Understanding Recent Developments in Growth Theory
Lars Weber
Brandenburg University of Technology Cottbus
Faculty Mechanical Engineering, Electrical Engineering and Economical Engineering
Chair of Macroeconomics
P.O. Box 10 13 44
03013 Cottbus / Germany
Phone: +49 355 69-3967 Fax: +49 355 69-3624
Email: lars.weber@ tu-cottbus.de
Abstract
The growth theory has, through so-called ‘endogenous’ or new growth theory taken on
decisive impulses. This contribution delivers an overview of the various extensions
without going into detail about the mathematical observations and the main focus on
supply-side orientated approaches. The main goals of the growth theory are to
understand the exponential climb of the population’s income or also the per-capita
increase and to divert from the extensions for policy makers. The paper uses stock-flow-
graphs to visualize the major loops. Because changes tend to be incremental I adopt
standard textbook models first. All models are used in economic teaching with additional
simulation and extensions. Later on students learn to modify those models.
Key words
endogenous growth theory, system dynamics, model, simulation, visualization
1 Introduction...
Exponential inl is multiply underestimated .
11
1.2 Technical progress is a key growth factor...
2 General Structure of Growth (Systems a, “
2.1 Stocks and flows...
2.2 Arrows create caus
2.3 Exponential growth has a positive total effect on the variable...
3 Solow-Model... veewosuaannaavseawwoseuereveee
3.1. Land A grow exogenous and exponentially
3.2 Convergence thesis
4 AK-Model ou.
4.1 Overcoming the decreasing marginal productivities ..
4.2 Divergence thesis..............
4.3 Introduction of the human capital factor. 5
5 Uzawa-Lucas-MOodel....eesessessesseeseesesseessesssesseseesessssueesusesssesuessessesseaseas LO
5.1 Education as a stock
5.2. Humane capital stock can explain missing convergences .
6 Romer-Model........eesecsees
6.1 R&D-Models in general
6.2 Schumpeter’s ideas are the basis for the R&D-Model
6.3 The goods “knowledge” cannot be excluded..
6.4 The good “knowledge” has no rivalry .
6.5 Structure of the Romer-model..........
6.6 Knowledge increases the amount of intermediate goods
7 Jomes-MOdel .....eseesessessesssesseessessesnesssessesseessessuesseesseeseeseesneesesesessessessesseesseasness
7.1 Theory and empiricism do not always agree ...
7.2 Integration of the education sector in the Romer- Model
8 Summary...
9 References..
1 Introduction
At first sight, the current economic problems of today seem to have very little to do with
growth politics. Numerous times it is emphasized that the economic trend is paralyzed,
due to the insignificant demand for consumer goods, or the challenges presented in
foreign trade. However, if one examines this more exactly, for example in globalization,
demographic change and unemployment have much to do with resilient economic
growth. Thus, without steady and appropriate economic growth, we will hardly be able
to master the challenges of globalization or dismantle the numbers of unemployment on a
larger scale.
1.1 Exponential growth is multiply underestimated
It is said again and that growth is not everything. However growth is a decisive size in
order to increase the wealth of a nation. An entirely simple example should clarify this:
There are two countries. Exponential growth follows at the same time the formula:
(1) A,=A,-e"' , with Ao=Beginning amount and n = Growth rate of country A and
(2) B,=B,-e°* , with Bo=Beginning amount and g = Growth rate of country B.
One country A is exactly twice as "rich" as country B. It counts:
(3) Ay =2-By
The stronger country A grows at a slower rate than country B: n <g. The difference of
both rates amounts to 1%. How long would it take for the poorer country to catch up to
the richer country? Answer: circa 70 years. How long does it take if the growth difference
amounts to 2%? Now it would only take around 36 years. And at 5% about 15 years until
country B becomes equal. It is important to understand the dynamics behind these
examples. Over time even only small difference in the growth rate makes a large
different.
1.2 Technical progress is a key growth factor
For a long time, growth theories have been paid less attention. However, the neoclassical
growth model from Robert Solow (1956) changed this. Many economists, also notable
noble prizewinners, dedicate themselves to the growth theory. Technical development is
now seen as an influential size behind economic growth. Solow’s model included an
explicitly technical progress, but was only integrated as an exogenous factor into the
model. Joan Robinson (1962) noticed correctly, that technical progress does not fall “like
Manna” from the sky. Therefore, the newer growth theories try to describe the technical
progress within the models.
It is the goal of this contribution to deliver an overview of the over the supply-side
growth theories, also named endogenous or new growth theory. Nevertheless there are
yet further current growth disciplines, for example, the empirical growth research or
growth theories with an emphasis on demand-side. These also entail new aspects, thus the
concept of newer growth theories can be misleading.
Starting with the model from Robert Solow, we can explain the AK-Model, the
Uzawa-Lucas-Model, and the Romer-Model and in conclusion the Jones-Model. The
main consideration in the representation lies at the same time on the comparative
analysis. The exact proof for this balance can be found in the cited literature.
2 General Structure of Growth (Systems Archetypes)
In general the exponential growth follows the form: A Stock K that changes over time
through the sum of the in- and outflows per time t with K =K,—K,,. The growth is
exponential if the net change of K depends on the stock K and a certain factor multiplied
becomes: K =K, ,-n. A certain share of the stock K leads to an increase if n>0. This
basic growth scenario is in figure 1 represented.
figure 1 - basic scenario of exponential growth
2.1 Stocks and flows
The stock K is represented as a rectangle. K is determined by the changes in K(Punkt),
which one can observe on the thick straight arrow. K(Punkt) represents the change in
stock K from t to t+1. In addition, there can also be auxiliary variables and constants.
Auxiliary variables use the calculation between stocks, flows, constants and other
variables. They change in each period t. In the graph, they border on the circular.
Constants are independent of the time. They are exogenous. In this example, there are no
assisting variables, however a constant: n.
2.2 Arrows create causal links
The arrows show the relation of the variables, stocks and flows to each other. A “+” on
the arrow tip means identical direction. In figure 1, an increasing growth rate n would
lead K(Punkt) climbing. A “-” sign symbolizes climbs towards the opposed direction,
therefore a variable increases, which leads to a decrease in the dependent variable. The
strength of this relationship is mathematically determined and cannot be indicated
explicitly in this representation. The large advantage of a graph instead of a pure
mathematical representation is to be sure that the observer can optically recognize, which
relations form the model.
2.3 Exponential growth has a positive total effect on the variable
When the total effect of a loop is reinforcing, represented by a large “+”, then the
statement of the effect direction is in the middle of the loop. Within the total effect of the
loop is negative because of an uneven number of negative polarities amounts to a
marking of “-”. Generally the model structure determines the behavior and same
structures evoke similar patterns of behavior. Y et we will see that the simple exponential
basic structure can be regained in all introduced growth models.
3 Solow-Model
The Solow-model with Harrod-neutral technical progress and population growth consists
of three important factors:
= the capital stock K,
= the population or also the labor forces supply L and finally
= the technical progress A.
The population’s income Y is a Cobb-Douglas-production function with
Y =K*%.(A-L)“ and consists of the three stocks. Alpha is the production elasticity
and amounts to about 0.30.
3.1L and A grow exogenous and exponentially
The labor force grows exogenous and exponentially with the rate n. The technical
progress also grows exogenous and exponentially with the rate g. Formally, it is
represented by:
(4) L=L,-e™*
(5) A=A,-e%*
A constant share S of the income Y is saved:
(6) S=s-Y ;s =Savings.
Investments I are the change of the capital stock over time, therefore K . Because the
identity I = S in a closed economy counts, we can formulate the first order equation of
the capital stock with:
(7) K=l=s-Y
The formulation of the capital stock therewith not exogenous is purported, but rather
diverted out of the model. Depreciations are thereby considered within the net
investments.
The model includes exponential growth patterns at three places:
= In the growth of the technical progress A,
= In the growth of the population L and
= In the growth of the capital stock K.
This becomes especially evident if we represent the structure of the Solow-model
graphically (figure 2). The picture is comparable with figure 1, however it includes
several of these simple growth patterns.
aS
Cam
figure 2 - Solow-Model with technical progress and population growth
>
J. Robinson expressed criticism that in this model the technical progress is not fully
explained, stating that one can recognize this because the technical progress surly
influences the population’s income Y, and yet no variable except the growth rate g
changes the progress. Therefore, it is only explained exogenously.
The model aims at a related steady state on k= (with k = capital intensity in
efficiency units). The variables still grow exponentially, however the ratio k=
remains constant k*. One can show that the equilibrium Y and K with the growth rates of
the technical progress and population growth grows: gY =gK=gL +gA. Interesting is that
the per-capita income at the steady state are still growing with the same rate as the
technical progress. The statement is clear - steady wealth increasing growth is doable
only through technical progress.
3.2 Convergence thesis
If one compares two countries with different initial values of the capital stock than
follows, according to the Solow-model, that the capital stocks will approach each other
over time. When the growth rate of the population and the technical progress are equally
large and the saving ratio is identical than steady state will be the same. This is
designated as an absolute convergence. When the saving ratios or the growth rates of the
population or of technical progress differ, then they simply approach each other. This is
designated as a relative convergence. Therefore, it is only a matter of time before the
wealth of these countries adjusts itself. This is due to the decreasing boundary
productivities in the production factors. In 1992 Mankiw/Romer/Weil presented a famous
study, which empirically examined this approach and within the approach confirmed the
core statements of the Solow-model.
4 AK-Model
4.1 Overcoming the decreasing marginal productivities
The Rebelo model (1991) also called AK-Model goes a step further and builds on the
Solow-model. The decreasing marginal productivities of the input factors should be
overcome. Thereby, adjustments of these countries would be explicable, and also a
durable divergence or also overtaking would be possible.
4.2. Divergence thesis
The drift is designated as a divergence thesis. Rebelo undertakes some decisive
variations from the Solow basic model. First the technical progress A is no longer
Harrod-neutral and laborsaving, but rather Hicks neutral. This means that it no longer
functions as productivity factor for work L, but rather as a total productivity factor.
Secondly, a human capital factor H is introduced. It takes the place of the technical
progress for the Harrod-neutral. Thus, the new production function reads:
(8) Y=A-K*-(H-L)*
4.3 Introduction of the human capital factor
This is similar to the Solow-model. However, the decisive step is the determination of the
human capital factor. It is no longer exogenous, but rather becomes endogenous and is
described in the model with:
Verbally this means that the efficiency of the factor work L is determined by factor H.
H is larger when the capital intensity grows. For example, through the uses of a computer
(stands for K) at a workplace (stands for L) the output of the work can be increased. Use
of H -t in the new and summarized production function includes:
(10) Y =A-K%; with 0 <<1.
Other equations, like the growth of A and L or the movement equation of the capital
stock are identical to the Solow-model.
figure 3 - AK-Model
Figure 3 shows the structure of the AK-model. Preceding this, the Solow-model comes,
by which a new calculation shows the difference in human capital. The capital stock K is
once again doubly reinforced. The consequence of this is the description of overcoming
the decreasing marginal productivity of the capital. The model no longer shows a steady
state. The AK-model is attributed the endogenous growth theory. On the first glance this
is remarkable, because the technical progress A is still declared exogenous and reinforces
only the accumulation process. However, the AK-model is an attempt to get around the
decreasing marginal productivity in capital and for the first time the meaning of the
human capital is underlined. Although the human capital cannot yet be calculated, it
nevertheless shows that a connection exits with the capital stock. This is explained
therewith that the capital stock is composed of real capital and human capital.
5 Uzawa-Lucas-Model
The Uzawa-Lucas model does not try to integrate the human capital as one of two stocks
K orL, but rather it models it directly as an own stock. The weakness of the AK-Model is
that the human capital is not directly accumulative. Uzawa presented this idea in 1965.
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Lucas took up this model and developed it further. Usually it is represented as an
Uzawa-Lucas-Model.
5.1 Education as a stock
The capital stock increases through net investments. However, how does one increase
human capital? The answer is short: through investments in education. The labor force
supply L is replaced by the human capital. The labor force supply is described
quantitatively and is no longer qualitative. Therefore, it is no longer an absolute quantity
for labor forces, but rather a quality. Macro economically the human capital H is the labor
force supply weighted with the average qualification level. By doing this the working
hours u of an average worker can now be calculated into the production sector Y and in
(1-u) for the advanced education divide. Thus, the amount of education enlarges the
human capital stock. Thus, the human capital appears:
(11) H =u-H +(1-u)-H
The variation of the human capital stock results from:
(12) H =(1-u)-H,,-B; B = technology parameter of the education sector.
A high productivity B yields a faster increase in human capital H. The original model
also included the depreciation of the human capital, for example through the retirement
from professional life or outward migration from the economic system. This not
explicitly observed in this contribution, however, but H is rather understood as a net
increase. The human capital stock grows consequently exponentially with:
(13) H, =H,-e®
Next to the productivity B, yet the spent time (1-u) and the initial value Ho, the human
capital also is important.
The remaining time u of the human capital flows, as already indicated, towards the
production sector. The people’s income Y can be described again in a similar form to the
Cobb-Douglas- production function:
(14)Y =A-K*%-(u-H)'*
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A constant part s of the people’s income directs over the identity I = S to increase the
capital stock with
(15) K =I=s-Y
The technical progress A is Hicks-neutral and grows exponentially in accordance with:
(16) A, = A, -e
figure 4 - Uzawa-Lucas-Model
Figure 4 represents the Uzawa-Lucas-Model graphically. A gain the basic resemblance to
the previous models is evident also here - three stocks K, H and A lead to the growth of
Y. All three grow exponentially. The model includes a development sector and a final
production sector. An important difference is in the growth of the human capital. This is
determined through the size of (1-u) and is changeable, thus, u is still exogenous.
However, u becomes “endogenous” through the amount of time spent. Each economy can
determine how much time is designated to education. Lastly, the entire human capital is
the power that determines the per-capita-wealth Y /H.
5.2 Humane capital stock can explain missing convergences
With the Uzawa-Lucas-model the absence of a convergence of the countries can also be
explained. The reason for this can be that the human capital stock is too slight or also
12
the growth of H is too little. This new knowledge comes vis-a-vis the AK-model. There
the absence of convergence is due to the savings.
6 Romer-Model
6.1 R&D-Models in general
All previously introduced models are similar, in that they offer a good explanation for
growth pattern, however the technical progress A is always exogenous. The work of
Grossman-Helpman changed this. They incorporated an intermediate goods sector,
borrowed from Dixit and Stiglitz. The model from Grossman-Helpman is not explicitly
introduced here because Romer published a further development, which connects the
ideas from Grossman-Helpmans with the extension of the Solow model. Thus, the
Romer-model is introduced. In addition to this, there are other models that can be
summarized by the “R&D-Model”. The basic idea behind was that research and
development sets stimuli for economic growth. In the following, the origins of these
theories are also clarified.
6.2 Schumpeter’s ideas are the basis for the R&D-Model
This group of models is constructed on the basis of Schumpeter’s ideas. Already in the
1920's, Schumpeter explained that competition represents a sequence of innovation and
transfer processes. Firms support research and development in order to secure themselves
by creating a monopoly over time. Through a transfer process, they encourage
customers from other companies to try their new product. By doing such they enable the
chance for disbursed research costs and development costs and have a pioneer advantage
and competitive advantage in comparison to other companies. Other firms also have a
reason to use R&D because they can generate the innovation for a positive transfer
processes.
6.3 The goods “knowledge” cannot be excluded
Human capital and knowledge (from now onward synonymous for technical progress)
are different, in that human capital is tied to the person. Here one can assume that people
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decide about themselves. Knowledge, however, manifests itself in the Schumpeterian
understanding for new products and is registered through the patents in abstract form.
Thus, the non-excludability no longer works automatically, but rather becomes
accessible for a larger group. Knowledge is therefore marked with external effects
(spill-over). It also deals with the free-rider-problem.
6.4 The good “knowledge” has no rivalry
The marginal costs of knowledge are not dependable and that knowledge can be ended by
patent protection, the feature of knowledge has no rival. Comprehensively, according to
Schumpeter’s characteristics, knowledge also serves a public good.
6.5 Structure of the Romer-model
The Romer model is divided into three sectors:
= aresearch and education sector A,
= aintermediate goods sector x and
= aconsumer goods sector Y.
For simplification, a constant is used as an input for work L. It is seen L=Lo. The work
can be used in two sectors. Either in the R&D-sector (A) or in the final product
sector (Y): L=L,+L,. At the same time, the variable z determines the share of work in
the R&D-sector and (1-z) shows the remaining share in work for the consumer goods
sector free:
(17) Ly =z-L
(18) Ly =(1-z)-L
In the R&D-sector, knowledge is accumulated through the use of work. The variation of
the knowledge is given:
‘ 1
(19) A= Au ‘ Ly (3
14
The productivity parameter is g. The growth of the knowledge follows therewith, again
the well-known exponential growth scheme:
ao AA)
6.6 Knowledge increases the amount of intermediate goods
The decisive step in Romers model is the linking of knowledge to the consumption goods
sector. It is supposed that with knowledge, procedure innovations are carried out, i.e. the
manufacture of a product occurs through the specialization of singular intermediate good.
The amount of the intermediate goods x(j) is identical to the knowledge A. Thereby, the
capital K is divided amongst all intermediate goods with x -*. One can describe the
intermediate goods sector x then as follows:
(21 x =fx(i)'di=A- x * Ke. ate
) “J J) Q= A} >
The intermediate goods sector moves toward the production function:
(22) Y =x*-L,"%
(23) Y =K*-Ab*.L.°* =K*-(A-L, )?
The last equation is almost identical with the well-known production function from the
Solow-model. The difference exists, however, in the share of work, which flows towards
the consumption goods sector. A gain it applies, that the share s from the people’s income
is saved and that identity of the capital stock S=I increases.
(24) K =I =s-Y
15
figure 5 - Romer-Model
Figure 5 shows the total Romer-model as a stock-flow-diagram. Because the labor forces
offer L as a constant and exogenous, the Romer-model consists of only two stocks, the
knowledge A and the capital stock K. With x, the intermediate goods sector is marked,
and leads directly into the consumption good sector. Again the exponential growth
pattem of the two stocks is clear. The behavior remains, therefore, comparable to the
other models, but nevertheless the explanation for exponential growth is unrelated. It is
readily conceivable that the labor force L grows at an exogenous rate n (such as in the
Solow model). If this was the case, the structure would be very similar to the Solow-
model, however other information can be diverted out of the Romer-model.
The growth rate of the R&D-sector is determined through the supply of labor forces (i.e.
engineers). The consumption goods sector Y and the R&D-sector A compete for the labor
forces L. For the first time, the Romer-model explains how innovations can influence an
economic system by a productivity increase. The innovation process is explained in the
model, even though the Schumpeterian transfer process is not yet implemented, because
older technologies are not squeezed out of the market. It should not be unexpected that in
the model the acceptance of an imperfect competition and the existence of monopolistic
competition are possible. This represents an inconsistency with the neo-classical
acceptance of perfect competition.
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7 Jones-Model
7.1 Theory and empiricism do not always agree
In spite of the brilliance of Romer’s theoretical extensions, it is difficult to find empirical
proof for the validity of the theory. Between the growth of the knowledge and the number
employed in the R&D-sector, a connection would have to exist after Romer’s idea. In
1995 Jones criticized that that unfortunately this cannot happen. Amold (1997, S. 222)
presents the basic problem very appropriately:
State of the things is, that we have an implausible model with appropriate
empirical implications (the Uzawa-Lucas-model with growth through
human accumulation) and a plausible model with doubtful empirical
implications (the Grossman-Helpman-Romer-model with growth through
R&D).
Moreover the problem existed that in the R&D-sector constant return on scales were
present. Usually neo-classical models assume decreasing return of scales. In 1995 Jones
presented a model of a semi-endogenous growth, which combined the Romer-model with
the Uzawa-Lucas-model.
7.2 Integration of the education sector in the Romer-Model
The previously presented Romer model integrates an education sector. Instead of the
constant labor force supply L, the human capital sector H is introduced. The quantity in
the foreground therefore no longer exists, but rather the quality of the labor forces.
Education increases the humane capital stock. Beside it is the population growth,
which grows at the rate n. The human capital stock can consist of three areas:
= Education enlarges the supply of human capital
" Activities in the R&D-sector increase the amount of knowledge or
= Work in the consumption goods sector supplies the people’s income Y .
This is formally written:
' Translated from German into English
17
(25) H =H, +H, +Hy with
(26) Hy =w-H_, with w = share in education
(27) H, =(1-w)-z-H_ ,with z =share in the R&D-sector
(28) H, =(@—w)-(1-z)-H
The human capital grows exponentially with:
(29) H, =H,-e"™
The R&D-sector remains basically identical with the Romer-extension, however Jones’
allows scales effects of the human capital and the already available knowledge. This is
expressed through the exponents » and y. The change over time of knowledge is
therefore:
A , Ll
(30) A= HE AG
Through the exponents, one can see the exponential effect on the stock of knowledge,
which can be over or under proportional.
figure 6 - J ones-model of the semi-endogenous growth
18
The intermediate goods sector and the consumption goods sector are similar to the
Romer-model. From that the stock-flow-diagram in figure 6 arises the three stocks K, H
and A evidently. All three grow exponentially. However, in the Jones-model human
capital growth and accumulation of technical progress (knowledge) are simultaneously
possible. The growth of the human capital, such as in the Uzawa-Lucas-model, is
dependent on the economic system and how much time will be devoted to development.
The Jones-model can present empirical observations sooner and harmoniously (see
criticism Romer-model) by using decreasing return on scales in the R&D-sector.
8 Summary
With the graphic representations, it is possible to recognize the uniform exponential
growth patterns. The behavior of the model is similar to all of the extensions; however
the statements on the growth are very different. In a further step the teaching would
include also simulations on those models because there are essential for effective systems
thinking (see also Sterman 2002, 525). But the curriculum often does not offer time for
these extensions. So we have to deal with this time constraints. The challenge would be to
shift the amount near future of the scientific time-scale. This means we have to reconsider
what is necessary for students to emancipate them for their future challenges.
The goal of this contribution was to represent different extensions of the endogenous
growth theory by means of comparing them. Constructions of the Solow-model, the
AK-model and the Uzawa-Lucas-model were used and introduced as representations of
the human capital approaches. Following the Romer-model was employed as an example
for the R&D-models. The Jones-model represents a type synthesis of the two streams in
the new growth theory. Ideas of the human capital theories and the R&D-models were
connected. The overview stood in the foreground, thus mathematical representations were
carried out in a limited fashion. For further information consultation in the recommended
literature would be suggested. Importantly it is to be shown that after the Solow-model,
which forms the point of departure in the economic growth education, yet many further
interesting extensions are available. With the graphic representations, it is possible to
recognize the uniform exponential growth patterns. The behavior of the model is similar
to all of the extensions; however the statements on the growth are very different.
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