chapter 7 the solow model with scarce natural resources
so far our presentation of growth theory has been a bit like 'playing Hamlet without
the Prince of Denmark'. The theory of economic growth was initiated by the great
classical economists. Adam Smith. Thomas Malthus and David Ricardo. They
thought that the presence of land as a fixed factor in production implied a severe constraint
on the economy's long-term growth potential. In their time agriculture accounted
for a much greater share in total production than today, so it is no wonder that the
economists of the time considered land to be essential. However, today land is still in fixed
supply and remains an important input to aggregate production. Not only is agriculture
still of importance. but any kind of production requires at least some space and hence
land. Moreover, if we interpret the services of 'land' in a broad sense to include all the life
support services of the natural environment - such as its ability to generate clean air and
water and to absorb the waste products of human activity - it should be clear that 'land' is
vital for economic activity.
The analysis of the classical economists taught them that the presence of an irreplacable
factor in fixed supply would imply a tendency towards long-run decline in
income per capita and eventually stagnation at a low income level. In modern terms their
argument was the following.
Assume there is some population growth and disregard technological progress for a
moment. Assume further that the economy manages to build up capital at the same speed
as the population increases. Note that this L~ exactly what we found in the basic Solow
model's steady state: capital and labour were growing at the same rate (the capital- labour
ratio was constant). Because of constnnt returns to cnpitnl and labour, this implied that output
was also growing at this rate. leaving output per worker constant (possibly at a high level).
Asswne now, realistically, that there are three inputs in the aggregate production
function: capital. labour and land. Suppose further that land is in llxed supply. From the
replication argument this production function should have constant returns to capital,
labour and land, implying diminishing returns to the combinntion ofcnpital nnd labour. Now.
as the inputs of capital and labour increase proportionally. as we assume they do, while
land stays fixed, output will grow less than proportionally to labour and capitaL and
hence output per worker will decline in the long run. In a nutshell, this is the classical
argument why income per capita has to decline in the long run as a consequence of population growth. The ultimate root of this decline L~ the diminishing returns to capital
and labour arising from the presence of a fixed factor, land.
This is not even the end of the story according to the classical economists. As income
per capita falls, savings per capita would also fall, so capital would not really be able to
increase at the same speed as labour in the long run. This would imply an even faster
economic decline. Furthermore, with decreasing incomes, population growth would also
eventually be brought to a halt. Fertility might stay unchanged, but because of the
miserable living conditions of workers, mortality would be so high that the labour force
would stay constant. This would eliminate population growth, the original source of
declining incomes, but in the 'end state' income per capita would have stagnated at a
subsistence level. In the words ofMalthus:
Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in
an arithmetical ratio. A slight acquaintance with numbers will show the immensity of the first
power in comparison of the second ... This implies a strong and constantly operating check on
population from the difficulty of subsistence ... its effects are ... Among mankind, misery and
vice. (from Chapter I of Thomas Malthus' famous book, An Essay on Population, 1798).
The labelling of economics as the 'dismal science' arose from this idea. The economic
history of the last 200 years suggests that the pessimism of the classical economists was
unwarranted. Income per capita has been growing at rates around 2 per cent per year
over very long periods in many Western countries. Thls does not mean that the classical
economists were wrong in their reasoning. only that there were perhaps some other
factors to which they did not pay enough attention. In fact, the modelling of thls chapter
will show that the classical economists were in some sense righ t: land does imply a growth
drag such that population growth leads to negative growth in income per capita in tile
afJsence of tecill'lological progress.
The possibilities of prolonged technological progress and birth control was not
emphasized by the classical economists. As we saw in Chapter 5, and in Chapter 6 as well,
technological growth is a source of long-run growth in income per capita. Fixed natural
resources and technological progress seem to be a.vo countervailing influences on longrun
growth . Which one will be the strongest in the long run is a crucial issue and a central
theme in this chapter.
We will study a Solow growth model that also includes a fixed factor, 'land·, as an
input in the aggregate production function . This will enable us to take a stand on the
essential issue whether fixed land or technical progress has the strongest influence on
long-run growth. The model we consider comes from an article by the economL~t William
D. Nordhaus, 1 tal<ing up a discussion about the limits to growth provoked by the
pessimistic views expressed in some famous books from 19 72 and 199 2 known as the
'Limits to Growth' reports. 2
Like Nordhaus, we will go one step further and also present a model v.rith exhaustible
natural resources. If a fixed factor such as land, the supply of which stays unchanged as it is used in production. can imply a drag on growth, one should expect that exhaustible
resources such as oil and gas, which disappear as they are used. can imply even more of a
growth drag. Our model will show that this intuition is indeed correct. We will again use the
model to address the essential question: is the growth drag so strong that it will ultimately
bring growth in income per capita to a halt despite continued technological progress?
the Prince of Denmark'. The theory of economic growth was initiated by the great
classical economists. Adam Smith. Thomas Malthus and David Ricardo. They
thought that the presence of land as a fixed factor in production implied a severe constraint
on the economy's long-term growth potential. In their time agriculture accounted
for a much greater share in total production than today, so it is no wonder that the
economists of the time considered land to be essential. However, today land is still in fixed
supply and remains an important input to aggregate production. Not only is agriculture
still of importance. but any kind of production requires at least some space and hence
land. Moreover, if we interpret the services of 'land' in a broad sense to include all the life
support services of the natural environment - such as its ability to generate clean air and
water and to absorb the waste products of human activity - it should be clear that 'land' is
vital for economic activity.
The analysis of the classical economists taught them that the presence of an irreplacable
factor in fixed supply would imply a tendency towards long-run decline in
income per capita and eventually stagnation at a low income level. In modern terms their
argument was the following.
Assume there is some population growth and disregard technological progress for a
moment. Assume further that the economy manages to build up capital at the same speed
as the population increases. Note that this L~ exactly what we found in the basic Solow
model's steady state: capital and labour were growing at the same rate (the capital- labour
ratio was constant). Because of constnnt returns to cnpitnl and labour, this implied that output
was also growing at this rate. leaving output per worker constant (possibly at a high level).
Asswne now, realistically, that there are three inputs in the aggregate production
function: capital. labour and land. Suppose further that land is in llxed supply. From the
replication argument this production function should have constant returns to capital,
labour and land, implying diminishing returns to the combinntion ofcnpital nnd labour. Now.
as the inputs of capital and labour increase proportionally. as we assume they do, while
land stays fixed, output will grow less than proportionally to labour and capitaL and
hence output per worker will decline in the long run. In a nutshell, this is the classical
argument why income per capita has to decline in the long run as a consequence of population growth. The ultimate root of this decline L~ the diminishing returns to capital
and labour arising from the presence of a fixed factor, land.
This is not even the end of the story according to the classical economists. As income
per capita falls, savings per capita would also fall, so capital would not really be able to
increase at the same speed as labour in the long run. This would imply an even faster
economic decline. Furthermore, with decreasing incomes, population growth would also
eventually be brought to a halt. Fertility might stay unchanged, but because of the
miserable living conditions of workers, mortality would be so high that the labour force
would stay constant. This would eliminate population growth, the original source of
declining incomes, but in the 'end state' income per capita would have stagnated at a
subsistence level. In the words ofMalthus:
Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in
an arithmetical ratio. A slight acquaintance with numbers will show the immensity of the first
power in comparison of the second ... This implies a strong and constantly operating check on
population from the difficulty of subsistence ... its effects are ... Among mankind, misery and
vice. (from Chapter I of Thomas Malthus' famous book, An Essay on Population, 1798).
The labelling of economics as the 'dismal science' arose from this idea. The economic
history of the last 200 years suggests that the pessimism of the classical economists was
unwarranted. Income per capita has been growing at rates around 2 per cent per year
over very long periods in many Western countries. Thls does not mean that the classical
economists were wrong in their reasoning. only that there were perhaps some other
factors to which they did not pay enough attention. In fact, the modelling of thls chapter
will show that the classical economists were in some sense righ t: land does imply a growth
drag such that population growth leads to negative growth in income per capita in tile
afJsence of tecill'lological progress.
The possibilities of prolonged technological progress and birth control was not
emphasized by the classical economists. As we saw in Chapter 5, and in Chapter 6 as well,
technological growth is a source of long-run growth in income per capita. Fixed natural
resources and technological progress seem to be a.vo countervailing influences on longrun
growth . Which one will be the strongest in the long run is a crucial issue and a central
theme in this chapter.
We will study a Solow growth model that also includes a fixed factor, 'land·, as an
input in the aggregate production function . This will enable us to take a stand on the
essential issue whether fixed land or technical progress has the strongest influence on
long-run growth. The model we consider comes from an article by the economL~t William
D. Nordhaus, 1 tal<ing up a discussion about the limits to growth provoked by the
pessimistic views expressed in some famous books from 19 72 and 199 2 known as the
'Limits to Growth' reports. 2
Like Nordhaus, we will go one step further and also present a model v.rith exhaustible
natural resources. If a fixed factor such as land, the supply of which stays unchanged as it is used in production. can imply a drag on growth, one should expect that exhaustible
resources such as oil and gas, which disappear as they are used. can imply even more of a
growth drag. Our model will show that this intuition is indeed correct. We will again use the
model to address the essential question: is the growth drag so strong that it will ultimately
bring growth in income per capita to a halt despite continued technological progress?
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