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) 22 84 06 09
E-mail: fulvio.castellacci@tik.uio.no
* 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.
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 19911999 points out the relevance of the model for explaining the recent performances of
technological activities and productivity growth.
1
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
(“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
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
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
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 capita2, 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
Exogenous
variables
Innovative activity
Primary education;
government
expenditure
World demand;
world productivity
World demand;
nominal wages;
level of education
Growth
mechanism
The technology-gap
dynamics determine
the knowledge stock
and the external
competitiveness
Interactions between
capital accumulation,
innovative activity and
human capital
Interactions between
productivity, demand
and exports
Export-led growth with
interactions between
innovative activity,
ext. competitiveness,
demand and
productivity-gap
Verspagen
(1993)
Amable
(1993)
Targetti and
Foti (1997)
Ledesma
(1999)
Table 1: Overview of the models of technology-gap and cumulative growth
17
25
59
114
Countries in
the sample
1965-94
1950-88
1960-85
1960-85
Period
1- [level of labour
productivity relative to
the US]
Log of the GDP/worker
ratio between the leader
and the followers
Percentage of the US
level of real GDP
per worker
GDP per capita
Initial technology-gap
proxied by
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.
Aggregate demand
Qi = αXi,
(1)
where: i = l, f,
and: α > 0
The first equation assumes that the rate of growth of demand Qi in both countries depends
on the rate of growth of exports Xi. 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
Exports
Xi = βPi + λZ + γKi + φ(I/O)i,
(2)
where: i = l, f,
and: β < 0, γ > 0, λ > 0, φ > 0
The second equation describes the rate of growth of exports Xi as a function of two sets of
factors. On the one hand, exports depend inversely on national prices Pi, 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 Ki (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).
Prices
Pi = Wi – APi,
(3)
where: i = l, f.
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 (Wi) and the rate of growth of
average productivity (APi).
Average productivity
APi = εQi + ηKi + σ(I/O)i,
(4)
where: i = l, f,
and: ε > 0, η > 0, σ > 0.
This fourth equation describes the rate of growth of average productivity as dependent on
three factors. First, the growth of output Qi, 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
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 Ki,
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.
Knowledge stock
Kl = ζIl,
Kf = ζIf + θGe-G/δ,
(5)l
(5)f
Where: ζ > 0, θ > 0, δ > 0.
The only source of growth for the knowledge stock in the leader country (Kl) 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. 129-130), 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
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
Fig.1b: Knowledge spillovers as a three dimensional function
of the gap and of the intrinsic capability.
Technology-gap
G = ln(kl/kf)
(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 Kl > Kf, it is zero if Kl = Kf, and it is negative if Kl <
Kf, in which case the follower has overtaken the leader’s position.
Innovative activity
Ii = aHi + bKi,
(7)
where: i = l, f,
and: a > 0, b > 0.
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
variable Hi, which represents the level of education and the human capital of the working
population; and on the knowledge stock Ki 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 Ki), 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 5f, 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
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
15
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 = Kl – Kf = ζ(Il – If) - θGe-G/δ,
(8)
whose solution is given by the following condition:
dG/dt = 0 ⇒ Il – If = (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:
Il – If = (Hl – Hf)[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
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
Substituting equation (10) in (9), we thus obtain the following:
dG/dt = 0 ⇒ (Hl – Hf)[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 (Hl – Hf). 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 (Hl – Hf) 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 (Hl – Hf) 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
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
From equations (1), (2), (3), (5) and (7), we obtain the demand regimes for the leader (DRl)
and the follower (DRf) respectively, that is:
DRl:
(13)l
Ql = {Wl(αβ) + Z(αλ) + (I/O)l(αφ) + Hl[αγζa/(1 - bζ)]} - αβAPl,
DRf:
(13)f
Qf = {Wf(αβ) + Z(αλ) + (I/O)f(αφ) + Hf[αγζa/(1 - bζ)] + Ge-G/δ[αγθ/(1 - bζ)] } - αβAPf,
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 (PRl) and
the follower (PRf) respectively, that is:
PRl:
(14)l
APl = {σ(I/O)l + [ηζa(1 - bζ)]Hl} + εQl,
PRf:
(14)f
APf = {σ(I/O)f + [ηζa(1 - bζ)]Hf + ηθ Ge-G/δ} + εQl,
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 PRf (DRf) 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 PRf (DRf)
line will increase (decrease) up to a maximum (minimum) value (again in correspondence
20
to G = δ), before decreasing (increasing) again as the gap closes. As a consequence,
following the dynamics of the technology-gap, the PRf (DRf) 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 DRf and PRf 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
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:
APl = Wl[αβε/(1 + αβε)] + Z[αλε/(1 + αβε)] + (I/O)l[αφε/(1 + αβε)] +
+ Hl[aζ(η + αγε)/(1 - bζ)(1 + αβε)],
(16)l
APf = Wf[αβε/(1 + αβε)] + Z[αλε/(1 + αβε)] + (I/O)f[αφε/(1 + αβε)] +
+ Hf[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 Hl (positively) and Wl (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 APf 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
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:
APl – APf = (Wl – Wf) [αβε/(1 + αβε)] + [(I/O)l – (I/O)f] [αφε/(1 + αβε)] +
+ (Hl – Hf)[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
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
(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, 1991-1999
Some empirical evidence on the experience of 26 OECD countries in the period 1991-1999
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-gap3),
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
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
Jp
Austria
1
Switzerland
Nor
Den
Belgium
It
Differences
0,5
New Zeal
Neth
Ger
0
US
Ire
CanAustralia
Por
Sp
Gr
Fr
Fin
-0,5
Tur
UK
Hun
-1
Pol
Czech
Kor
-1,5
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
Table 2: some indicators on technological and productivity performances
in OECD countries in the 90s.
Countries
Average
productivity
increases
Human
capital
US
Japan
Germany
France
Italy
UK
Canada
Australia
Austria
Belgium
Czech Rep.
Denmark
Finland
Greece
Hungary
Ireland
Korea
Netherlands
New Zealand
Norway
Poland
Portugal
Spain
Sweden
Switzerland
Turkey
1,714
1,25
2,18
1,38
1,7
1,78
0,95
2,04
0,71
1,91
0,16
2,65
2,785
1,15
4,28
4
4,82
0,53
0,94
2,71
5,58
2,52
1,72
2,65
0,8
3,02
92,0
97,0
98,0
99,0
79,0
86,0
101,0
82,0
104,0
103,0
88,0
109,0
116,0
99,0
79,0
101,0
90,0
120,0
89,0
103,0
81,0
68,0
105,0
90,0
90,0
54,0
Initial
Knowledge stock
increases
knowledge stock
2,62
2,88
2,42
2,45
1,14
2,15
1,6
1,62
1,49
1,58
1,23
1,74
2,21
0,48
0,98
1,2
2,3
2
1,02
1,73
0,82
0,58
0,91
3,39
2,74
0,44
3,95
2,025
0,33
0,16
-0,35
0,4
3,5
5,9
3,55
0,95
-1,66
5,11
9,2
6,76
-3,98
15,2
14,6
3,825
6,43
4,2
4,67
4
1,65
5,1
0,9
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
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
Labour
productivity
Human capital
Initial
knowledge
stock
Knowledge
stock increases
Cluster
composition
and
correspondance
to the outcomes
of the model
A
B
C
D
E
F
-0,673
0,510
2,080
1,704
0,068
-0,289
0,110
-2,297
-0,951
0,146
1,312
-0,021
-0,54
-1,470
-0,970
0,120
0,33
1,290
-0,147
0,248
-0,864
2,480
0,342
-0,518
(2)
Austria
Canada
Czech Rep
Greece
Italy
New Zeal.
US
Japan
(3)
Portugal
Turkey
(3)
Hungary
Poland
(3)
Australia
Belgium
Spain
(3/5)
Ireland
Korea
(5)
Denmark
Finland
Norway
(5)
France
Germany
UK
(4)
Netherland
(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
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
stock has been less dynamic than all the other followers of the previous clusters. Though
having similar technological performances, the productivity growth largely differs between
the countries belonging to this cluster, determining different situations: fast convergence
for Sweden (case 5 of the model), slow convergence for Germany, France and UK (again
case 5), divergence for Switzerland (case 4).
On the whole, the cluster analysis seems rather encouraging for the model of technologygap and cumulative growth presented in the previous sections. It points out that the
experience of OECD countries in the 90s in terms of technological and productivity
performance is rather heterogenous. Some countries seem to adapt very slowly to the new
technological regime based on ICTs (clusters A, B and C); some others appear to be much
more dynamic (clusters D and E); and some more advanced countries seem to be already
very close to the technological frontier (European countries in cluster F). As predicted in
the model, such process of technological catching-up has been faster the higher the level of
human capital at the beginning of the period. Moreover, it is not necessarily true that a
more dynamic technological performance determines faster productivity growth rates, as it
is clear from the case of the Netherlands, Switzerland and a small group of diverging
countries in cluster A. The model presented in this paper explains these diverging paths in
terms of the difficulty for these countries to activate and sustain a process of cumulative
growth based on the interactions between productivity and demand growth.
Finally, notice that we do not find in our sample neither countries that are falling behind
(case 1), nor countries that are already overtaking the leaders’ position (case 6 and 7). In
order to find falling behind countries, it would be sufficient to enlarge the sample to
include some LDCs; in order to investigate the case of overtaking, on the other hand, we
should consider a much longer historical period than the one analysed here.
30
5. Conclusions
The model presented in this paper has tried to combine the technology-gap approach to
economic growth with the Kaldorian idea of cumulative causation. After presenting the
structural form of this two-country macroeconomic model in section 2, its analytical
properties have been considered in section 3. These properties show that combining the
technology-gap approach with the Kaldorian idea of cumulative growth allows for a large
set of possible outcomes. If the follower is unable to catch up in technological terms,
divergence in productivity will be the necessary conclusion; but if the follower is able to
close the technology-gap, not necessarily it will also be able to steadily converge in terms
of productivity growth. 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; while the
cumulative growth mechanism 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.
This large set of possible outcomes is generated by a model whose structural form is rather
simple. In fact, in this first formulation we have considered important to keep the
specification as simple as possible, in order to build up a framework to investigate the main
relationships between the technology-gap approach and the idea of cumulative growth.
Nevertheless, the structural form of the model can be refined along the following five lines.
First, the internal component of aggregate demand can be introduced in the specification, as
already done in a recent work (Castellacci, 2001); in fact, it is reasonable to assume that
private consumption and gross investment determine, together with the exports, the rate of
growth of demand. Their introduction could then improve the description of the cumulative
growth mechanism.
Second, the rate of growth of wages is an exogenous variable of our model; it would then
be possible to specify it as an endogenous variable, following the idea that part of the
productivity increases can be redistributed to the wage perceivers.
Third, it is necessary to improve the specification of the innovative activity (equation 7).
On the one hand, we could introduce the idea that innovation is at least partly demand-led
31
(relationship known as “Schmookler-effect”); on the other hand, it would be reasonable to
assume that the relationships between innovative activity and knowledge stock is not
necessarily linear, but it would probably be better represented by a non-monotonic function,
following the idea that it becomes more and more difficult to produce successful
innovations when a country is getting closer to the technological frontier.
Fourth, the intrinsic capability to assimilate knowledge spillovers (δ) is an exogenous
parameter in our specification; nevertheless, we may follow the suggestions by Amable
(1993) and define it as an endogenous variable of the model, in order to “escape from the
limitations of a static theory of the social capability” (ibidem, p. 12).
Finally, in the model we have assumed for simplicity that all the parameters that define the
cumulative growth mechanism (α, β and ε) are the same for the two countries. This means
that we have supposed the same interaction between growth of productivity and growth of
demand for the leader and the follower country, so that it is the dynamics of the
technology-gap (together with the exogenous variables) to determine the final outcome in
terms of convergence/divergence in productivity growth. However, we could reasonably
think that these parameters differ in the two countries, in which case the final outcome
would also depend on the different abilities of the countries to activate and sustain this
process of cumulative growth.
The next step is therefore to improve the model along these five lines. The purpose of the
future work is to build up a more realistic framework, and to investigate how the
conclusions presented here would change with different and more elaborated specifications.
32
Appendix: data definitions and sources
Initial real GDP/capita (Fig.7): Real GDP divided by total population, from “World
Development Indicators 1997”, World Bank.
Average productivity of labour increases (Table 2): Real GDP per person employed,
average annual percentage change, 1991-1997, from “OECD Historical Statistics, 19601997”, 1999 edition.
Human capital (Table 2): Secondary school enrolment ratio in 1990, from “World
Development Indicators 1997”, World Bank.
Initial knowledge stock (Fig. 7 and Table 2): Gross domestic expenditure on R&D as a
percentage of GDP in 1993, from “Main Science and Technology Indicators”, OECD, 1999
edition.
Knowledge stock increases (Table 2): Gross domestic expenditure on R&D, compound
annual growth rate 1993-1999 at constant prices, from “Main Science and Technology
Indicators”, OECD, 1999 edition.
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