PÉTER CSILLIK1 – TAMÁS TARJÁN2
Schumpeterian development of the leader-follower
countries in the 19th–20th Century
The technology-follower countries follow an S-shaped growth path on the long-run, having
normalized by the technology leader’s trend-line. It has been possible to devise several growth
models that follow S-shaped paths to the long-term steady state, but all use naïve-expectation
or short-term consumption optimization, not the rational Ramsey method of consumption
optimization that has been known and used most widely for eighty years.
The present paper contains two new findings related to the Ramsey type optimization.
It shows
(1) that the Ramsey type optimization is not capable, in conjunction with the AghionHowitt aggregate productivity parameter regulation, of modeling any S-shaped path
but proves that
(2) it is possible to model the S-shaped transitional dynamics with a slightly modified
Aghion-Howitt type regulation by taking the capital into account, as well.
Introduction
According to the growth theory of Schumpeter, economic growth is due to innovation but
these innovations can be materialized only in the case of the presence of a bank which
grants credit to finance the necessary investments. The Behavioural Macroeconomics
(Akerlof) is the most recent research branch of the modern growth theory that is joining to
the behavioural micro models. A pioneer on the field there was Ramsey (1928) who
supposed the rational behaviour of the consumer when he decided on his consumption.
Later on this model became known as Ramsey-Cass-Koopmans (1965).
The New Growth Theory has mostly been brought into the word in order to the nowadays
most important factor of production, the technological progress, not being any longer an
exogenous endowment but an endogenous motive power of the macro modeling subject
to the Ramsey-type rational behaviour.
It is well known and studied that the diffusion of innovation follows an S-shaped curve in
time, i.e. the number of new adopters plotted as a frequency histogram against time follows
a Rogers’ bell curve.3
But it is not so well known that the long-term growth path, of the technology follower
countries, normalized by the trend of the technological leader, follow S-curves, as well.
Several models have been developed that can model the S-shaped curve of growth, but
all work with naive expectation or myopic optimization of consumption and not with
Ramsey-type one being the best known and most common during the last eight decades.
Related to the latter, the present paper contains two new results:
1) It proves that the Ramsey model matched to the aggregate productivity
parameter regulation of Aghion-Howitt is not able to model S-shaped curves.
1
Budapest Business School, Research Centre, C.Sc.(Economics), senior research fellow
Budapest Business School, Research Centre, C.Sc.(Economics), senior research fellow, E-mail: [email protected]
3
Since the pioneering work of Rogers [1962] on the diffusion of technology and innovations there is an abundance of literature on it.
2
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BUDAPESTI GAZDASÁGI FÔISKOLA – MAGYAR TUDOMÁNY ÜNNEPE, 2011
2) Based on the heuristics of the aggregate parameter regulation of Aghion-Howitt
but having involved the capital as well, a slightly modified aggregate productivity
parameter regulation than that of Aghion-Howitt, is already able to model the
S-shaped transitional dynamics.
The central issue of the paper is to answer the question when and under which conditions
an S-shaped growth paths may be modelled by the help of the “new growth theory”4?
Such growth models will be discussed here where the so called “aggregate productivity
parameter” (A) is one, while the other is the augmented notion of capital (K) encompassing
both the physical + human capital:
This (A) having become autonomous and new factor of production associated with R&D
and innovation due to the parameter regulation of Aghion-Howitt [1992] (based on
Schumpeter's Creative Destruction paradigm) has for nowadays become the most
important research branch of the endogenous growth theory. This branch is formed by
the models with improvements in the quality of products that are called shortly as
Schumpeterian growth theory while the models with an expanding variety of products of
Romer [1990], where the aggregate productivity varies in the function of the extent of the
variety of products.
Model elements and reasoning are borrowed from both branches in the paper. Jánossy
argues straight for the Schumpeterian approach even a quarter century before them when
had proved that “The diffusion of the new technology is realized through the qualitative
change of labour”5.
The paper proves that the regulation of Aghion-Howitt matched with the Ramsey model
is unable to model S-shaped paths. Finally, being aware of this important negative result, it
will be proved that a slightly modified regulation based on the heuristics of that of AghionHowitt encompassing (together with A) the capital K, as well, is already able to perform
S-shaped transitional dynamics6.
Definition of the technology follower country
First, a technological leader country is supposed to exist that defines the frontiers, in its
both factors of production [i.e. per capita capital (kF) and aggregate productivity parameter
(AF)] and that follow exponential paths with the same constant rate x of growth.
Let k≡ke–xt denote the discounted capital value [with same rate of x as above for the
technological leader] of a given country with capital and k* the value of k in its “steadystate“, that supposed to exist, as well.
4
5
6
See Barro, R. J.–Sala-I-Martin, X. [1995] and Dietmar [1995].
See Jánossy [1969]
Let us remark that several growth models have been developed for the S-curves to reach steady states (transitional dynamics), but all work with naive
expectation or myopic optimization of consumption and not with Ramsey-type one being the best known and most prevailing during the last eight
decades and in the heuristics of parameter regulation of Aghion-Howitt in Csillik-Tarján [2007], [2009], and [2010].
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CSILLIK PÉTER – TARJÁN TAMÁS
A country is considered to be a technology follower if the aggregate productivity
parameter (A) of a follower country never surpasses the frontier parameter (AF) of the
technological leader and its capital level at starting point is less than its discounted value
in the steady sate,
i.e. A≤AF and k(0) < k*.
[It is important to remark that the index F in AF isn’t associated with the initial of the word
Follower but to the Frontier!]
Definition of the S-shaped growth path
A growth path y≡y/yF (yF≡ext) of a technology follower country is called S-shaped;
if the path y (t) is twice continuously differentiable, d[ln y (0)]/dt < 0, d2[ln y (0)]/dt2 > 0 and
for appropriate 0 < tm < ti time points: d[ln y (tm)]/dt = 0, d2[ln y (ti)]/dt2 = 0, hold and the
first and second order derivatives, according to time, change their signs in points tm and
ti, respectively.
In words, the above tells us that the path ln y (t) first forms a U-turn, where the minimum
in time is denoted by tm and after that there is a point of inflexion in ti where a left turn is
followed by a right turn.
Now, it will be squarely demonstrated, on the basis of historical statistics, that during the
last 138 years, from 1870 (starting from the second industrial revolution) the so called
technology follower countries pursue S-shaped paths, in line with the above definition.
It’s important to mention that the above definition of the S-shaped paths – in some sense
– incorporates both modified versions of the Jánossy hypothesis7: the 2nd ‘modified Janossy’
hypothesis and the 3rd ‘reverse Janossy’ hypothesis, as well, because both modified versions
are linear approximation of an S-shaped path in some segments of time.
Trend analysis by ‘French curve ruler’
Let us consider the three charts below made of PPP-based real GDP per capita data of
Maddison [2010]. On the first two ones, the factual data are shown for 12 technology
follower OECD countries. As a first approach to demonstrate the S-shaped curves, 3rd-order
polynomials as trendlines are fitted to the factual time series (as the simplest odd curves
among trend-curves) to perform our trend analysis by using charting software instead of the
traditional ‘French curve rulers’. The factual time series and the 3rd-order polynomial trends
are plotted on the following three charts. On the Chart 1 the six OECD countries are found
that started in 1870 at a level below 70%, compared to the US, on Chart 2 the other six
countries that started between the levels of 70-85%, compared to the US, as technology
followers. The above defined S-shaped form is the most characteristic on the first chart, i.e.
in case of the OECD countries that have had a gap at the start more than 30%, compared
to the US (see Chart 1). The countries in the first hundred years describe only a U-shaped
pattern. There are four of the six paths nearly parallel to each other while Japan starts as the
poorest and ends among the bests. Spain does the same, but just inversely.
7
See the above “Brief summary of the definitions of Jánossy hypothesises”.
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BUDAPESTI GAZDASÁGI FÔISKOLA – MAGYAR TUDOMÁNY ÜNNEPE, 2011
Chart 1
Six OECD countries being poorer than the US with more than 30 per cent, in 1870
y≡ y/yF factual data and polynomial trends
The S-shaped character holds only for the 3rd-order polynomial trend of Sweden – among
the further six (see Chart 2 below) technology follower OECD countries having started
between the levels of 70-85% – while the other five describe a U-pattern during the last
138 years. However the factual time series data (that assign the 3rd-order polynomial
trends) are becoming strictly horizontal or even declining during the last two decades; thus
bearing the promise of an S-shaped future path, as well.
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CSILLIK PÉTER – TARJÁN TAMÁS
Chart 2
Six OECD countries being poorer than the US with 15– 30 per cent, in 1870
y≡ y/yF factual data and polynomial trends
Finally, the remaining five OECD countries at the start having been richer than the US, that
are not, of course, considered to be technology followers from 1870 like the above twelve
countries. Nevertheless they have shown some form of behaviour of the followers after
WW2. In spite of this we think interesting to show them also as the above twelve, in the
same frame of reference with their polynomial trends.
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BUDAPESTI GAZDASÁGI FÔISKOLA – MAGYAR TUDOMÁNY ÜNNEPE, 2011
Chart 3
Five OECD countries being richer than the US, in 1870
y≡ y/yF factual data and polynomial trends
Summarizing the above three charts it may be concluded that the plotting of Maddison‘s
historical time series and the simple, so called trend analysis by “French curve ruler”
provides sufficient grounds for the hypothesis that the growth paths of the technology
followers, normalized by the US (long-term exponential) trend, form S-shaped paths.
This S-shaped path as a stylised fact has to mean that the role of technology following just
after the second industrial revolution, for a long time and for a lot of countries, has entailed
a disadvantage/decline compared to the technological leader that could only later on turn
to an advantage regarding the growth rate or the speed of catching up. The question thus
arises obviously that the Leader-Follower models of economic growth (including the most
recent endogenous models, as well) are they able to grasp and reflect this phenomenon?
In growth theory for more than eight decades the Ramsey model is the most prevailing
while in the endogenous growth theory the Aghion-Howitt-type “aggregate productivity
parameter”-regulation that is rooted in the (nearly also eighty year-old) Schumpeterian
growth theory. This “aggregate productivity parameter”-regulation has a strange limitation
as being self-regulated, while the regulation of such a parameter may depend not only on
A itself but on the entire economy, as well. The present paper will prove that a model
based on that two above pillars is unable to model the S-shaped path and what is more
the followers tend to their steady state with decreasing growth rate, but a slight (and
according to the above, necessary) modification the Aghion-Howitt-regulation will already
be able to pursue an S-shaped path.
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CSILLIK PÉTER – TARJÁN TAMÁS
In case of the Ramsey model, is the regulation of Aghion-Howitt able to
model S-shaped curves?
In growth theory the appearance of technical advance as a separate factor of production
starts by Arrow‘s [1962] „learning-by-doing” hypothesis. – Similar phenomenon is studied
by Jánossy [1966] as well, in a chapter of his book entitled “The machine as master of man”.
– This new factor of production, after having introduced the notion of the so called
“technology frontier” (see Nelson-Phelps [1966]) and the parameter regulation of (AghionHowitt [1992]) based on the Schumpeterian “Creative Destruction” paradigm, becomes
the most important research branch of endogenous growth theory. This branch is formed
by the models with improvement in quality of products, that is called shortly as
Schumpeterian growth theory while the models with an expanding variety of products of
Romer [1990], where the aggregate productivity varies in the function of the extent of the
variety of products. The “identification of the state of technology with the number of
varieties of products should be viewed as a metaphor; it selects one aspect of technical
advance and thereby provides a tractable framework to study long-term growth.”8 The
models with improvements in the quality of products use “another metaphor in which
progress shows up as quality improvements for an array of existing kinds of products.”9
The two approaches may be viewed as complementary to each other. Model elements
and reasoning are borrowed from both branches in the paper.
Endogenous Ramsey model
Let (y≡Y/L) denote the per capita output and assume a neoclassical production function,
y = F(k, A),
where the “aggregate productivity parameter” (A) and the per capita (physical+human)
capital (k) are the two factors of production. [Y = F(K, L·A); y ≡ Y/L, k ≡ K/L.]
Let the per capita consumption be denoted by c(t), the wage income per adult person by
w(t), the real rate of return by r(t) and finally the household’s net assets per person by a(t).
Let the Capital market equilibrium be supposed to be true:
a(t) ≡ k(t), for ∀ t.
Firms maximize profits
L·F(k, A)–L·w–L·(r+d)k=max(k, L)
∂F(k, A)/∂k = r+d
F(k, A)–(r+d)k=w
[Let us remark that ∂w/∂k=0 but ∂w/∂A=∂F(k,A)/∂A doesn’t necessarily annul.]
Each household wishes to maximize overall utility, U, as given by
U≡∫0∞u(c)e–pt dt = max (c)
subject to
–the flow budget constraint for the household
8
9
See Barro, R. J.–Sala-I-Martin, X. [1995] p. 213.
See in the same page.
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BUDAPESTI GAZDASÁGI FÔISKOLA – MAGYAR TUDOMÁNY ÜNNEPE, 2011
ȧ=w+ra–c, [Let us remark that in case of Capital market equilibrium the flow budget
constraint for the household: ȧ=w+ra–c, transforms equivalently into the economy’s
resources constraint: k̇=y–δk–c]
– the aggregate productivity parameter regulation
Ȧ=Z(k, A),
– the transversality condition
[a(t)·exp(–dx)]=0
Let the shortened forms of partial derivatives of Z(k, A) be introduced as follow:
∂a≡∂Z/∂a, ∂A≡∂Z/∂A,
Then the Hamiltonian expression (with zero population growth) is:
J=u(c)e–ρt+ν(w+ra–c)+μZ;
∂J/∂C=0 => ν= u’(c)e–ρt => ν̇[u”(c)ċ–ρu’(c)]e–ρt =>
=> ν̇/v=u”(c)ċ/u’(c)–ρ => ċ/c=–(ν̇/v+ρ)/θ(c); where θ(c)≡–[u”(c)c]/u’(c).
∂J/∂a=νr+μ∂a=–ν̇ => ν̇/ν = –r–(μ/v)∂a,
∂J/∂A=ν∂F/∂A+μ∂A=μ̇ [introducing a notation of η≡μ/ν] => η̇=μ̇/ν-ην̇/ν=
=–∂F/∂A – η∂A+η(r+η∂k)
The transversality condition
t
lim[a(t)·exp(–∫0r(x)dx)]=0
t∞
The modified Keynes-Ramsey rule
ċ/c=(r–ρ+η∂k)/θ(c);
[Let us remark that if specially ∂k≡∂Z(k, A)/∂k=0 holds, i.e. in an other (equivalent) interpretation Z(k, A) doesn’t depend on k but only on A and thus Z(k, A)=Z(A); then the above
equation reduces to the so called Keynes-Ramsey rule10: ċ/c=(r–ρ)/θ.]
The law of motion of k(t), c(t), y(t),
(o) y=kα A(1–α)
(i) r=α y/k–δ
(ii) k̇=y–δk–c
(iii) ċ/c=(r–ρ+η∂k)/θ
(iv) η̇=–(1–α)y/A+η(r–∂A)+η2∂k;
(v) Ȧ=Z(k, A)
Let us introduce for the capital to output ratio the notations of κ≡k/y&y*≡ ω ext then the
above (o)–(v) equations in ‘steady state’ have the forms (o*)–(iv*) below:
(o*) 1=κ*α(A/y)*(1–α) => (y/A)* =κ*α/(1–α)
(i*)
r*=α/κ*–δ
(ii*) x=1/κ*–δ–(c/k)*
(iii*) xθ =r*–ρ+η*∂k* => θ ={r*–ρ+η*∂k*}/x
(iv*) 0=–(1–α)κ*α /(1–α) +η*(r*–∂A*)+η*2∂k* ;
10
See e.g. Barro–Sala-i-Martin [1995] p. 65. equation (2.10).
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CSILLIK PÉTER – TARJÁN TAMÁS
The determination of η* is as follows:
If ∂k*≡(∂Z/∂k)*≠0, then the second order equation below holds for η*:
η*= 0.5{∂A*–r*±[(r*–∂A*)2+4∂k*(1–α)κ*α/(1–α)]0,5}/∂k*
Finally the transversality condition
Let us remark that if r*>x holds and since lim r=r* & lim κ=κ* hold, as well, there exist
t∞
t∞
an ε and a T such that r(t)≥r*–ε>x, κ(t)≤κ*+ ε & y(t)≤(ω+ε)ext hold for all t≥T>0.
κy·exp(-∫1Tr(x)dx)≤(κ*+ε)(ω+ε)ext·exp[–(r*–ε)(t–T)] =
=(κ*+ε)(ω+ε)exp[(r*–ε)T]·exp[(x–r*+ ε)t] → 0, because (x–r*+ε)< 0.
Thus if r*>x holds the transversality condition holds, as well:
lim[k(t)·exp(–∫10r(x)dx)]=lim[κy·exp (–∫10r(x)dx)]=0.
t∞
t∞
The regulation of Aghion-Howitt isn’t able to model S-shaped curves (subject to a Ramsey
model)
Lemma 1: If ∂k≡∂Z/∂k=0 and at the start k(0)<k* holds, then (strictly) monotonically
decreases in [0,∞).
Proof: E.g. Barro–Sala-i-Martin [1995] p. 90. APPENDIX 2C: „PROOF THAT γk DECLINES
MONOTONICALLY IF THE ECONOMY STARTS FROM k(0)<k*”, where Dk≡ke–xt.
Definition: The parameter regulation of Aghion-Howitt being called as inverse measure of
distance to the technology frontier, AF11:Z≡A[x+q(AF/A–1)], where q>0.
Lemma 2: In case of parameter regulation of Aghion-Howitt, γA
 monotonically decreases
in [0,∞).
Proof: Since, by definition A≤AF, holds γA=Z/A=[x+q(AF/A–1)]≥x holds, as well.
≡Ae–xt monotonically increases and A e–xt is (supposed to be)
Thus γA–x≥0 implies that A
F
constant. Therefore γA
=γA–x=q(AF/A–1) monotonically decreases in [0,∞). Q.E.D.
Lemma 3: In case of parameter regulation of Aghion-Howitt if at the start, k(0)<k* holds,
then γy (strictly) monotonically decreases in [0,∞).
Proof: According to Lemma 1 γk (strictly) monotonically decreases and from Lemma 2 γA

monotonically decreases in [0,∞). Thus their linear combination: γy = αγk + (1–α) (strictly)
monotonically decreases in [0,∞) . Q.E.D.
Theorem: In case of parameter regulation of Aghion-Howitt the technology follower path
y≡ye–xt cannot be S-shaped one, because it has neither point of minimum, tm nor point of
inflexion, ti.
* holds, thus from the
Proof: In the case of a technology follower country at start, k(0)<k
above Lemma 3 γy (strictly) monotonically decreases in [0,∞), thus it has no point of
11
Where x ≡ µn (γ –1), is the long-term growth of the US, µn is the frequency of the leading-edge innovations and q ≡ µm is that of the imitation.
See footnote 10, as well.
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BUDAPESTI GAZDASÁGI FÔISKOLA – MAGYAR TUDOMÁNY ÜNNEPE, 2011
inflexion, ti. Since in „steady state” γy*=0 holds, thus all during [0,∞) γy necessarily positive,
i.e. γy >0. Thus the path γy (strictly) monotonically increases in [0,∞) and so has no point
of minimum, tm either. Q.E.D.
The modified parameter regulation of Aghion-Howitt is able to model
S-shaped curves
Among the aggregate productivity parameter regulations of a)–f) apart from that of AghionHowitt’s d) –as it can be seen, Ȧ depends solely on A and the technology frontier, AF of
it– in all the other cases, apart from A, Ȧ depends on: K̇/K, h, Y/L and finally on K/L. Thus
the fact that the change of technology, Ȧ, depends on the output, Y, from Conlisk [1967]
hasn’t any new, we now add to it only that similarly to A the output y, as well, has got a
frontier yF,12 and the per capita output, y being an appropriate proxy of the expanding
variety of products of a follower, then a similar heuristics to the metaphor of Romer and
Aghion-Howitt one arrives to the parameter regulation, the so called „inverse measure of
distance to the output frontier, yF”:
·
A= μn(γ–1)A+μm(yF/y–1)A,
where the left μn(γ–1)A is the change attributed to the own innovation, while the right
change, μm(yF /y–1)A is coming from imitation. ‘In the former case the country is making
a leading-edge innovation that builds on and improves the leading-edge technology in its
industry. In the latter case the innovation is just implementing technologies that have been
developed elsewhere’13
If for μm, [μm≡μn(γ–1)/(1–yF)] is chosen, one gets just the above equation, the so called
‘inverse measure of distance to the output frontier, yF’.14
Ȧ/A=μn(γ–1)+μm(yF/y–1)=μm(1–yF)+μm(yF/y–1)=μm yF(1/y–1).
Let the aggregate productivity parameter15 regulation have now the form of
Z/A≡x+q(ωvF/v–1), where v≡kβA(1–β).
Thus the partial derivatives are:
∂k≡∂Z/∂k=qβ ω vF A/(vk) and ∂A≡∂Z/∂A=x–q+qβ ω vF/v,
and the parameter values let be as follow:
α
0.8415
δ
0.08
ρ
0.12
κ*
4
θ
0.252
12
y*
0.85
q
x
0.0186 0.0186
<
<
r*≡ α/κ*–δ
0.130375
Let us remark that the technology frontier AF generally means the frontier of Aus (aggregate productivity parameter of the US).
See Aghion, P. – Howitt, P. [2006] p. 7.
14
It is important to remark that the latter derivation has got an additional advantage, i.e. in case of the Aghion-Howitt‘s parameter regulation the stipulation of growth stemming from both the leading-edge own innovations and imitations are necessary than at the cited latter derivation for the followers is enough to assume that imitate/adapt only if their „steady state” are sufficiently far from and parallel with that of the technological leader.
According to the factual data the „steady state”-s of followers ranges between 70-90 per cent of that of the leader the US.
15
Where x ≡ µn (γ –1), is the long-term growth of the US, µn is the frequency of the leading-edge innovations and q ≡ µm is that of the imitation.
See footnotes 10 and 12, as well.
13
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CSILLIK PÉTER – TARJÁN TAMÁS
Since the condition r*> x obviously holds for the above Z, the transversality condition holds,
as well.
On the chart below (Chart 4), find the fitting (to Japan/US per capita GDP data) of the
S-shaped model path of the modified parameter regulation of Aghion-Howitt; where the
values of the model parameters are as follow:
Chart 4
Fitting of S-shaped model path to Japan/US per capita GDP data
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