Download “New” Growth Theory

Barebones Outline of “New” Growth Theory
revised 2006
Background to the classroom presentation and discussion, Part D
© David M. Nowlan 2005
As we saw, neoclassical growth theory predicts that the growth rate of output per capita
will converge across regions and that only the levels of output per capita might vary
because of conditioning variables.
The reason for this neoclassical outcome is that the only augmentable factor of
production, capital, is subject to diminishing returns to scale and technical change is
given, it’s exogenous, so cannot be influenced. A region therefore can’t pull itself up by
its own bootstraps.
The idea, however, that technical change is unrelated to investment, especially
investment in research, in innovation and in education, is not very intuitive. In many
ways, it seems more reasonable to assume that technical progress is embedded in new
capital and higher levels of education, including “learning by doing” on the job and
learning by association with others. If these are the principal sources of technical change,
then it may well be the case that investment in capital, physical and human, does not
entail diminishing returns as in the neoclassical model.
If new techniques or new products are continuously embedded in new capital, i.e., in
investment, then additional investment, even with a constant labour input, may create
constant (or even rising) additions to output, rather than falling incremental additions
(which occur in the neoclassical world where there is no change in the nature of a unit of
capital, only a disembodied, exogenous force that renders all capital more productive).
Economies of scale associated with investment of various kinds may also give rise to
constant or increasing returns to capital alone.
Such economies of scale might be associated with research and advanced education and
learning: a new idea or invention, once conceived, doesn’t disappear when it is first
applied. It isn’t used up but rather is available to be used over and over again without
limit. The original cost of the invention is a one-time or fixed cost that gives rise to
subsequent economies of scale as it is exploited many times. Similarly, people investing
in advanced education create externalities across clusters or across industries as their
knowledge and abilities become applied not simply narrowly to their jobs but, through
learning by association – think of cross-fertilization of ideas in a cluster or in a city – but
to other activities as well. Education also builds on itself, with each generation of
learners absorbing the knowledge of their predecessors and then expanding it, another
externality.
These ideas, which initially, in the mid-1980s, sprang from a disenchantment with the
predictive power of the neoclassical model, lead us to the “new” growth theory. The
essence of the new growth theory is that productivity-improving changes in the economy
do not float down exogenously but are embedded in investments of various sorts,
research, invention, innovation and education. Because of this, investment and
endogenous action to increase investment may not encounter diminishing returns;
rather it may have a lasting effect not just on the level of economic well being (i.e.,
output per capita) but also on its rate of growth.
Let’s look at this in a very simple model. Suppose, as in the neoclassical growth model,
output is a multiplicative function of capital and labour:
Y  K  L
In keeping with the idea that productivity improvements, technical change, is embedded
in capital and that capital is not subject to diminishing returns, let   1 (the same
assumptions could entail   1 , but to keep it simple I’ll work with   1 ). We will
keep the labour exponent at less than 1:   1 . Thus,
equation 1
Y  KL
With this as our new growth model, we see that the marginal product of capital
expanding against a constant labour force is itself constant, not diminishing. Equation 2
shows this: the marginal product of capital is constant, with a constant labour input.
Y
MPK 
 L
equation 2
K
Notice from equation 1, that if you double the amount of capital, with labour constant,
output doubles. This is also an implication of equation 2. Thus the output relationship in
this new growth model exhibits increasing returns to scale.
Notice that, with increasing returns to scale, productive factors, K and L, cannot all be
paid their marginal products. In the case of equation 1 and 2, if K were to be paid its
marginal product, then its share of output would be given by
MPK K L K Y

 1
Y
Y
Y
In this case, the returns to capital would equal the whole of output, with nothing left for
labour, so capital must in this economy receive less than its marginal product. This is
consistent with the notion that capital investment, which includes investment in research
and innovation and, in this case, education, creates a positive externality. Thus private
investors may receive private returns that are less than their contribution to the economy
and so the level of investment, of various types, in market equilibrium may be less than
optimal. An opening exists for the argument that basic research yielding public
knowledge and investment in education should be subsidized.
Before leaving the production relationship of equation 1, let’s look at the implications for
output growth, GY , and output per capita, GY  GL . From 1,
equation 3a
GY  GK  GL
equation 3b
GY  GL  GK  (1   )GL
2
Suppose that the labour force is a given constant size, so that GL  0 . From 3a, the
growth rate of output is equal to the growth rate of capital, and so therefore must the
growth rate of output per capita, shown in 3b, also equal the growth rate of capital.
Unlike the neoclassical world, if we can somehow raise the growth rate of capital, we can
also raise, permanently, the growth rate of output and output per capita.1
Let me turn now to a slightly expanded version of a production function that might
underlie a new growth model, one that differentiates between physical capital, K, and
human capital, H. This production function is still multiplicative. K and H are involved,
but we’ll include a labour term, L , representing unskilled labour, workers stripped of
their educational or similar qualifications which are captured in H .2
Yt  K t H t Lt
Equation 4
In the new growth theory, the capital factors, K and H in this equation, are together not
subject to diminishing returns. That is,     1 .
In order to make comparison with the neoclassical model as clear as possible, suppose
that we have     1 and   0 . Having the marginal product of unskilled labour set
to zero may well reflect the situation in many regions with an excess of unskilled labour,
high unemployment but a shortage of educated workers. Or, it may simply reflect the
fact that the marginal product of a labour unit is entirely the result of some amount of
education or skill associated with a person; a person with absolutely no education or skill
would not be employable.
These assumptions allow equation 4 to be written in a form that looks a lot like the
neoclassical production functions we have worked with:
Yt  K t H t1 L0t  K t H t1
Equation 5
Now, differentiate the log form of equation 5, as we did with the neoclassical model, to
get the new growth equation:
GY  GK  1   GH
Equation 6
The saved resources of this region may be put into augmenting the physical capital stock
and/or increasing the amount of human capital. Suppose that a proportion s of the
1
Recall that in the neoclassical model, an increase in the savings rate would result in a temporary increase
in the rate of growth of output. Output per capita would rise permanently, but the rate of growth of output
per capita would ultimately return to its original level – the conditional convergence result.
2
How to measure human capital is a question about which much has been written. For our purposes, we
can think of it simply as the aggregate number of years of schooling in the economy.
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region’s output is invested in physical capital and another proportion e is invested in
human capital. Then each of these two factors will experience the following rates of
growth:
sY
eY
, as in the neoclassical world, while GH 
.
K
H
GK 
Equation 7
From equation 6, we see that the growth rate in output is a weighted average of the
growth rate of physical capital and of human capital. So, GY must be between GK and
GH , unless GK  GH in which case
GY  GK  GH .
Equation 8
Equation 7 represents equilibrium growth. As with the neoclassical model, if the
economy is out-of-equilibrium, it will move towards equilibrium. If, for example,
GK  GH , then GY  GK and GY  GH . A look at equation 7 will tell us that in these
circumstances, GK will be falling and GH increasing. This will continue until they are
both equal.
Re-write the expressions for factor growth in the following ways:


1



sY s K  H 1
H
GK 

 s 
K
K
K
Equation 9
eY e K  H 1
K
GH 

 e 
H
H
H 
Equation 10
In equilibrium, GK  GH , so from equations 9 and 10 we have
H
s 
K
1
K
 e 
H

Equation 11

H
Multiply both sides of equation 11 by   to get
K
1 
H
s 
K



H
K  H
   e    . From this,
K
H K 
4
H e
H

s   e , or
K s
K
Equation 12
Using equation 8 and either 9 or 10 along with 12, we get an expression for growth in
output in long-run equilibrium:
1 
e
GY  s 
s
 s e1
Equation 13
This result is very different from the neoclassical result. In the neoclassical model, the
growth rate of output in equilibrium was fixed at the growth rate of efficiency units of
labour. Since both the growth rate of the labour force and the rate of technical change
were determined by circumstances outside the model, the growth rate of output was
similarly beyond the reach of any variables in the model. With the new growth model,
however, equation 13 tells us that the rate of growth of output is a function of savings and
investment rates. Technical change is embodied in investment and therefore can be
varied.
If you want the region to grow faster, invest a higher proportion of output. This can raise
growth permanently, so any region investing more than another similar region can
permanently grow faster than the other.
Notice that if either s or e is zero, then growth, in long run-equilibrium, will be zero.
This has a neoclassical flavour and reflects the fact that both physical capital and human
capital taken separately have diminishing returns to scale (at least in equation 5), so you
can’t use just one to increase the growth rate. This helps explain why a region or a
country cannot expect to get growth results by investing heavily in say higher education
without also investing in the tools with which the educated people need to work, whether
these tools are computers or research space.3
Another example helps underline the difference the new perspective makes for our
understanding of regional growth. Take two regions with equal levels of output, equal
savings rates and equal rates of population growth. Suppose one region has a relatively
Y
larger population, that is, a larger L . This region will be poorer, in the sense that
will
L
3
Notice also that an optimal balance between e and s , in equilibrium, can be worked out
from equation 13. Take the derivative of GY with respect first to s and then to e to get:
1    G . If   1    , then it’s better
GY

GY
 s 1e1  GY and
 1   e  s 
Y
s
s
e
e
s
e
 1   
to invest in physical capital, but if
then it’s better in invest in human capital.

s
e
5
Y
would
L
grow faster in the poorer region and ultimately catch up with the richer region – an
example of absolute convergence. In the new-growth model, with technology the same
in the two regions and with equal savings and investment rates in physical and human
capital, output will grow at the same rate in both regions. Since population growth rates
are the same in both regions, the poorer region will therefore remain relatively poor and
never catch up with the richer region.
be smaller. In the neoclassical model, with equal savings and investment rates,
This outcome may not hold, however, if labour is mobile between the regions. If labour
migrates from the region with the lower output per capita to the region with the higher,
then the richer region will experience higher population growth rates and, therefore,
lower growth rates of output per capita. The reverse will occur in the initially poorer
region. Thus, with migration, the neoclassical convergence results reappear, at least in
this example.
Notice, however, that this reappearance of convergence requires a special set of
circumstances. The technological parameters of the growth model, in particular the
exponents on K and H,  and  , need to be the same, which implies that an additional
unit of investment has the same effect on output in the two regions. This in turn likely
requires first a relatively easy flow of ideas and new technologies among regions or
countries, second an industrial structure that with equal ease (or difficulty) in the two
regions can translate the ideas into productivity-enhancing investment, and third an
equally well trained labour force that can work with the new ideas and new technologies.
These factors then have to be combined with the labour mobility previously commented
upon.
The movement of ideas and inventions across regions or countries is very likely to have
become easier with the rapid growth of international firms’ activities in many regions and
countries. Basic research may take place principally in one location but may be applied
in many locations. Countries and regions that are part of this global loop will benefit as
well as the country in which the inventive activity took place. Similarly, people gain
education in many countries other than their home country and often then make use of
that training back home. So, with globalization we might expect to see more evidence of
convergence in output per capita growth rates than the originators of the new growth
theory had thought likely.
We should find that countries that are part of the global production and education system,
with migration possible, will show a tendency towards convergence, whereas those that
are not part of this system, usually because of underdeveloped institutions of government,
education and health, will continue to languish. This indeed is exactly the way some
researchers read the empirical evidence of global growth over the last half of the 20th
century.
We will explore some further determinants of technological change in class.
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