The
AK model of economic growth is an endogenous growth model used in the theory of
economic growth, a subfield of modern macroeconomics. In the 1980s it became
progressively clearer that the standard neoclassical exogenous growth models
were theoretically unsatisfactory as tools to explore long run growth, as these
models predicted economies without technological change and thus they would
eventually converge to a steady state, with zero per capita growth.
The
neo-classical approaches to economic growth were largely considered to be
unsatisfactory due to several inherent flaws. These models view improvements in
total factor productivity (technological progress) to be the ultimate source of
growth in output per worker, but they do not provide an explanation as to where
these improvements come from. In the language of economists, long-run growth is
determined by something that is exogenous in the model. Diminishing returns to
the accumulation of capital, which plays a crucial role in limiting growth in
the neoclassical model, is an inevitable feature of an economy in which the
other determinants of aggregate output, namely technology and the employment of
labour, are both given. However, there is a class of model in which one of
these other determinants is assumed to grow automatically in proportion to
capital, and in which the growth of this other determinant counteracts the
effects of diminishing returns, thus allowing output to grow in proportion to
capital.
These
models are generally referred to as AK models, because they result in a
production function of the form Y = AK, with 'A' constant. The AK model is
actually considered the first version of endogenous growth theory. However, the
earlier version of this model go back to Harrod (1939) and Domar (1946) who
assumed an aggregate production function with fixed coefficients. Frankel
(1962) developed the first AK model with substitutable factors and knowledge
externalities, with the purpose of reconciling the positive long-run growth
result of Harrod Domar with the factor substitutability and market clearing
features of the neoclassical model. The Frankel model showed a constant savings
rate, whereas Romer (1986) developed an AK model with intertemporal consumer
maximization. The idea that productivity could increase as the result of
learning-by-doing externalities, was put forth by Arrow (1962). Then Lucas
(1988) developed an AK model where the creation and transmission of knowledge
occurs through human capital accumulation. Similarly, we can cite a number of other
models which have followed the AK framework. Hence, it is important here to
examine this approach and its contribution to economic theory.
The Cross-Country Difference in Growth
There
are large differences in per capita income across countries. Of total world
income, 42 per cent goes to those who make up the richest 10 per cent of the
world's population, while just 1 per cent goes to those who make up the poorest
10 per cent (World Bank, World Development Indicators, 2014). This points
towards not only unequal distribution of world income across different
countries but also differences in their growth rates. The key sources of these
differences can be numerous depending upon the national policies and
institutions. Hence, it is very important to understand how some countries can
be so rich while some others are so poor as the income differences have major
welfare consequences. The differences in growth rate across economies have
actually widened the income inequalities. Acemoglu (2007) has indicated that even
in the historically brief post- war era, the world has witnessed tremendous
differences in growth rates across countries and these have ranged from
negative growth rates to average rates as high as 10 per cent a year. During
this period some of the countries have grown at a faster pace, some at a slower
rate and some stagnated after growing for a short period. It is being believed
that much of these differences in economic growth cannot be wholly attributed
to the post-war era alone as during this period, the "world income
distribution" has been more or less stable, with a slight tendency towards
becoming more unequal Further, the Maddison data has suggested that much of the
divergence took place during the 19th century and early 20th century. It is
important to observe that the process of rapid economic growth started in the
19th, or perhaps in the late 18th century and then takes off in Western Europe,
while many other parts of the world do not experience the same sustained
economic growth. The high levels of income today in some parts of the world are
owed to this process of sustained economic growth, and this process of
differences in economic growth has also caused the divergence among nations.
This divergence took place at the same time as a number of countries in the
world started the process of modern and sustained economic growth. Therefore
understanding modern economic growth is not only interesting and important in
its own right, but it also holds the key to understanding the causes of
cross-country differences in income per capita today. The endogenous growth
theories largely owe these differences or divergences in economic growth to the
institutions, policies, technologies along with the levels of investments and
other transitional dynamics. These theories also pointed out the importance of
investment in human capita along with that of physical capital to explain these
divergences. These theories also point out that the technology differences
across countries include both genuine differences in the techniques and in the
quality of machines used in production, but also differences in productive
efficiency resulting from differences in the organization of production, from
differences in the way that markets are organized and from potential market failures
and how the human factors handle these technologies with effects on productive
efficiency.
Hence,
a detailed study of "technology", physical capital and human capital
as correlates of economic growth is necessary to understand both the world-wide
process of economic growth and cross-country differences. It is important to
examine the sources of income differences among countries that have (free)
access to the same set of technologies, but do not generate sustained long-run
growth. A full analysis of both cross-country income differences and the
process of world economic growth requires models in which technology choices
and technological progress are endogenized. Hence, we can start with simple AK
model.
Simple AK Model
As
we have already discussed that the first version of endogenous growth theory
was AK theory, which did not make an explicit distinction between capital
accumulation and technological progress. In effect it lumped together the
physical and human capital whose accumulation is studied by neoclassical theory
with the intellectual capital that is accumulated when innovations occur. An
early version of AK theory was produced by Frankel (1962), who argued that the
aggregate production function can exhibit a constant or even increasing marginal
product of capital. This is because, when firms accumulate more capital, some
of that increased capital will be the intellectual capital that creates
technological progress, and this technological progress will offset the
tendency for the marginal product of capital to diminish. In the special case
where the marginal product of capital is exactly constant, aggregate output Y
is proportional to the aggregate stock of capital K:
Y
= AK
Where A is a positive constant that reflects the level of technology and 'K' here is taken in a broader sense as it includes physical as well as human capital. This model shows
constant marginal product to capital (as`MP_k=\frac{dY}{dK}=A`) indicating that long run growth is possible. Thus, AK model is a simple way of illustrating endogenous growth. Assuming a closed economy, the savings are equal to investment under conditions of full employment.
Since
savings are the function f income and capital depreciates at a constant rate
i.e. '8' the change in capital stock can be traced through following equations.
I
= S = s.Y = s.AK
and, since capital depreciates at a constant rate, the change in capital stock i.e. K can be expressed as K = s. Y - d . K. This change in capital stock can also be represented by a diagram given below.
In
this figure Y-axis show output per worker while the X-axis show the capital
stock. The line Y=AK having a constant slope shows the constant marginal
productivity of capital; the line S = sY is the gross investment line while the
line dK
shows the depreciation line or the total replacement investment. The difference
between the gross investment line and the replacement line i.e. area between
S=s.Y line and dK
line shows net investment in the economy which is positive and increasing.
The
growth of capital stock can be found by dividing both sides of the equation
showing change in capital stock with 'K', we get:
`\frac{K^'}K=s.\frac YK-\delta`
Since, Y = AK, i.e. Y / K = A therefore, above equation can be rewritten as
`\frac{K^'}K=s.A-\delta`
As,
growth of output is equal to the growth of capital stock, Further, assuming
that . A>d
the growth of capital stock as well as growth of output i.e, showing that the
economy will be ever increasing as compared to the Solow model.
The AK Model and the Solow Model Compared for Rising Saving Rate
.jpg "AK Model")
Figure
2 compares the impact of rate of change in savings upon the growth of income.
The top part of the diagram shows the levels of income and the bottom part
shows the growth rate of the same. In the upper part, we can see that a once
for all increase in saving rate in to time period leads to an ever growing
income curve (shown as In y) in case of AK model while in case of Solow model,
the income increases initially but ultimately reaches at the same level after tl.
This can be observed through the angle 'g'. In case of Solow type growth path, as
savings increase or say, due to exogenous change in technology in to time
period, the income curve immediately and its slope rises as we can see that the
size of angle 'g'
increases from 0 to g1
but after t1 time period, it again comes back to the previous level
i.e. g0.
However, in case of AK type growth the increase in income is forever, shown by
an ever increasing curve and once for all increase in the size of angle g0
to g1.
The growth path can better be elaborated through growth rate of income in the
lower segment of the diagram.
AK Model with Human Factor
In
its more realistic form, we can also add labour as an input along with capital.
In this context, first of all, we can discuss Arrow's model with knowledge
spillovers. In this model, the production function for final output can be
written as
Y
= B. Kα L1-α
-----------(1)
which
is a Cobb-Douglas type production function showing constant returns to scale
with inputs K and L. In a model with technology and population growth as
exogenous factors, the population, equal to labour input L, can be normalized
to one and the individual firm takes total factor productivity B as given.
However, we suppose that B is in fact endogenously determined. Specifically,
the accumulation of capital generates new knowledge about production in the
economy as a whole. In particular, we assume that
B
= AK1-α
--------------(2)
where,
A is constant and is greater than zero i.e. A > 0
That
is, an incidental by-product of capital accumulation by firms in the economy is
the improvement of the technology that firms use to produce. Technological
progress, modelled as a by-product of capital accumulation, is external to the
firm. Combining the two preceding equations gives
Y
= A.K. L1-α -----------
(3)
This
is exactly the AK model above, noting that L = 1. However, in further
formulation of the AK model, we can include human capital as a separate
variable having a positive effect upon the level of output. Thus, more skilled
labour force will be assumed to produce more output than an unskilled
individual, and the total stock of such "skills" is called human capital.
Crucially, human capital can be accumulated through education. Thus, both types
of capital can be accumulated-this turns out to imply that the model has similar
properties to the AK model. In this perspective, we can have a production function
of the following type:
Yt
= At. Kt1-α
Ht 1-α ------(4)
Where
Kt is physical capital and Ht is human capital. So. growth is determined by:
`\frac{Y_t^'}{Y_t}=\frac{A_t^'}{A_t}+\alpha\frac{K_t^'}{K_t}+\left(1-\alpha\right)\frac{H_t^'}{H_t}`
Assuming
that like physical capital, human capital also depreciates for given
attainments. This can be understood in this way that if a person does not
updates its knowledge, the knowledge accumulated so far depreciates in a
dynamic economy. For simplifying the analysis, let us assume that both the
physical as well as human capital depreciates at the same rate, then we can
easily derive the equations for accumulation of each type of capital stock.
In
sum, AK model gives a new framework for the long run growth of the economies.
However, there are still some reasons to doubt the predictions about long-run
growth generated by this class of models. The first line of criticism is
related with the non-accumulable factors. In the real world, there are factors
of production that are in fixed supply, such as land, or that cannot simply be
accumulated indefinitely such as energy. Remember that the AK model results are
of a knife-edge variety: Any move away from all factors being accumulable, and
we move back to the Solow model results. Moreover, similar treatment to all
type of human capital is also criticised by many as they say that the strict
parallel between human capital and physical capital in the model just described
is probably not completely accurate. For instance, not all expenditures on
education will produce the same effect on output. The marginal boost to
aggregate output of primary teaching is altogether different to that of higher
education; training the non- skilled informally and the formal training of the
professional also differ in their marginal returns. By clubbing all these
different types of human capital together hardly proposes an effective policy
suggestion for countries with a varied structure of human capital. Another
source of the difficulties faced by the AK model is that it does not make an
explicit distinction between capital accumulation and technological progress.
In effect it just lumps together the physical and human capital.
Discussion of AK Model
Non-Accumulable
Factors : In the real world, there are factors of
production that are in fixed supply, such as land, or that cannot simply be
accumulated indefinitely such as energy. Remember that the AK model results are
of a knife-edge variety: Any move away from all factors being accumulable.
Treatment
of Human Capital : The strict parallel between human
capital and physical capital in the model just described is probably not
completely accurate. For instance, not all expenditures on education will
produce the same effect on output. The marginal boost to aggregate output of
teaching someone how to read and write is presumably greater than that of
masters in economics! Thus, there may be limits to which one can increase
growth just by boosting educational enrollment.
The AK model and Policy Debates
1.
The fact that savings rate can affect the growth rate (and in a big way) made
the AK model very popular in policy discussions.
2.
It makes government policy potentially very important for growth.
3.
In a famous paper, Lucas (1990) called tax cuts on savings as the "largest
genuinely free lunch I have seen in 25 years in this business."
4.
Even today when candidates fiercely debate tax policy, an important part of
discussion revolves around growth
5.
King and Rebelo (1990, JPE): The "welfare effect "of a 10 percent
increase in income tax is 40 times larger in an (AK) endogenous growth model
(65% of consumption) than it is in a neoclassical growth model (1.6% of
consumption)
6.
Stokey and Rebelo (1995) and Lucas (1990) argue that if endogenous growth models
are calibrated to plausible values the effect on welfare is not likely to be
large
7.
Because if tax differences are so important for growth, how come countries like
Sweden with extremely high tax rates grow as fast as the US?
Shortcomings of the AK model
1.
Growth is the outcome of accidents---actions that are completely unintentional.
2.
Externalities must be substantial: For example, the capital bought by an
investor contributes twice as much to others production than to his/her own.
Same for human capital: Your education benefits others more than it benefits
you.
3. Alternatively stated, the Social return on many types of investments far exceed their private return. If externalities are really that big, individuals will typically find a way to capitalize on them (A doctor will not distribute advise on the street, etc.)
भारतीय अर्थव्यवस्था (INDIAN ECONOMICS)
व्यष्टि अर्थशास्त्र (Micro Economics)
समष्टि अर्थशास्त्र (Macro Economics)
अंतरराष्ट्रीय व्यापार (International Trade)
