Technology-gap and cumulative growth: models, results and performances

Transcription

1 Paper to be presented at the DRUID Winter Conference, Aalborg, January 17-19, 2002 Technology-gap and cumulative growth: models, results and performances Fulvio Castellacci Centre for Technology, Innovation and Culture (TIK), Forskningsparken, Gaustadallen 21, Po. Box 1108, 031 Oslo, Norway Phone: (+47) * I would like to thank all the participants to the AITEG meeting in Madrid and to the ETIC Final Conference in Strasbourg who gave helpful comments and suggestions on a previous draft of the paper. All the remaining errors are mine only.

2 Technology-gap and cumulative growth: models, results and performances Abstract Addressing the question why productivity growth rates differ between countries from a dis-equilibrium standpoint, in the present paper it is explored the possibility to compound in a single formalization two different but complementary theories of technical change and macroeconomic growth, that is the Kaldorian idea of cumulative causation and the technology-gap approach to economic growth. In order to investigate the complementarities between these two approaches, a two-country macroeconomic model of technology-gap and cumulative growth is presented, built on previous contributions by Verspagen (1993), Amable (1993) Targetti and Foti (1997) and Ledesma (1999a). The analytical solutions of the model for the growth rates of productivity and demand, and the dynamics of the technology-gap show the existence of a large set of possible outcomes: the follower country can fall behind, partly or totally catch up, or overtake the leader. Moreover, even if the follower is able to close the technology-gap, not necessarily it will also be able to close the growth rate differential. The empirical evidence on the experience of 26 OECD countries in the period points out the relevance of the model for explaining the recent performances of technological activities and productivity growth. 1

3 1. The Kaldorian idea of cumulative causation and the technology-gap approach : two complementary theories of macroeconomic growth Why do productivity growth rates differ between countries? In the last fifteen years there has been a renewed interest and a great effort to answer this question, after many scholars realized that the neoclassical theory of growth (Solow, 1956) was not able to explain the persistent differences in growth rates and in productivity levels between countries and regions observed in the world. The most recent answer within the neoclassical approach has been given by new growth theory models (Romer, 1986 and 1990; Lucas, 1988), which assuming a general equilibrium setting and neoclassical microfoundations allow for the possibility of divergent growth as a result of an endogenous process of national accumulation of knowledge and technology, in which national specific factors may explain why some countries grow faster than others. A more complete answer has been given by a large set of theories, which approach the analysis of convergence/divergence from a much broader perspective, stressing the fact that economic growth is a complex process of transformation, not of simple transition on a steady state growth path. Such a process of transformation or structural change over time is shaped by the interactions between technology, institutions and social factors. The complexity of these interactions makes modelling exercises more difficult, and many different theoretical models have pursued such dis-equilibrium approach. Instead of stressing the differences between these explanations of economic growth as a disequilibrium process, this paper explores their complementarities and the possibility to link them. More precisely, the theoretical model presented in the next sections tries to combine in a single formalization two complementaries theories of macroeconomic growth, that is the Kaldorian idea of cumulative causation and the technology-gap approach to economic growth. Built on the original ideas of Myrdal and Kaldor, and further developed by Dixon and Thirlwall (1975) and Boyer (1988), the Kaldorian cumulative growth model is based on two main elements: a causal link of a Keynesian kind between growth in demand and growth in production, and a process of interaction between growth of demand and growth of productivity. The latter is developed through two distinct causal sequences. On the one hand, it is assumed the existence of Kaldor-Verdoorn returns to scale, through which the causal connection between growth in demand and growth in productivity is made explicit 2

4 ( productivity regime 1 ). On the other hand, the same productivity growth determines growth in demand, through the positive influence that it may have on exports by raising the price competitiveness of national products on foreign markets ( demand regime ). With regard to the causal relationship from growth in demand to growth in productivity ( productivity regime ), its sources can be different: static increasing returns to scale; the deepening of the division of labour due to the expansion of the market; technical advances embodied into specific equipment and machine tools; learning by doing and, by extension, learning by using. Recent works by Fingleton and McCombie (1998), Pini (1996; 1997) and Vivarelli (1995) show that Verdoorn law can explain a part of the average productivity increases, both at a national and regional level in many OECD countries. On the other hand, the causal relationship from growth in productivity to growth in demand ( demand regime ) is based on an external causation mechanism, built on the medium- and long-term dynamics of the foreign component of the aggregate demand, exports. These are influenced by external exogenous factors, such as the evolution of foreign markets and the price and non-price competitiveness of foreign goods, and by internal factors, such as the dynamics of the exchange rate and productivity gains. The latter affects the competitiveness of national products on foreign markets, besides the results of the innovation process, which influence the non-price competitiveness of national products. In this Kaldorian approach to cumulative growth, technological change takes on an important role in the determination of the dynamics of production, demand and productivity. The intensity, bias and results of the innovation process, together with the dynamic returns to scale, not only trace the growth path of labour productivity, but also set off important external mechanisms by stimulating the export dynamics. However, the interactions between productivity and demand are not the only mechanisms of growth. Many countries have been able to catch-up by imitating and using the technologies developed in more advanced countries; the diffusion of knowledge and technologies from abroad is therefore an important and complementary source of growth that needs to be taken into account. Building on the seminal contributions by Gerschenkron (1962) and Abramovitz (1986), technology-gap studies to economic growth have shown that the domestic capability to absorb knowledge spillovers from abroad is a key factor in order to explain growth rate 1 The productivity regime and demand regime functions have been originally proposed by Boyer (1988) in order to formalize these relationships. 3

5 differentials over time and space. According to the technology-gap theory (Fagerberg 1987 and 1994; Verspagen 1991 and 1993), the succesful adoption and use of new technologies is a costly activity, that requires investment in indigeneous capabilities, capital equipment, infrastructure, etc. Without a sufficient level of such investments, a country is unlikely to benefit from backwardness, and risk of falling behind relative to the technology leaders, rather than catching up (Verspagen, 1991). Thus, following this perspective, economic growth may be seen as the outcome of three sets of factors. First, the technologies developed in the country by its internal innovative activity. Second, the potential for exploiting more advanced technologies developed elsewhere (diffusion of international technologies); this potential depends on the technological backwardness of the country, but it is also affected by its technological congruence and social capability (Abramovitz, 1986), without which this process of technology diffusion may be hampered. Third, a set of complementary and structural factors affecting to what extent this potential is realized. Among these factors, Abramovitz (1994, p.26) points out the facilities that laggard countries have for learning about more advanced methods, for appraising them and for acquiring them ; the determinants of resource mobility, because they facilitate the process of structural change required by the aggregate productivity growth; and the macroeconomic conditions that govern the intensity of use of resources and the volume of investment activity, which influence the rate at which more advanced technology is incorporated into production. While the technology-gap approach focuses on the possibility that technological catchingup may lead to higher productivity growth, the Kaldorian cumulative model points out that any given increase in the rate of growth of productivity may activate a process of cumulative causation by interacting with the dynamics of the aggregate demand. Considering jointly these two dis-equilibrium approaches, their complementarities may be expressed as follows. A country lagging behind the technological frontier can benefit from the process of catching up through the diffusion of innovation elsewhere created, if it has some basic structural characteristic to exploit this potential. This catching up process leads to the introduction of new technologies and thus to increases in the average productivity. But once an initial increase in productivity is realized, it can raise the price competitiveness, thus stimulating exports and aggregate demand. In turn, the overall growth in demand can itself spur further increases in the average productivity through the existence of Kaldor- 4

6 Verdoorn returns to scale, thus possibly activating a cumulative causation process of economic growth. The potential engines of growth are both, the country s internal innovative activity, and the catching up process through the imitation of more advanced technologies developed elsewhere. However, the success of this mechanism in terms of catching up and convergence depends on the way in which the increases in productivity interact with the dynamics of demand, thus leading to a cumulative causation mechanism. On the other hand, such an ideal convergence pattern may not be realized when a country lacks the structural capability to exploit the potential for diffusion developed elsewhere (as pointed out by the technology-gap literature), or when its internal interactions between institutions, distribution and aggregate demand components lead to a vicious rather than a virtuous pattern of economic growth (as pointed out in the Kaldorian literature). Four previous models of technology-gap and cumulative growth have already been set up, whose most relevant features are outlined in table 1. The first is the one by Verspagen (1993, chapters 5-6), that presents a model in which the basic idea of the technology-gap approach is formalized through non-linear knowledge spillovers that flow toward the backward country. This outstanding non-linear representation of the knowledge spillovers determines the dynamics of the technology-gap and the final outcome in terms of growth rate differentials. Nevertheless, the possibility that an increase in productivity leads to a process of cumulative growth through the interactions with the growth of exports and demand is represented in a way that is rather different from the one typical in the Kaldorian cumulative growth model. Consequently, the outcomes of this model are determined by the dynamics of the technology-gap, while the role played by the cumulative causation mechanism is not clearly articulated. The second is the one by Amable (1993), which presents a linear model in which the cumulative causation mechanism is represented through the interactions between endogenous investments, innovative activity and human capital. The third is the simpler linear model by Targetti and Foti (1997), based on the relationships between productivity, demand and exports. Finally, the fourth model is the one by Ledesma (1999a), which refines the cumulative mechanism presented by Amable by including the exports. The formalization presented in the next sections builds up on these four previous models, and proposes two refinements. First, the interactions between technology-gap, demand and 5

7 productivity are set in a non-linear framework, as in Verspagen (1993). Second, an effort is done to show that it is the technology-gap and not the productivity-gap to open up the possibility of catching-up through the international diffusion of knowledge and technologies. In fact, as reported in the last column of table 1, in all these four models the technological distance from the leader is proxied by the relative GDP per capita or the relative labour productivity. Nevertheless, the use of such proxies for the technology-gap can be misleading, because the idea of catching-up is that a country grows faster if it is able to exploit the potential advantages coming from its technological backwardness, not from its low average wealth. In, fact, suppose to compare two countries with the same level of technological development, but with different GDP per capita levels. These could differ for many different reasons not linked to technological aspects, such as different resource and population endowments, different institutional and political contexts, or different economic policies. In this case, if we measured the initial level of technological development of the countries with the GDP per capita 2, we would start from the incorrect assumption that the technology-gap between the two is large. Therefore, in the model presented in the next sections, an effort will be done to stress the distinction between the technology-gap and the productivity- or GDP per capita-gap. Although being aware of the fact that such abstract concepts as knowledge stock and technology-gap are rather hard to measure, we believe that it is necessary and possible to proxy them by some better indicators than the initial GDP per capita or labour productivity. In this paper it is presented a first preliminary attempt in this direction.. 2 This last point is further considered in section 4. 6

8 Table 1: Overview of the models of technology-gap and cumulative growth Growth mechanism Exogenous variables Countries in the sample Period Initial technology-gap proxied by Verspagen (1993) The technology-gap dynamics determine the knowledge stock and the external competitiveness Innovative activity GDP per capita Amable (1993) Interactions between capital accumulation, innovative activity and human capital Primary education; government expenditure Percentage of the US level of real GDP per worker Targetti and Foti (1997) Interactions between productivity, demand and exports World demand; world productivity Log of the GDP/worker ratio between the leader and the followers Ledesma (1999) Export-led growth with interactions between innovative activity, ext. competitiveness, demand and productivity-gap World demand; nominal wages; level of education [level of labour productivity relative to the US]

9 2. The model The structural form of a two-country macroeconomic model of technology-gap and cumulative growth is presented in this section. The first country is the technological leader (l), whose growth rate depends on the innovative activity internally developed, and on the Kaldorian cumulalative growth mechanism generated by the interaction between productivity and demand growth. The second country is the technological follower (f), for which a potential additional source of growth is the diffusion of innovation created by the leader. However, as pointed out in the technology-gap literature, the catching up through knowledge spillovers from abroad is far from being an automatic process, as it requires capability to imitate and to use the new technologies developed in the leader country. As it will be shown in section 4, if the follower is not able to imitate the leader, it will fall behind, and the technology-gap between the two countries will increase. On the other hand, if the follower is able to imitate the leader, it will start to catch up in technological terms, and therefore it will grow at a faster rate than the leader. Then, the purpose of the model is to show that combining the technology-gap approach with the idea of cumulative growth enriches the set of possible outcomes in terms of economic growth and convergence/divergence. Let us now consider the seven equations of the model for both countries. where: i = l, f, and: α > 0 Aggregate demand Q i = αx i, (1) The first equation assumes that the rate of growth of demand Q i in both countries depends on the rate of growth of exports X i. Then, following Kaldor (1957) and Thirlwall and Dixon (1979), exports are assumed to be the most important component of aggregate demand, whose growth over time is therefore export-led. A possible extension of the model to include the internal components of demand (investments and consumption) is considered in section 5. 8

10 where: i = l, f, and: β < 0, γ > 0, λ > 0, φ > 0 Exports X i = βp i + λz + γk i + φ(i/o) i, (2) The second equation describes the rate of growth of exports X i as a function of two sets of factors. On the one hand, exports depend inversely on national prices P i, and directly on world demand Z. On the other hand, non-price factors are key elements to explain exports performance. Therefore, this is also assumed to depend on the stock of knowledge K i (as an indicator of the country s ability to compete in quality) and on the investment-output ratio (I/O) i (as a proxy for capital accumulation, following the suggestion by Fagerberg (1988) that the capacity of a country to deliver in the international markets depends on its growth of physical equipment and infrastructures). where: i = l, f. Prices P i = W i AP i, (3) Assuming that prices are set in imperfectly competitive markets, and that the pricing rule is a constant mark-up on unit labour costs, then the rate of growth of prices Pi is given by the difference between the rate of growth of money wages (W i ) and the rate of growth of average productivity (AP i ). where: i = l, f, and: ε > 0, η > 0, σ > 0. Average productivity AP i = εq i + ηk i + σ(i/o) i, (4) This fourth equation describes the rate of growth of average productivity as dependent on three factors. First, the growth of output Q i, according to the idea that an higher extension of production can lead to dynamic economies of scale due to increased specialisation (Young, 1928) and embodied technical progress (Kaldor, 1957). This relationship, known 9

11 as Verdoorn-Kaldor mechanism, supports the cumulative growth idea, according to which the growth of output leads to growth in average productivity, which in turn determines higher price competitiveness (equation 3), higher exports (equation 2) and finally a higher rate of growth in demand (equation 1). The second factor affecting the growth of average productivity is the knowledge stock K i, because a higher knowledge stock leads not only to a higher degree of product differentiation and quality that affect exports (equation 2), but also to the introduction of process innovations that positively affect productivity itself. Finally, the third factor considered in the equation is the investment-output ratio (I/O) i, as a proxy for the technical progress embodied in new machines and equipments which can also lead to a higher productivity. Where: ζ > 0, θ > 0, δ > 0. Knowledge stock K l = ζi l, K f = ζi f + θge -G/δ, (5) l (5) f The only source of growth for the knowledge stock in the leader country (K l ) is the innovative activity internally developed. In other words, it is assumed here that the leader has no advantages from the knowledge developed in the follower; on the other hand, the latter can exploit the higher knowledge developed in the leader country, through a process of imitation and catching up. In order to formalize the idea that this process is far from being automatic, knowledge spillovers are introduced in equation 5f as a potential additional source of growth for the follower. The interpretation is the same as in the original formulation by Verspagen (1993, pp ), that is the following. The term θg represents the potential spillovers, which increase as a linear function of the technology-gap G, according to the idea that the higher the technological distance the higher is the potential for catching up for the follower country. Nevertheless, the extent to which this potential is realized depends on the term e -G/δ, that represents the learning capability to assimilate knowledge spillovers. This learning capability, in turn, depends on two factors: it is a positive function of the intrinsic capability (δ); and a negative function of the technologygap G itself, because it is assumed that for a given intrinsic capability to assimilate spillovers, the overall capability will diminish with the technological distance. Then, the 10

12 spillovers term θge -G/δ, depending on the potential spillovers (θg) and on the learning capability (e -G/δ ), it is a non-monotonic function of the gap, given the parameters θ and δ. This is shown in the two graphs in fig.1. In fig.1a, knowledge spillovers are sketched as a function of the gap. The interpretation of this graph is that when the gap is very high the learning capability is low, and therefore the knowledge spillovers are also low (this is the situation in which the follower country is not able to exploit the potential spillovers by imitating the leader); on the other hand, when the gap becomes smaller the learning capability increases, and the knowledge spillovers increase (this is the situation in which the follower is able to exploit its backwardness). Finally, when the gap is completely closed (G = 0) the potential for catching up has been entirely realized, and the spillovers are therefore zero. Moreover, the graph shows two possible spillovers functions S and S according to two different values of the intrinsic capability δ, in order to stress the fact that an increase in this parameter (due for example to an active policy in education, investment in infrastructure, etc. ) has the effect of increasing the knowledge spillovers for any given technological distance G. This is more evident in fig.1b, in which knowledge spillovers are represented as a three dimensional function of the gap and the intrinsic capability. Fig.1a: Knowledge spillovers as a function of the gap (source: Verspagen, 1993) 11

13 Fig.1b: Knowledge spillovers as a three dimensional function of the gap and of the intrinsic capability. Technology-gap G = ln(k l /k f ) (6) The technology-gap (or technological distance ) G between the leader and the follower country is defined as the natural logarithm of the knowledge stock ratio; according to this specification, the gap G is positive if K l > K f, it is zero if K l = K f, and it is negative if K l < K f, in which case the follower has overtaken the leader s position. where: i = l, f, and: a > 0, b > 0. Innovative activity I i = ah i + bk i, (7) This last equation describes the dynamics of the innovative activity, internally developed by both countries through the innovation system. It is assumed to depend on an exogenous 12

14 variable H i, which represents the level of education and the human capital of the working population; and on the knowledge stock K i itself, following the idea that the higher is the stock of existing knowledge the more successful will be the R&D sector in creating new products and processes. This formulation is rather simple and not satisfactory. In fact, although innovative activity is partly endogeneized (through the term K i ), a better specification should take into account a set of other endogenous factors. Possible extensions of the model are discussed in section 5. The main idea of the model is that the economic growth of the leader country is determined by its internal innovative activity, and by the ability of the resulting productivity increases to start a process of cumulative causation, through the interactions between productivity and aggregate demand. On the other hand, the follower has a potential additional source of growth, that is the possibility to imitate the superior knowledge and technology developed in the leader country. Fig. 2 summarizes the main relationships presented in the model for the follower. In the lower part of the diagram is represented the idea of technology-gap growth : the follower country lags behind the technological frontier, and has therefore the possibility to catch up through a process of imitation; if it is able to absorb and use the knowledge spillovers coming from the leader, its knowledge stock will increase (equation 5 f, second term), with the twin consequences of closing the gap (equation 6) and fostering internal innovative activity (equation 7, second term). This, in turn, will increase the knowledge stock (equation 5f, first term) and further close the gap (again in equation 6). So, the technologygap part of the model shows the possibility that knowledge spillovers and the (endogenous) innovative activity lead to a process of cumulative catching up in technological terms, in the sense that higher spillovers determine a higher knowledge stock and higher innovation, and in turn higher innovation determines a higher knowledge stock, a smaller gap and thus higher spillovers. However, this catching up process is conditional to the learning capability of the country, meaning that if this capability is too low this process of cumulative catching up will not take place, and the follower will fall behind. In the upper part of the diagram in fig. 2 the Kaldorian idea of cumulative growth is represented, according to which an increase in average productivity leads to a higher price competitiveness (equation 3), and therefore a better exports and output performance (equation 2 and 1 respectively). In turn, a higher rate of growth of output determines a higher rate of growth of productivity 13

15 through the Verdoorn-Kaldor mechanism (equation 4, first term). If all these relationships hold, the process of growth is self-sustained, and therefore cumulative. The novelty of this model is that technology-gap and cumulative causation mechanism simultaneously determine the rate of growth of productivity, output and knowledge stock for the follower country. Hence, the Kaldorian cumulative mechanism can accelerate the growth in productivity for a follower country that is closing the technology-gap, leading to a faster convergence. On the contrary, if the follower is unable to catch up in technological terms, it will also be unable to activate this cumulative growth mechanism, and divergence in productivity will be the (unfortunately common) outcome. 14

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17 3. Dynamics, analytical solutions and stability of the model In this section the following analytical properties of the model are discussed: first, the dynamics of the technology-gap, in order to find out under which conditions the follower is able to catch up in purely technological terms; then, the effects of such dynamics on productivity and demand; finally, the implications in terms of growth rate differential between the leader and the follower. 3.1 The dynamics of the technology-gap The dynamics of the technology-gap can be examined as follows. From equation (6), (5) l and (5) f, we can derive the differential equation that describes such dynamics, that is: dg/dt = K l K f = ζ(i l I f ) - θge -G/δ, (8) whose solution is given by the following condition: dg/dt = 0 I l I f = (θ/ζ)ge -G/δ. (9) Before analysing this condition, it should be noted that the left hand side of the equation is endogenous, as innovative activity is endogenous for both countries. Thus, using equations (5) and (7), we can rewrite the left hand side as follows: I l I f = (H l H f )[a(1 - bζ)] - Ge -G/δ [bθ(1 - bζ)]. (10) Equation (10) describes the difference between the innovative activity developed in the two countries as a function of the difference between human capital in the two countries, and as a function of the gap. In order for this equation to be stable, let us assume that: 1 - bζ > 0 bζ < 1, (11) that requires that the interrelationship between innovation and knowledge stock must not be too strong, otherwise the growth of the latter would be unstable and therefore explosive. If condition (11) holds true, then the differential innovation between the leader and the 16

18 follower increases with the (exogenous) difference between the human capital in the two countries; moreover, it is a non-monotonic function of the gap, as shown in fig. 3. The interpretation of the graph is straightforward: when the gap is too high, the follower is not able to sustain the described process of cumulative catching up, and its innovative activity will be much lower than the one of the leader, with the obvious consequence that the difference between the two is high and increasing. On the other hand, if the follower is able to catch up, its innovative activity will increase more than the one of the leader due to the interactions between knowledge spillovers, knowledge stock and the innovation itself. Consequently, the difference between the innovative activities in the two countries will be smaller and decreasing down to a minimum value (in correspondence to the value G = δ), before increasing again for very low values of the gap. Fig.3: Difference between the innovative activity in the leader and the follower country as a function of the technological distance. 17

19 Substituting equation (10) in (9), we thus obtain the following: dg/dt = 0 (H l H f )[a(1 - bζ)] - Ge -G/δ [bθ(1 - bζ)] = (θ/ζ)ge -G/δ. (12) Equation (12) describes the dynamics of the technology-gap. While we have already analysed its left hand side (LHS), the right hand side (RHS) describes a scale transformation of the knowledge spillovers that flow from the leader to the follower country, that we have already considered in equation 5 and in fig. 1. Then, also the RHS is a non-monotonic function of the gap, with a maximum in correspondence to G = δ. The dynamics of the technology-gap is shown in fig. 4, in which we sketch together the RHS and the LHS of equation (12). When the LHS is higher than the RHS, the gap tends to increase; on the other hand, when the LHS is smaller than the RHS, the gap tends to decrease. Only when the LHS equals the RHS, the gap will be steady. The interpretation of such dynamics is straightforward: only when the advantage for the leader in terms of higher innovative activity are exactly counterbalanced by the advantage for the follower in terms of knowledge spillovers, there will be no tendency for the gap to increase/decrease. In every other point, one of these two forces is stronger than the other, and the gap will change. However, given the RHS curve, the existence and stability of equilibria depend on the position of the LHS curve, which in turn depend on the difference (H l H f ). We have three possible cases. If this difference is positive, we have the curve LHS, with two intersection points with the RHS curve. Among these two equilibria, B is unstable, because the gap tends to increase (decrease) if we start from a point on the right (left) of it. On the other hand, the equilibrium A is stable, and it will be reached whatever the starting point is. Notice that even if this equilibrium is reached, the corresponding value of the gap is positive: in this case the final outcome in technological terms is partial catch up. The second case is when the difference (H l H f ) is zero, in which we have the curve LHS. In this case we have only one possible equilibrium (the point C), which is stable. As it is clear from fig. 4, the corresponding value of the gap is now zero, meaning that in this case the follower is able to completely catch up in technological terms. Finally, if the difference (H l H f ) is negative, we have the curve LHS, with only one (unstable) equilibrium D. In this situation, the corresponding gap is negative, meaning that the follower has overtaken the technological leadership. 18

20 Fig.4: the dynamics of the technology-gap. 3.2 Demand and productivity regimes The next question is: if a country is able to close the technology-gap, is it necessarily able to close the growth rate differential? The relationships between the growth of demand and average productivity need to be considered here. As we have seen in section 1 and 2 above, the Kaldorian mechanism of cumulative growth is based on the idea that a higher rate of growth of productivity leads to a higher growth of demand, via price competitiveness and exports; and in turn a higher rate of demand growth leads to a higher growth of productivity via the Verdoorn-Kaldor effect. These relationships have been originally formalized by Boyer (1988) through the demand regime (DR) and productivity regime (PR) equations. Let us obtain them in our model for both countries, in order to investigate their relationships with the dynamics of the technology-gap described above. 19

21 From equations (1), (2), (3), (5) and (7), we obtain the demand regimes for the leader (DR l ) and the follower (DR f ) respectively, that is: DR l : Q l = {W l (αβ) + Z(αλ) + (I/O) l (αφ) + H l [αγζa/(1 - bζ)]} - αβap l, (13) l DR f : Q f = {W f (αβ) + Z(αλ) + (I/O) f (αφ) + H f [αγζa/(1 - bζ)] + Ge -G/δ [αγθ/(1 - bζ)] } - αβap f, (13) f where the terms in brackets define the intercepts of the linear functions, while the term (- αβ) defines their slopes. Notice that the parameter β is assumed to be negative, so the slope is positive and the same for both countries. From equations (4), (5) and (7) we obtain the productivity regimes for the leader (PR l ) and the follower (PR f ) respectively, that is: PR l : AP l = {σ(i/o) l + [ηζa(1 - bζ)]h l } + εq l, (14) l PR f : AP f = {σ(i/o) f + [ηζa(1 - bζ)]h f + ηθ Ge -G/δ } + εq l, (14) f where the terms in brackets define the intercepts of the linear functions, while the positive parameter ε defines the slope, which is the same for both countries. Observing equations (13) l, (13) f, (14) l and (14) f, and plotting them on the same graph in fig. 5, we notice the following two properties. First, while the PR and DR lines for the leader do not change their position when the gap changes, the same is not true for the PR and DR lines for the follower. In fact, as it is clear from equations (13) f and (14) f, the y-intercepts of these lines are non-monotonic functions of G, because they both contain the term Ge -G/δ. This means that when the technology-gap is very high the PR f (DR f ) line has a very low (high) y-intercept, and therefore their intersection point determines an equilibrium with a very low rate of growth of productivity. But if the follower is able to catch up in technological terms, the gap will decrease and consequently the y-intercept of the PR f (DR f ) line will increase (decrease) up to a maximum (minimum) value (again in correspondence 20

22 to G = δ), before decreasing (increasing) again as the gap closes. As a consequence, following the dynamics of the technology-gap, the PR f (DR f ) line will both shift upward (downward) up to a maximum (minimum) value, and then gradually shift downward (upward) again for very low values of the gap. Their final position determines the equilibrium for the rates of growth of productivity and demand. As it is clear from the two possible cases depicted in fig. 5, the productivity growth for the follower in such equilibrium can be greater (point F ), equal or smaller (point F) than the one of the leader (point L). Therefore, as it will be shown in section 3.3, a follower that is able to catch up in technological terms not necessarily converge in terms of productivity growth. On the other hand, if the follower is falling behind in technological terms, it will necessarily diverge also in terms of productivity growth. Fig.5: Productivity and demand regimes for the leader and the follower countries. The second property of the model is that, while the dynamics of the technology-gap affect the y-intercepts of the DR f and PR f lines in the way just described, it does not affect the slopes of the DR and PR equations for neither of the two countries. Both these slopes are 21

23 positive, and therefore the stability condition in order to have stable growth is that the slope of the DR line is higher than the one of the PR line, that is: ε < - 1/αβ αβε < 1. (15) The interpretation of this condition is the same as the original one by Boyer (1988), i.e. the cumulative mechanism: productivity prices exports demand productivity must not be too strong, otherwise it will generate unstable and explosive growth. Equation (15) and (11) define together the stability conditions for our model. 3.3 Growth rate differential, convergence and divergence Given the catching up process in technological terms and the interactions between productivity and demand that we have described so far, what is the dynamic of the growth rate differentials between the two countries? In order to answer this question, let us first obtain the average productivities for both countries as a function of all the exogenous variables of the model, that is: AP l = W l [αβε/(1 + αβε)] + Z[αλε/(1 + αβε)] + (I/O) l [αφε/(1 + αβε)] + + H l [aζ(η + αγε)/(1 - bζ)(1 + αβε)], (16) l APf = W f [αβε/(1 + αβε)] + Z[αλε/(1 + αβε)] + (I/O) f [αφε/(1 + αβε)] + + H f [aζ(η + αγε)/(1 - bζ)(1 + αβε)] - Ge -G/δ [αγθε/(1 - bζ)(1 + αβε)]. (16) f Equations (16) l and (16) f are shown in fig. 6 as a function of the gap, and the interpretation is the following. The leader s productivity does not depend on the gap, but only on the exogenous variables (I/O) l, Z and H l (positively) and W l (negatively). On the other hand, the follower s productivity is not only a function of these exogenous variables, but also a non-monotonic function of the gap. For high technological distances AP f is rather low, because the follower is not able to exploit the potential knowledge spillovers (which in turn would increase the knowledge stock, the innovative activity and hence the productivity): in this case the follower s productivity growth will be rather slow. But if the follower is able to exploit these spillovers, then we will observe over time decreasing values of the gap, increasing knowledge stock, higher innovation and faster productivity growth, which in 22

24 turn will interact with the growth of demand, leading to a virtuous circle. In this case, the follower can grow faster than the leader and eventually converge to the same rate of growth of productivity. Then, as it is obvious from fig. 6, it is the dynamics of the technology-gap to determine the dynamics of the growth rate differential in this model. More precisely, the difference between equations (16) l and (16) f gives the dynamics of the growth rate differential as a function of all the exogenous variables and as a function of the technology-gap, that is: AP l AP f = (W l W f ) [αβε/(1 + αβε)] + [(I/O) l (I/O) f ] [αφε/(1 + αβε)] + + (H l H f )[aζ(η + αγε)/(1 - bζ)(1 + αβε)] - Ge -G/δ [αγθε/(1 - bζ)(1 + αβε)]. (17) Fig.6: Rates of growth of productivity for the leader and the follower countries as a function of the gap. Let us suppose that the conditions for stability (11) and (15) hold true, so that all the denominators in equation (17) are positive. If this is the case, the productivity growth rate 23

25 differential between the leader and the follower can be positive or negative, because it depends at any time on: (i) the wage differential in a negative way, because wages affect negatively price competitiveness and exports performance (equations (3) and (2)); (ii) the difference between the investment-output ratios in a positive way, because we have assumed that this ratio positively affects exports and productivity of each country (equations (2) and (4)); (iii) the difference between human capital in a positive way, because this variable supports innovative activity (equation (7)); (iv) the knowledge spillovers Ge -G/δ in a negative way, because high spillovers push the productivity in the follower country, reducing the growth rate differential. In particular, in the equilibrium point where dg/dt = 0, equation (17) can assume positive, zero or negative values, depending on the exogenous variables and on the value of G. In fact, as outlined in section 4.1, not necessarily the gap in the equilibrium will be entirely closed: it will be positive (zero) (negative) if the difference between the human capital between the leader and the follower is positive (zero) (negative). Then, depending on the values of these exogenous variables and on the value of G in the equilibrium, the model generates the following seven possible cases: 1. falling behind in technological terms and divergence in productivity; 2. partial catching-up in technological terms and divergence in productivity; 3. partial catching-up in technological terms and convergence in productivity; 4. total catching-up in technological terms and divergence in productivity; 5. total catching-up in technological terms and convergence in productivity; 6. technological overtaking and divergence in productivity; 7. technological overtaking and convergence in productivity. This leads us to the main implication of the model. If the follower is falling behind in technological terms, the growth rate differential will increase over time, and divergence in productivity growth will be the outcome (case 1). If the follower is able to catch up in technological terms, it will tend to grow faster than the leader; however, the extent to which such transitional dynamics will lead to total or only partial convergence in productivity growth in the final equilibrium, depends on the exogenous variables of both countries 24

26 (cases 2 to 7). In simple words, if a follower wants to steadily grow faster than the leader, it must have a higher investment-output ratio, better human capital and/or lower wages than the leader. Then, as a general conclusion, combining the technology-gap approach with the Kaldorian idea of cumulative growth allows for a large set of possible outcomes. Far from being the only necessary conclusion, convergence in productivity rates of growth is only one of the possible outcomes of the model. From a pure theoretical point of view, the main implication of our specification can be summed up in the following way: technological catching-up is a necessary but not sufficient condition for convergence in productivity growth. On the other hand, cumulative growth can be source of divergence (if the follower is not able to imitate) or convergence (if the follower is able to catch-up in technological terms). Thus, contrary to many common interpretations, the technology-gap model does not necessarily imply convergence; and the Kaldorian idea of cumulative causation does not necessarily imply divergence. 4. The experience of OECD countries, Some empirical evidence on the experience of 26 OECD countries in the period can point out the relevance of such a model for explaining the recent performances of technological activities and productivity. As already considered in section 1, all the four previous models of technology-gap and cumulative growth used the initial GDP per capita or the labour productivity as a proxy for the initial technology-gap. However, as shown in fig. 7, if we compare at the beginning of the 90s the GDP per capita with the R&D/GDP ratio (as an alternative proxy for the knowledge stock and thus for the technology-gap 3 ), we notice that the two indicators are rather different for some countries. This implies that in our OECD sample there are countries with an initial level of GDP per capita relatively higher than their initial knowledge stock (Japan, Switzerland, Austria, Belgium, Denmark and Norway), and countries with an initial level of GDP per capita relatively smaller than their initial knowledge stock. (Czech Republic, Korea and Sweden). Therefore, using GDP 3 The R&D/GDP ratio is far from being a satisfactory proxy for the knowledge stock and for the technology-gap. It is certainly not easy to find a good indicator to measure such abstract concepts, but the point stressed here is that it is possible to find a better proxy than the initial GDP per capita. It is then necessary in future works to construct a more complete indicator, that take into account simultaneously variables of input and output of the innovative process. 25

27 per capita as a proxy for the initial technology-gap could lead to overestimate (underestimate) the importance of the process of technological catching up in generating growth for those countries whose initial GDP per capita is relatively higher (lower) than the initial knowledge stock. Fig. 7: Difference between real initial GDP/capita and initial knowledge stock at the beginning of the 90s 2 1,5 1 Jp Austria Belgium Den Nor Switzerland Differences 0,5 0-0,5 US Ger Fr It UK CanAustralia Fin Gr Hun Ire NethNew Zeal Pol Por Sp Tur -1 Czech -1,5 Kor Sweden -2 Taking into account this possible bias, four basic indicators are considered for this period in Table 2: 4 the average annual rate of growth of labour productivity; the secondary school enrolment ratio in 1990 (as a proxy for the human capital at the beginning of the period); the R&D/GDP ratio in 1993 (as a proxy for the initial knowledge stock, and therefore for the initial technology-gap); the average annual rate of growth of the R&D/GDP ratio (as a proxy for the increases in the knowledge stock). 4 For the definition and source of each indicator, see the Appendix. 26

28 Table 2: some indicators on technological and productivity performances in OECD countries in the 90s. Countries Average productivity increases Human capital Initial knowledge stock Knowledge stock increases US 1,714 92,0 2,62 3,95 Japan 1,25 97,0 2,88 2,025 Germany 2,18 98,0 2,42 0,33 France 1,38 99,0 2,45 0,16 Italy 1,7 79,0 1,14-0,35 UK 1,78 86,0 2,15 0,4 Canada 0,95 101,0 1,6 3,5 Australia 2,04 82,0 1,62 5,9 Austria 0,71 104,0 1,49 3,55 Belgium 1,91 103,0 1,58 0,95 Czech Rep. 0,16 88,0 1,23-1,66 Denmark 2,65 109,0 1,74 5,11 Finland 2, ,0 2,21 9,2 Greece 1,15 99,0 0,48 6,76 Hungary 4,28 79,0 0,98-3,98 Ireland 4 101,0 1,2 15,2 Korea 4,82 90,0 2,3 14,6 Netherlands 0,53 120,0 2 3,825 New Zealand 0,94 89,0 1,02 6,43 Norway 2,71 103,0 1,73 4,2 Poland 5,58 81,0 0,82 4,67 Portugal 2,52 68,0 0,58 4 Spain 1,72 105,0 0,91 1,65 Sweden 2,65 90,0 3,39 5,1 Switzerland 0,8 90,0 2,74 0,9 Turkey 3,02 54,0 0,44 6,34 Source: see the Appendix In order to compare such empirical evidence with the basic predictions of the model, a cluster analysis has been performed, using as inputs the four variables reported in Table 2. After having arbitrarily fixed the number of clusters to six, and having standardized all the variables, a so called K-means clustering algorithm has been applied. The results of the cluster analysis are reported in table 2. Notice that as the data were standardized, a value of zero corresponds to the sample mean, and plus (minus) one corresponds to one standard deviation above (below) the mean. In the last row of the table, the composition of each 27

29 cluster is reported, with the correspondance between each group of countries and the basic predictions of the model (cases 1 to 7, section 3.3). Table 3: A Cluster Analysis of OECD countries in the 90s Clusters Standardized variables A B C D E F Labour productivity -0,673 0,510 2,080 1,704 0,068-0,289 Human capital 0,110-2,297-0,951 0,146 1,312-0,021 Initial knowledge -0,54-1,470-0,970 0,120 0,33 1,290 stock Knowledge stock increases -0,147 0,248-0,864 2,480 0,342-0,518 Cluster composition and correspondance to the outcomes of the model (2) Austria Canada Czech Rep Greece Italy New Zeal. (3) Australia Belgium Spain (3) Portugal Turkey (3) Hungary Poland (3/5) Ireland Korea (5) Denmark Finland Norway (4) Netherland US Japan (5) France Germany UK (5) Sweden (4) Switzerland Cluster A: the ten countries included in this group show similar technological characteristics (substantial initial technology-gap, slow knowledge stock increases, human capital on the average of the sample), but rather different productivity performances. In fact, the countries corresponding to the case 2 of the model have disappointing productivity trends although they are slowly closing the gap; while the countries corresponding to case 3, besides closing the technology-gap, seem to be able to grow faster than the leaders. The composition of this first cluster confirms the basic prediction of the model that not necessarily the process of technological catching-up leads to convergence in productivity growth. Cluster B: Portugal and Turkey belong to this second group, homogenous in terms of fast convergence in productivity, very high initial technological distance from the leaders, slow 28

30 increases in the knowledge stock, and human capital below the average of the sample. This cluster corresponds to the case 3 of the model, because these countries are able at the same time to activate a slow process of partial catching up and economic convergence. Cluster C: this group is formed by Hungary and Poland, two countries homogenous in terms of very fast convergence in productivity, high initial technological distance from the leaders, rather slow knowledge stock increases, and human capital below the average of the sample. Even this cluster corresponds to the case 3 of our model, because these countries are (only partly) catching-up in technological terms and growing much faster than the leaders. Cluster D: Ireland and Korea are in all the respects the most dynamic countries in the sample. Although they still had at the beginning of the 90s a substantial technology-gap, their process of catching-up seems to be much faster than any other follower in OECD countries; the above average level of human capital could lead us to suppose that this process will go on in the present decade. Moreover, this process of partial (soon total?) catching-up determines very fast convergence in productivity growth. Cluster E: the countries belonging to this group did not have a large technological distance from the leaders at the beginning of the 90s, but their process of technological catching-up has been rather dynamic. These Northern European countries have the required social capability and human capital to take advantage of the new innovation-based growth regime. Nevertheless, their productivity performances are rather heterogenous: above the average of the sample for the Scandinavian countries (corresponding to case 5 of the model), but rather disappointing for the Netherlands (outcome 4 of the model). Here again, as predicted by the model, technological catching-up is not a sufficient condition for convergence in productivity growth. Cluster F: this final group has a rather heterogenous composition. On the one hand, we have the traditional technological leaders US and Japan; on the other, the most technologically advanced EU economies (Germany, France, UK, Sweden and Switzerland), whose technological performances were already very good at the beginning of the 90s. All these European countries already had no significant technology-gap from US and Japan at the beginning of the period, and therefore their increase in knowledge 29