macro chapter 8 Productive externalities and endogenous growth
Productive externalities and endogenous growth
although the Solow models studied so far are quite successful in accounting lor
many important aspects of economic growth, they have one major limitation: by
treating the rate of technological change as exogenous. they leave the economy's
long-run tsteady state) growth rate unexplained. Solow models therefore belong to the
class of so-called exogenous growth theories.
In this part of the book we will try to answer a very big question that remains
unanswered: how can we explain the rate of technological change which is the source of
long-run growth in income per capita? The search lor an answer to this fundamental
question takes us to the modern theory of elldogellous growth where the long-run rate of
growth in GDP per person is truly endogenous.
A model that ;explains' the long-run rate of growth in GDP per worker is one that
endogenizes the underlying rate of technical change, that is, makes this rate depend on
basic model parameters. Hence. by an endogenous growth model we mean a model in
which the long-run grm.vth rate of technology depends on basic model parameters such as
the investment rates in physical and human capital. the population growth rate, or other
fundamental characteristics of the economy. An endogenous growth model therefore
allows an analysis of how economic policies that affect these basic parameters will ailect
long-run growth in income per capita. This is an important, and some will say the defining.
feature of endogenous growth models: structural economic policy has implications
for grm.vth in output per capita in the long run.
In this chapter and the two following ones we \>Viii study endogenous growth theory.
The models to be presented can be divided into two categories. In both categories there will
be aggregate production functions involving a variable :1 1 that describes ;technology', but
there will be no assumption of exogenous technological progress such as A,+ 1 = ( 1 + g)A ,.
where g is exogenous.
One category contains models that include an explicit description of how technological
progress, A,+ 1 - A, in period t, is produced through a specific production process
th at requires inputs of its own. Since we think of the production of technological progress
as arrising from research and development, we call such models R&D-based models of
e11dogenous growth. These are the subject of the next two chapters.
The other category does not have an explicit production process for technological
improvement. but assumes that the A1 of the individual llrm depends positively on the
aggregate use of capital. or of output. because of so-called 'productive externalities'. This
implies that the aggregate production function, as opposed to that of the individual firm ,
will have increasing returns to scale. As we will see, this will result in growth in GDP per
worker in the long nm without any exogenous technological progress being assumed.
The models in this category are referred to as endogenous growth models based on productive
externalities, and they are the subject of this chapter.
8.1 Growth model with productive externalities
In Chapter 3 we explained why growth in income per worker had to vanish in the long
run according to the basic Solow model. The explanation was related to constant returns to
capital and labour and the associated diminishing returns to capital alone. Let us tal<e the
explanation once more in a way that is well suited to our present purposes.
The production function was F1 - Kf L:-a. exhibiting constant returns to K1 and L1
and diminishing returns to K1 alone (c1 < 1). Consequently there were also diminishing
returns to capital per worker in the production of output per worker: y1 = k;'. Now, assume
that there is some growth in the labour force, say at 1 per cent per year. If capital also
increases by 1 per cent per year, as in the steady state of the basic Solow model, then
because of constant returns to capital and labour, output will increase by 1 per cent per
year. Hence output per worker will be constant.
How could there possibly be growth in output per worker? With the production
function of the basic Solow model, only if capital increases by more than 1 per cent per
year. If capital increases at a given constant rate of more than 1 per cent per year. say at
2 per cent. then each year output will also increase by more than 1 per cent and hence
there '.vill be growth in output per worker. Indeed, the (approximate) growth rate of
capital per worker, gf, '.viii be constant and equal to 1 per cent. and the (approximate)
growth rate in output per worker. gf. will be gf = ag ~. Hence, there '.vill be a constant and
positive growth rate in output per worker. However. the formula gy = ag ~ already reveals
the problem. Since a< 1, the growth rate of income per worker is smaller than the
assumed grm>Vth rate of capital per worker. Therefore the continued constant growth rate
in capital per worker cannot be sustained by savings. What happens is that as long as
capital increases faster than labour there will be more and more capital per worker and
this implies, due to diminishing returns, that additional units of capital per worker create
less and less additional output per worker, and hence, less and less additional savings per
worker. As a consequence, growth in capital, and in GDP, per worker will have to cease in
the lon g run.
This reasoning suggests that if there were increasing returns to capital and labour,
then long-run growth in GDP per worker would be possible without exogenous technological
progress. If both capital and labour were increasing at a rate of 1 per cent per year.
then, simply because of increasing returns, output would be increasing by more than
1 per cent per year. And in this case growth would not have to cease in the long run , since it would be unnecessary to build up more and more capital per worker to sustain grm.vth
and hence diminishing returns would not be a problem. 1
Increasing returns to scale at the aggregate level therefore seems to be a potential
source of endogenous growth.
although the Solow models studied so far are quite successful in accounting lor
many important aspects of economic growth, they have one major limitation: by
treating the rate of technological change as exogenous. they leave the economy's
long-run tsteady state) growth rate unexplained. Solow models therefore belong to the
class of so-called exogenous growth theories.
In this part of the book we will try to answer a very big question that remains
unanswered: how can we explain the rate of technological change which is the source of
long-run growth in income per capita? The search lor an answer to this fundamental
question takes us to the modern theory of elldogellous growth where the long-run rate of
growth in GDP per person is truly endogenous.
A model that ;explains' the long-run rate of growth in GDP per worker is one that
endogenizes the underlying rate of technical change, that is, makes this rate depend on
basic model parameters. Hence. by an endogenous growth model we mean a model in
which the long-run grm.vth rate of technology depends on basic model parameters such as
the investment rates in physical and human capital. the population growth rate, or other
fundamental characteristics of the economy. An endogenous growth model therefore
allows an analysis of how economic policies that affect these basic parameters will ailect
long-run growth in income per capita. This is an important, and some will say the defining.
feature of endogenous growth models: structural economic policy has implications
for grm.vth in output per capita in the long run.
In this chapter and the two following ones we \>Viii study endogenous growth theory.
The models to be presented can be divided into two categories. In both categories there will
be aggregate production functions involving a variable :1 1 that describes ;technology', but
there will be no assumption of exogenous technological progress such as A,+ 1 = ( 1 + g)A ,.
where g is exogenous.
One category contains models that include an explicit description of how technological
progress, A,+ 1 - A, in period t, is produced through a specific production process
th at requires inputs of its own. Since we think of the production of technological progress
as arrising from research and development, we call such models R&D-based models of
e11dogenous growth. These are the subject of the next two chapters.
The other category does not have an explicit production process for technological
improvement. but assumes that the A1 of the individual llrm depends positively on the
aggregate use of capital. or of output. because of so-called 'productive externalities'. This
implies that the aggregate production function, as opposed to that of the individual firm ,
will have increasing returns to scale. As we will see, this will result in growth in GDP per
worker in the long nm without any exogenous technological progress being assumed.
The models in this category are referred to as endogenous growth models based on productive
externalities, and they are the subject of this chapter.
8.1 Growth model with productive externalities
In Chapter 3 we explained why growth in income per worker had to vanish in the long
run according to the basic Solow model. The explanation was related to constant returns to
capital and labour and the associated diminishing returns to capital alone. Let us tal<e the
explanation once more in a way that is well suited to our present purposes.
The production function was F1 - Kf L:-a. exhibiting constant returns to K1 and L1
and diminishing returns to K1 alone (c1 < 1). Consequently there were also diminishing
returns to capital per worker in the production of output per worker: y1 = k;'. Now, assume
that there is some growth in the labour force, say at 1 per cent per year. If capital also
increases by 1 per cent per year, as in the steady state of the basic Solow model, then
because of constant returns to capital and labour, output will increase by 1 per cent per
year. Hence output per worker will be constant.
How could there possibly be growth in output per worker? With the production
function of the basic Solow model, only if capital increases by more than 1 per cent per
year. If capital increases at a given constant rate of more than 1 per cent per year. say at
2 per cent. then each year output will also increase by more than 1 per cent and hence
there '.vill be growth in output per worker. Indeed, the (approximate) growth rate of
capital per worker, gf, '.viii be constant and equal to 1 per cent. and the (approximate)
growth rate in output per worker. gf. will be gf = ag ~. Hence, there '.vill be a constant and
positive growth rate in output per worker. However. the formula gy = ag ~ already reveals
the problem. Since a< 1, the growth rate of income per worker is smaller than the
assumed grm>Vth rate of capital per worker. Therefore the continued constant growth rate
in capital per worker cannot be sustained by savings. What happens is that as long as
capital increases faster than labour there will be more and more capital per worker and
this implies, due to diminishing returns, that additional units of capital per worker create
less and less additional output per worker, and hence, less and less additional savings per
worker. As a consequence, growth in capital, and in GDP, per worker will have to cease in
the lon g run.
This reasoning suggests that if there were increasing returns to capital and labour,
then long-run growth in GDP per worker would be possible without exogenous technological
progress. If both capital and labour were increasing at a rate of 1 per cent per year.
then, simply because of increasing returns, output would be increasing by more than
1 per cent per year. And in this case growth would not have to cease in the long run , since it would be unnecessary to build up more and more capital per worker to sustain grm.vth
and hence diminishing returns would not be a problem. 1
Increasing returns to scale at the aggregate level therefore seems to be a potential
source of endogenous growth.
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