macro chapter 7 limits to growth scarce resources
Limits to growth 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?
7.1 solow model with land
The production function of the representative, proflt-maximizing ilrm will now include
three inputs: capital. labour and land. The consumers own the land (as well as the capital
and labour), and the amount of land is flxed and does not change with its use in production.
Consumers sell the services of capital. labour and land to the firm in competitive
markets where the real factor prices are denoted by r 1• w1 and vi' respectively. The consumers
as a group (or consumers and government together) save a fraction s of total
income which is the sum of interest earned on capital, wage income and rent from land.
We should note that although the llxed factor is referred to as 'land' for convenience. its
services could be interpreted broadly to include all the life support services of the natural
environment.
7.2 solow model with oil
Land is not the only natural resource of importance for aggregate production. Nonrenewable
resources such as oil, gas, coal, metals, etc .. are a lso important. Unlike land, th e
amount of which stays constant when used in production, non-renewable resources are
depleted as they are used. Intuitively this should imply even more of a drag on growth
than does land. We will investigate this intuition by analysing a Solow model with 'oil ',
using the term 'oil' generically for exhaustible resources. The model will abstract from
the presence of land. but in the next section we will consider a model with both land and
oil.
The underlying micro world one should now have in mind is like the one for the
Solow model with land, with one exception: the natural resource owned by consumers is
no longer a constant amount ofland but rather, at the beginning of any period t, the total
remai11ing stock. R,. of an exhaustible resource. oil. The part of this stock that is used as
energy input during period twill be denoted by E1
• An important resource depletion equation
will state that the stock of oil will be reduced from one period to the next by exactly the
amount used in production in the 11rst of the periods.
7.4 unlimited substitution
This chapter deals with the very big issue whether sustain ed growth in per capita income can
essentially go on for ever despite the limitntions given by the natural envirorunent. Our main
conclusions are repeated in the follm<Ving double statement:
• ~f underlying teclmological progress and population growth rates can be l!eld at levels that have
typically been seen over long (but recent) periods in Western countries, permanent economic
growth seems to be sustainable. that is, not in conjlict with nature's finiteness.
• ~f population growth rates are at the highest levels seen in developing countries. tile limited
natural resources imply a serious drag on growth that may eliminate most or all of tile positive
influence of technological progress on income per capita.
The second of these statements means that lor the poorest parts of the world the classical
('dismal science') views are still sadly relevant. Nevertheless the overall conclusion with
respect to the possibility of sustainable growth is optimistic because of the first of the
statements above.
The mainly optimistic conclusion is, of course, reached on the premises of the models
we have considered. It is now time to face one critical modelling assumption explicitly: our
Cobb-Douglas production functions assume that there are no limits to the technological
possibilities of substituting capital and technologically augmented labour lor scarce
natural resources. Indeed. we implicitly assumed that production can continue to grow
even as the input of natural resources becomes infinitely small relative to other inputs.
Much oft he public debate on the possible limits to growth is really about the validity of this
assumption. which is clearly not an innocent one.
Influenced by the economic h istory of Western countries over the last two centuries,
most economists tend to be technological optimists, believing that the substitution
possibilities in production are essentially unlimited in the long run. They point out that
whenever a particular natural resource becomes scarce, its relative price will tend to go
up, providing strong incentives for the development of alternative production techniques
and consumption patterns which rely less on the scarce factor. According to this
view the modelling assumption of unlimited substitution should not be taken literally,
but should be seen as representing the described incentive elfect. In other words, when
the world has only one ton of copper left it is not that production will literally still use
small amounts of the remaining copper in association with extremely sophisticated
labour in accordance with an old production function. Rather. copper will have been
replaced by a substituting product invented, as copper, because of its scarcity, became
extremely expensive and the economic gains from developing a substituting product
became very large. With just one or two types of natural resources in our production
function we cannot describe this process explicitly, but the assumption of unlimited
substitution can be seen as representing it.
The same (optimistic) economists also argue that the deterioration of the natural
environment caused by the polluting activities of firms and consumers can be held in
check by intelligent use of 'green' taxes and other economic instruments in environmental
policy.
On the other hand, many natural scientists and environmentalists argue that there
are in fact limits to the possibilities of substituting other factors for certain essential
raw materials and life support services offered by the environment. The more moderate
critics doubt that the market mechanisms will always ensure the development of new substitute
techniques in time to prevent serious disruptions stemming from environmental
degradation.
Passing a fully qualified judgement on these fundamental issues requires insigh t into
the natural sciences as well as into economics. With this chapter we do not pretend to
have given a definitive answer to the question: 'Are there limits to growth?' However, we
hope to have shed some light on the basic assumptions one needs to mal<e to warrant
either grmvth optimism or growth pessimism.
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?
7.1 solow model with land
The production function of the representative, proflt-maximizing ilrm will now include
three inputs: capital. labour and land. The consumers own the land (as well as the capital
and labour), and the amount of land is flxed and does not change with its use in production.
Consumers sell the services of capital. labour and land to the firm in competitive
markets where the real factor prices are denoted by r 1• w1 and vi' respectively. The consumers
as a group (or consumers and government together) save a fraction s of total
income which is the sum of interest earned on capital, wage income and rent from land.
We should note that although the llxed factor is referred to as 'land' for convenience. its
services could be interpreted broadly to include all the life support services of the natural
environment.
7.2 solow model with oil
Land is not the only natural resource of importance for aggregate production. Nonrenewable
resources such as oil, gas, coal, metals, etc .. are a lso important. Unlike land, th e
amount of which stays constant when used in production, non-renewable resources are
depleted as they are used. Intuitively this should imply even more of a drag on growth
than does land. We will investigate this intuition by analysing a Solow model with 'oil ',
using the term 'oil' generically for exhaustible resources. The model will abstract from
the presence of land. but in the next section we will consider a model with both land and
oil.
The underlying micro world one should now have in mind is like the one for the
Solow model with land, with one exception: the natural resource owned by consumers is
no longer a constant amount ofland but rather, at the beginning of any period t, the total
remai11ing stock. R,. of an exhaustible resource. oil. The part of this stock that is used as
energy input during period twill be denoted by E1
• An important resource depletion equation
will state that the stock of oil will be reduced from one period to the next by exactly the
amount used in production in the 11rst of the periods.
7.4 unlimited substitution
This chapter deals with the very big issue whether sustain ed growth in per capita income can
essentially go on for ever despite the limitntions given by the natural envirorunent. Our main
conclusions are repeated in the follm<Ving double statement:
• ~f underlying teclmological progress and population growth rates can be l!eld at levels that have
typically been seen over long (but recent) periods in Western countries, permanent economic
growth seems to be sustainable. that is, not in conjlict with nature's finiteness.
• ~f population growth rates are at the highest levels seen in developing countries. tile limited
natural resources imply a serious drag on growth that may eliminate most or all of tile positive
influence of technological progress on income per capita.
The second of these statements means that lor the poorest parts of the world the classical
('dismal science') views are still sadly relevant. Nevertheless the overall conclusion with
respect to the possibility of sustainable growth is optimistic because of the first of the
statements above.
The mainly optimistic conclusion is, of course, reached on the premises of the models
we have considered. It is now time to face one critical modelling assumption explicitly: our
Cobb-Douglas production functions assume that there are no limits to the technological
possibilities of substituting capital and technologically augmented labour lor scarce
natural resources. Indeed. we implicitly assumed that production can continue to grow
even as the input of natural resources becomes infinitely small relative to other inputs.
Much oft he public debate on the possible limits to growth is really about the validity of this
assumption. which is clearly not an innocent one.
Influenced by the economic h istory of Western countries over the last two centuries,
most economists tend to be technological optimists, believing that the substitution
possibilities in production are essentially unlimited in the long run. They point out that
whenever a particular natural resource becomes scarce, its relative price will tend to go
up, providing strong incentives for the development of alternative production techniques
and consumption patterns which rely less on the scarce factor. According to this
view the modelling assumption of unlimited substitution should not be taken literally,
but should be seen as representing the described incentive elfect. In other words, when
the world has only one ton of copper left it is not that production will literally still use
small amounts of the remaining copper in association with extremely sophisticated
labour in accordance with an old production function. Rather. copper will have been
replaced by a substituting product invented, as copper, because of its scarcity, became
extremely expensive and the economic gains from developing a substituting product
became very large. With just one or two types of natural resources in our production
function we cannot describe this process explicitly, but the assumption of unlimited
substitution can be seen as representing it.
The same (optimistic) economists also argue that the deterioration of the natural
environment caused by the polluting activities of firms and consumers can be held in
check by intelligent use of 'green' taxes and other economic instruments in environmental
policy.
On the other hand, many natural scientists and environmentalists argue that there
are in fact limits to the possibilities of substituting other factors for certain essential
raw materials and life support services offered by the environment. The more moderate
critics doubt that the market mechanisms will always ensure the development of new substitute
techniques in time to prevent serious disruptions stemming from environmental
degradation.
Passing a fully qualified judgement on these fundamental issues requires insigh t into
the natural sciences as well as into economics. With this chapter we do not pretend to
have given a definitive answer to the question: 'Are there limits to growth?' However, we
hope to have shed some light on the basic assumptions one needs to mal<e to warrant
either grmvth optimism or growth pessimism.
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