomenon and leave this issue for future work.

This assumption may also be justified if each innovation is interpreted as fairly specific. For example, “innovation” in this model would be represented by the specific innovation associated with iPhone 4S or smartphones instead of cell phones or information technology more generally.

Note that, in the next subsection, we assume that obsolete innovations stay “alive” in the sense that they continue to contribute to the current knowledge stock, although they are not explicitly traded in the marketplace. Whether they are traded or not is not important for our main story explained later.

One method for vertically capturing knowledge accumulation

*within the present setting*is to consider obsolescence of knowledge by assuming that knowledge accumulates as innovations*partially*replace old innovations, rather than simply being added to old innovations as in (8). For example, we can introduce a rate of knowledge destruction, say \(\delta \in [0,1]\), into (8), with which (9) would be revised to \(K^{i}(t+1)-K^{i}(t)=N^{i}(t)+\mu M^{i}(t)-\delta K^{i}(t)\). As long as the two countries have identical \(\delta \), we can demonstrate that our main result is robust to this extension. Otherwise, it would be possible to show a result similar to that of the present paper in a quality-ladder variant of the present model; see Furukawa (2014) for a formal explanation.Davids (2008) considered that a country that has technological leadership plays an initiating role in the development of new technologies across a wide variety of fields.

The model contains no zero-innovation equilibrium because the so-called Inada property is assumed in the constant elasticity of substitution utility function (1), which is standard in the literature.

Specialization pattern (2) implies that in equilibrium, the technology gap (\(K^{A} (t)/K^{B}(t)\)) is exactly equal to the wage ratio (\(w^{A}(t)/w^{B}(t)\)); otherwise, the countries would not be both engaging in R&D.

Here, we assume that the North is a country that innovates; however, if the North (the South) was defined as a country where the wage rate is higher (lower) as is also usual in the literature, these North/South labels could be misleading. Nevertheless, we use these labels because we can easily control the international wage differential in the present model by incorporating into the model an international differential in manufacturing productivity.

We do this by summing both sides of (12) over \(i\).

Note that the constant elasticity of substitution \(\theta ^{-1}\) is equal to the price elasticity of each good \(j\), which determines the markup ratio as \(1/(1-\theta )\).

Li (2001) argues that the evidence regarding whether there is any conventional value or a range of values for the elasticity of substitution is inconclusive. For example, Broda and Weinstein (2006) show that the elasticity of substitution is, on average, greater than two, but tends to decline over time and is actually less than two in some sectors (e.g., motor vehicles).

It is important to elaborate why the potential for an international wage differential is larger when \(\varTheta \) is larger. Given that the countries are identical except for \(K^{i}(t)\), the productivity gap in R&D (\(K^{A}(t)>K^{B}(t))\) is the only source for an international wage differential in our model. Thus, the potential for a wage differential is enhanced by a larger share of R&D investment, which naturally increases with the fundamental profitability of innovation \(\varTheta \). The larger the value of \(\varTheta \), the larger the potential for an international wage differential.

Note (7) with \(\varGamma ^{A}(t-1)\ne \emptyset \) and \(\varGamma ^{B}(t-1)=\emptyset \).

The formal definitions are:

$$\begin{aligned} a_{1}\equiv \begin{array} [c]{c} \frac{2L\varTheta }{1+\varTheta } \end{array} \ \text {and }a_{2}\equiv \begin{array} [c]{c} \frac{L\varTheta \left( 1+\varTheta \right) }{1+\varTheta \left( 2L+1\right) } \end{array}. \end{aligned}$$Only leading country \(A\) produces goods if and only if \(w^{A}(t)<w^{B}(t)\). This implies \(w^{A}(t)/K^{A}(t)<w^{B} (t)/K^{B}(t)\) with \(K^{A}(t)>K^{B}(t)\) in which only leading country \(A\) innovates. No labor demand exists in lagging country \(B\), which is inconsistent with the market-clearing condition.

The answer to the question of why the equilibrium wage differential between countries exists for \(\varTheta <1\) is similar and symmetric to that of the question why of the wages are equated for \(\varTheta <1\). See the discussion after Lemma 1.

See Furukawa (2014) for the formal proof. In Furukawa (2014), we also derive growth rates in the three regimes, which are all different in general. Specifically, the growth rate in the North–North regime can be higher or lower than the rates in other regimes, depending on the parameters and international technology ratio \(\psi (t)\).

Country \(A\) is assumed to be a leading country in period \(t-1\). Since we focus on an identical equilibrium path, \(K^{A}(t-1)\) must satisfy \(K^{A}(t)=\left( L+1\right) K^{A}(t-1)\) by (32), given \(K^{A}(t)\).

We may also relate \(\mu \) to more basic factors such as culture, morals, and ethics. Furukawa and Yano (2014) address the importance of a lagging country’s basic morals in international knowledge spillovers from a market quality perspective (see Dastidar and Dei (2014) for recent papers in the field of market quality economics).

See Appendix for the derivations.

The formal analysis is available from the author upon request.

This issue is intensively investigated in the literature on innovation cycles (see Shleifer 1986).

See Furukawa (2014).

We show this in Furukawa (2014).

See Carolan et al. (1998).

By (2), we can derive \(x^{B}(t)=\left( 1-\theta \right) E(t)/(N(t)w^{B}(t))\). Substituting this into \(L=N(t)x^{B}(t)\) yields this expression.

By using the labor market condition for lagging country \(B\) and the free-entry condition, we can easily verify that

$$\begin{aligned} \frac{K^{A}(t)}{K^{B}(t)}>\frac{w^{A}(t)}{w^{B}(t)}=\frac{\theta }{1-\theta }>1. \end{aligned}$$holds.

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The author acknowledges the hospitality and support of Simon Fraser University, where some of the work in this paper was completed. He would like to thank the two anonymous referees and Timothy Kehoe (Co-Editor) for their helpful suggestions and valuable advice. He is also grateful to Taro Akiyama, Gadi Barlevy, Eric Bond, Kenneth Chan, Fumio Dei, Patrick Francois, Takamune Fujii, Sam Gamtessa, Tetsugen Haruyama, Jun-ichi Itaya, Ronald Jones, Seokho Kim, Kozo Kiyota, Takashi Komatsubara, Yoshimasa Komoriya, Jiang Li, Yan Ma, Sugata Margit, Junya Masuda, Tsukasa Matsuura, Hiroshi Mukunoki, Takeshi Ogawa, Takao Ohkawa, Masayuki Okawa, Rui Ota, Yuki Saito, Hitoshi Sato, Kenji Sato, Yasuhiro Takarada, Hirokazu Takizawa, Yoshihiro Tomaru, Eiichi Tomiura, Makoto Yano, Taiyo Yoshimi, Eden Yu, Ryuhei Wakasugi, and conference/seminar participants at the European Economic Association Annual Congress 2013, the Western Economic Association International Pacific Rim Conference 2013, the Canadian Economics Association Annual Conference 2014, the Japan Society of International Economics Chubu Spring Meeting 2014, Chukyo University, Chuo University, Kobe University, and Yokohama National University for their helpful comments and advice on earlier versions of the paper. The partial financial support of a Grant-in-Aid for Young Scientists (B) #23730198/26780126 is gratefully acknowledged.

## Appendix

### Appendix

*Proof of Lemma 1* The necessity has been shown in the text. Namely, \(\varTheta <1\) must hold if both countries produce, recalling that only the leading country innovates in equilibrium if both countries manufacture. Here, we prove sufficiency, i.e., both countries produce in equilibrium if \(\varTheta <1\). For this purpose, let us prove the contrapositive: we show \(\varTheta >1\) must hold if only one country manufactures. There are two cases. (a) If only leading country \(A\) manufactures in equilibrium, \(w^{A} (t)<w^{B}(t)\) must hold. With \(K^{A}(t)>K^{B}(t)\), this implies \(w^{A} (t)/K^{A}(t)<w^{B}(t)/K^{B}(t)\), under which only leading country \(A\) innovates owing to its lower R&D cost. There is no labor demand in country \(B\), so (a) cannot be an equilibrium. (b) If only lagging country \(B\) manufactures in equilibrium, \(w^{A}(t)>w^{B}(t)\) must hold. Case (b) has three sub-cases. (b-i) Only lagging country \(B\) innovates, which is inconsistent for the same reason as in (a). (b-ii) If only leading country \(A\) innovates in equilibrium, \(w^{A}(t)/K^{A}(t)<w^{B}(t)/K^{B}(t)\) must hold. The labor market condition (12) implies \(N^{A}(t)=LK^{A}(t)\) and \(w^{B}(t)=\left( 1-\theta \right) E(t)/L\).^{Footnote 40} The free-entry condition requires \(V^{A}(t)=0\) because innovation takes place in equilibrium for country \(A\), which implies \(w^{A}(t)=\beta \theta K^{A}(t)E(t)/N(t+1)\).^{Footnote 41} Noting \(N(t+1)=N^{A}(t)\) in this case, \(w^{A}(t)>w^{B} (t)\ \)implies \(\varTheta >1\), not \(\varTheta <1\). (b-iii) If both countries innovate in equilibrium, it must be by the free-entry condition that \(V^{A} (t)=V^{B}(t)=0\) in equilibrium, which implies \(w^{A}(t)/K^{A}(t)=w^{B} (t)/K^{B}(t)=\frac{\beta \theta E(t)}{N(t+1)}\).^{Footnote 42} Using this and \(x^{B} (t)=\frac{(1-\theta )E(t)}{N(t)w^{B}(t)}\) from (4), \(N^{B}(t)=( \varTheta K^{B}(t)-K^{A}(t) ) L/(1+\varTheta )\) is derived from (12). Since \(N^{B}(t)>0\), then \(\varTheta >1\) must hold, noting \(K^{A}(t)>K^{B}(t)\). We have proven that if \(\varTheta <1\), both countries must manufacture in equilibrium; thus, only the leader innovates.

We can easily prove Lemmata 2 and 3 in an analogous way (see Furukawa (2014) for a formal proof).\(\square \)

Derivations for (37)

In the North–North regime: there are two cases (A) and (B). (A) Assume \(\frac{\theta }{1-\theta }>\frac{K^{A}(t)}{K^{B}(t)}>1\) (i.e., \(\psi (t)\in \left( 0.5,\theta \right) \)), where country \(A\) is a leading country. The free-entry condition (27) becomes

Given the labor condition for leading country \(A\), we have

By the labor condition for the lagging country, \(L=N^{B}(t)/K^{B} (t)+(N^{A}(t)+N^{B}(t))x^{B}(t)\), with \(x^{B}(t)=\frac{(1-\theta )E(t)}{\left( N^{A}(t)+N^{B}(t)\right) w^{B}(t)}\), we thus have

in which \(w^{B}(t)\) is eliminated using (38). Noting \(M^{B}(t)=N^{A}(t)\) here, by (9), we have the dynamic system as follows:

Note that \(\theta >0.5\) holds in the North–North regime. (B) Assume \(\frac{1-\theta }{\theta }<\frac{K^{A}(t)}{K^{B}(t)}<1\) (i.e., \(\psi (t)\in \left( 1-\theta ,0.5\right) \)), where country \(B\) is a leading country. Due to the symmetry,

hold. We can also derive

In the full North–South regime, there are two cases (C) and (D). (C) Assume \(\frac{K^{A}(t)}{K^{B}(t)}>\frac{\theta }{1-\theta }>1\) (i.e., \(\psi (t)\in \left( \theta ,1\right) \)). The leading country innovates following

The lagging country receives spillovers \(\mu M^{B}(t)=\mu N^{A}(t),\ \)with \(\mu \le 1\).^{Footnote 43} Then, the knowledge dynamics are as follows:

(D) Assume \(\frac{K^{A}(t)}{K^{B}(t)}<\frac{1-\theta }{\theta }<1\) (i.e., \(\psi (t)\in \left( 0,1-\theta \right) \)). Due to the symmetry, \(N^{A}(t)=0\) and \(N^{B}(t)=LK^{B}(t)\), with \(\mu M^{A}(t)=\mu N^{B}(t)\). We can easily have

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Furukawa, Y. Leapfrogging cycles in international competition.
*Econ Theory* **59**, 401–433 (2015). https://doi.org/10.1007/s00199-014-0850-y

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DOI: https://doi.org/10.1007/s00199-014-0850-y