The Solow Neoclassical Growth Model
Following the seminal contributions of Solow (1956, 1957) and Swan (1956),
the neoclassical model became the dominant approach to the analysis of
growth, at least within academia. Between 1956 and 1970 economists refined
‘old growth theory’, better known as the Solow neoclassical model of economic
growth (Solow, 2000, 2002). Building on a neoclassical production
function framework, the Solow model highlights the impact on growth of
saving, population growth and technolgical progress in a closed economy
setting without a government sector. Despite recent developments in endogenous
growth theory, the Solow model remains the essential starting point to
any discussion of economic growth. As Mankiw (1995, 2003) notes, whenever
practical macroeconomists have to answer questions about long-run
growth they usually begin with a simple neoclassical growth model (see also
Abel and Bernanke, 2001; Jones, 2001a; Barro and Sala-i-Martin, 2003).
The key assumptions of the Solow model are: (i) for simplicity it is assumed
that the economy consists of one sector producing one type of
commodity that can be used for either investment or consumption purposes;
(ii) the economy is closed to international transactions and the government
sector is ignored; (iii) all output that is saved is invested; that is, in the Solow
model the absence of a separate investment function implies that Keynesian
difficulties are eliminated since ex ante saving and ex ante investment are
always equivalent; (iv) since the model is concerned with the long run there
are no Keynesian stability problems; that is, the assumptions of full price
flexibility and monetary neutrality apply and the economy is always producing
its potential (natural) level of total output; (v) Solow abandons the
Harrod–Domar assumptions of a fixed capital–output ratio (K/Y) and fixed
capital–labour ratio (K/L); (vi) the rate of technological progress, population
growth and the depreciation rate of the capital stock are all determined
exogenously.
Given these assumptions we can concentrate on developing the three key
relationships in the Solow model, namely, the production function, the consumption
function and the capital accumulation process.
The production function
The Solow growth model is built around the neoclassical aggregate production
function (11.16) and focuses on the proximate causes of growth:
Y = AtF(K, L) (11.16)
where Y is real output, K is capital, L is the labour input and At is a measure of
technology (that is, the way that inputs to the production function can be
transformed into output) which is exogenous and taken simply to depend on
time. Sometimes, At is called ‘total factor productivity’. It is important to be
clear about what the assumption of exogenous technology means in the
Solow model. In the neoclassical theory of growth, technology is assumed to
be a public good. Applied to the world economy this means that every
country is assumed to share the same stock of knowledge which is freely
available; that is, all countries have access to the same production function.
In his defence of the neoclassical assumption of treating technology as if it
were a public good, Mankiw (1995) puts his case as follows:
The production function should not be viewed literally as a description of a
specific production process, but as a mapping from quantities of inputs into a
quantity of output. To say that different countries have the same production
function is merely to say that if they had the same inputs, they would produce the
same output. Different countries with different levels of inputs need not rely on
exactly the same processes for producing goods and services. When a country
doubles its capital stock, it does not give each worker twice as many shovels.
Instead, it replaces shovels with bulldozers. For the purposes of modelling economic
growth, this change should be viewed as a movement along the same
production function, rather than a shift to a completely new production function.
As we shall see later (section 11.15), many economists disagree with this
approach and insist that there are significant technology gaps between nations
(see Fagerberg, 1994; P. Romer, 1995). However, to progress with our
examination of the Solow model we will continue to treat technology as a
public good.
For simplicity, let us begin by first assuming a situation where there is no
technological progress. Making this assumption of a given state of technology
will allow us to concentrate on the relationship between output per
worker and capital per worker. We can therefore rewrite (11.16) as:
Y = F(K, L) (11.17)
The aggregate production function given by (11.17) is assumed to be ‘well
behaved’; that is, it satisfies the following three conditions (see Inada,
1963; D. Romer, 2001; Barro and Sala-i-Martin, 2003; Mankiw, 2003).
First, for all values of K > 0 and L > 0, F(·) exhibits positive but diminishing
marginal returns with respect to both capital and labour; that is, ∂F/∂K
> 0, ∂2F/∂K2 < 0, ∂F/∂L > 0, and ∂2F/∂L2 < 0. Second, the production
function exhibits constant returns to scale such that F (λK, λL) = λY; that
is, raising inputs by λ will also increase aggregate output by λ. Letting λ
=1/L yields Y/L = F (K/L). This assumption allows (11.17) to be written
down in intensive form as (11.18), where y = output per worker (Y/L) and k
= capital per worker (K/L):
y = f (k), where f ′(k) > 0, and f ′′(k) < 0 for all k (11.18)
Equation (11.18) states that output per worker is a positive function of the
capital–labour ratio and exhibits diminishing returns. The key assumption of
constant returns to scale implies that the economy is sufficiently large that
any Smithian gains from further division of labour and specialization have
already been exhausted, so that the size of the economy, in terms of the
labour force, has no influence on output per worker. Third, as the capital–
labour ratio approaches infinity (k→∞) the marginal product of capital (MPK)
The renaissance of economic growth research 605
Figure 11.3 The neoclassical aggregate production function
y
k
y = f (k)
approaches zero; as the capital–labour ratio approaches zero the marginal
product of capital tends towards infinity (MPK→∞).
Figure 11.3 shows an intensive form of the neoclassical aggregate production
function that satisfies the above conditions. As the diagram illustrates,
for a given technology, any country that increases its capital–labour ratio
(more equipment per worker) will have a higher output per worker. However,
because of diminishing returns, the impact on output per worker resulting
from capital accumulation per worker (capital deepening) will continuously
decline. Thus for a given increase in k, the impact on y will be much greater
where capital is relatively scarce than in economies where capital is relatively
abundant. That is, the accumulation of capital should have a much more
dramatic impact on labour productivity in developing countries compared to
developed countries.
The slope of the production function measures the marginal product of
capital, where MPK = f(k + 1) – f(k). In the Solow model the MPK should be
much higher in developing economies compared to developed economies. In
an open economy setting with no restrictions on capital mobility, we should
therefore expect to see, ceteris paribus, capital flowing from rich to poor
countries, attracted by higher potential returns, thereby accelerating the process
of capital accumulation.
The consumption function
Since output per worker depends positively on capital per worker, we need to
understand how the capital–labour ratio evolves over time. To examine the
process of capital accumulation we first need to specify the determination of
saving. In a closed economy aggregate output = aggregate income and comprises
two components, namely, consumption (C) and investment (I) = Savings
(S). Therefore we can write equation (11.19) for income as:
Y = C + I (11.19)
or equivalently Y = C + S
Here S = sY is a simple savings function where s is the fraction of income
saved and 1 > s > 0. We can rewrite (11.19) as (11.20):
Y = C + sY (11.20)
Given the assumption of a closed economy, private domestic saving (sY) must
equal domestic investment (I).
The capital accumulation process
A country’s capital stock (Kt) at a point in time consists of plant, machinery
and infrastructure. Each year a proportion of the capital stock wears out. The
parameter δ represents this process of depreciation. Countering this tendency
for the capital stock to decline is a flow of investment spending each year (It)
that adds to the capital stock. Therefore, given these two forces, we can write
an equation for the evolution of the capital stock of the following form:
Kt+1 = It + (1− δ)Kt = sYt + Kt − δKt (11.21)
Rewriting (11.21) in per worker terms yields equation (11.22):
Kt+1/L = sYt /L + Kt /L − δKt /L (11.22)
Deducting Kt /L from both sides of (11.22) gives us (11.23):
Kt+1/L − Kt /L = sYt /L − δKt /L (11.23)
In the neoclassical theory of growth the accumulation of capital evolves
according to (11.24), which is the fundamental differential equation of the
Solow model:
k˙ = sf (k) − δk (11.24)
where ˙ k = Kt+1 /L – Kt /L is the change of the capital input per worker, and
sf(k) = sy = sYt /L is saving (investment) per worker. The δk= δKt /L term
represents the ‘investment requirements’ per worker in order to keep the
capital–labour ratio constant. The steady-state condition in the Solow model
is given in equation (11.25):
sf (k* ) − δk* = 0 (11.25)
Thus, in the steady state sf(k*) = δk*; that is, investment per worker is just
sufficient to cover depreciation per worker, leaving capital per worker constant.
Extending the model to allow for growth of the labour force is relatively
straightforward. In the Solow model it is assumed that the participation rate is
constant, so that the labour force grows at a constant proportionate rate equal
to the exogenously determined rate of growth of population = n. Because k =
K/L, population growth, by increasing the supply of labour, will reduce k.
Therefore population growth has the same impact on k as depreciation. We
need to modify (11.24) to reflect the influence of population growth. The
fundamental differential equation now becomes:
k˙ = sf (k) − (n + δ)k (11.26)
We can think of the expression (n + δ)k as the ‘required’ or ‘break-even’
investment necessary to keep the capital stock per unit of labour (k) constant.
In order to prevent k from falling, some investment is required to offset
depreciation. This is the (δ)k term in (11.26). Some investment is also required
because the quantity of labour is growing at a rate = n. This is the (n)k
term in (11.26). Hence the capital stock must grow at rate (n + δ) just to hold
k steady. When investment per unit of labour is greater than required for
break-even investment, then k will be rising and in this case the economy is
experiencing ‘capital deepening’. Given the structure of the Solow model the
economy will, in time, approach a steady state where actual investment per
worker, sf(k), equals break-even investment per worker, (n + δ)k. In the
steady state the change in capital per worker ˙ k = 0, although the economy
continues to experience ‘capital widening’, the extension of existing capital
per worker to additional workers. Using * to indicate steady-state values, we
can define the steady state as (11.27):
sf (k* ) = (n + δ)k* (11.27)
Figure 11.4 captures the essential features of the Solow model outlined by
equations (11.18) to (11.27). In the top panel of Figure 11.4 the curve f(k)
graphs a well-behaved intensive production function; sf(k) shows the level of
savings per worker at different levels of the capital–labour ratio (k); the linear
relationship (n + δ)k shows that break-even investment is proportional to k.
At the capital–labour ratio k1, savings (investment) per worker (b) exceed
required investment (c) and so the economy experiences capital deepening
and k rises. At k1 consumption per worker is indicated by d – b and output per
worker is y1. At k2, because (n + δ)k > sf(k) the capital–labour ratio falls,
capital becomes ‘shallower’ (Jones, 1975). The steady state balanced growth
path occurs at k*, where investment per worker equals break-even investment.
Output per worker is y* and consumption per worker is e – a. In the bottom
panel of Figure 11.4 the relationship between ˙ k (the change of the capital–
labour ratio) and k is shown with a phase diagram. When ˙ k > 0, k is rising;
when ˙ k < 0, k is falling.
In the steady state equilibrium, shown as point a in the top panel of Figure
11.4, output per worker (y*) and capital per worker (k*) are constant. However,
although there is no intensive growth in the steady state, there is extensive
growth because population (and hence the labour input = L) is growing at a
rate of n per cent per annum. Thus, in order for y* = Y/L and k* = K/L to
remain constant, both Y and K must also grow at the same rate as population.
It can be seen from Figure 11.4 that the steady state level of output per
worker will increase (ceteris paribus) if the rate of population growth and/or
the depreciation rate are reduced (a downward pivot of the (n + δ)k function),
and vice versa. The steady state level of output per worker will also increase
(ceteris paribus) if the savings rate increases (an upward shift of the sf(k)
function), and vice versa. Of particular importance is the prediction from the
Solow model that an increase in the savings ratio cannot permanently increase
the long-run rate of growth. A higher savings ratio does temporarily
increase the growth rate during the period of transitional dynamics to the new
steady state and it also permanently increases the level of output per worker.
Of course the period of transitional dynamics may be a long historical time
period and level effects are important and should not be undervalued (see
Solow, 2000; Temple, 2003).
So far we have assumed zero technological progress. Given the fact that
output per worker has shown a continuous tendency to increase, at least since
the onset of the Industrial Revolution in the now developed economies, a
model that predicts a constant steady state output per worker is clearly
unsatisfactory. A surprising conclusion of the neoclassical growth model is
that without technological progress the ability of an economy to raise output
per worker via capital accumulation is limited by the interaction of diminishing
returns, the willingness of people to save, the rate of population growth,
and the rate of depreciation of the capital stock. In order to explain continuous
growth of output per worker in the long run the Solow model must
incorporate the influence of sustained technological progress.
The production function (11.16), in its Cobb–Douglas form, can be written
as (11.28):
Y = AtKαL1−α (11.28)
where α and 1 – α are weights reflecting the share of capital and labour in the
national income. Assuming constant returns to scale, output per worker (Y/L)
is not affected by the scale of output, and, for a given technology, At0, output
per worker is positively related to the capital–labour ratio (K/L). We can
therefore rewrite the production function equation (11.28) in terms of output
per worker as shown by equation (11.29):
Y/L = A(t )(K/L) = A(t )K L − /L = A(t )(K/L)
0 0
1
0
α α α (11.29)
Letting y = Y/L and k = K/L, we finally arrive at the ‘intensive form’ of the
aggregate production function shown in equation (11.30):
y = A(t0 )k
α (11.30)
For a given technology, equation (11.30) tells us that increasing the amount
of capital per worker (capital deepening) will lead to an increase in output per
worker. The impact of exogenous technological progress is illustrated in
Figure 11.5 by a shift of the production function between two time periods (t0
⇒t1) from A(t0)kα to A(t1)kα, raising output per worker from ya to yb for a
given capital–labour ratio of ka. Continuous upward shifts of the production
function, induced by an exogenously determined growth of knowledge, provide
the only mechanism for ‘explaining’ steady state growth of output per
worker in the neoclassical model.
Therefore, although it was not Solow’s original intention, it was his neoclassical
theory of growth that brought technological progress to prominence
as a major explanatory factor in the analysis of economic growth. But, somewhat
paradoxically, in Solow’s theory technological progress is exogenous,
that is, not explained by the model! Solow admits that he made technological
progress exogenous in his model in order to simplify it and also because he
did not ‘pretend to understand’ it (see Solow interview at the end of this
chapter) and, as Abramovitz (1956) observed, the Solow residual turned out
to be ‘a measure of our ignorance’ (see also Abramovitz, 1999). While Barro
and Sala-i-Martin (1995) conclude that this was ‘an obviously unsatisfactory
situation’, David Romer (1996) comments that the Solow model ‘takes as
given the behaviour of the variable that it identifies as the main driving force
of growth’. Furthermore, although the Solow model attributes no role to
capital accumulation in achieving long-run sustainable growth, it should be
noted that productivity growth may not be independent of capital accumulation
if technical progress is embodied in new capital equipment. Unlike
disembodied technical progress, which can raise the productivity of the existing
inputs, embodied technical progress does not benefit older capital
equipment. It should also be noted that DeLong and Summers (1991, 1992)
find a strong association between equipment investment and economic growth
in the period 1960–85 for a sample of over 60 countries.
Remarkably, while economists have long recognized the crucial importance
of technological change as a major source of dynamism in capitalist
economies (especially Karl Marx and Joseph Schumpeter), the analysis of
technological change and innovation by economists has, until recently, been
an area of relative neglect (see Freeman, 1994; Baumol, 2002).
Leaving aside these controversies for the moment, it is important to note
that the Solow model allows us to make several important predictions about
the growth process (see Mankiw, 1995, 2003; Solow, 2002):
1. in the long run an economy will gradually approach a steady state equilibrium
with y* and k* independent of initial conditions;
2. the steady state balanced rate of growth of aggregate output depends on
the rate of population growth (n) and the rate of technological progress
(A);
3. in the steady state balanced growth path the rate of growth of output per
worker depends solely on the rate of technological progress. As illustrated
in Figure 11.5, without technological progress the growth of output
per worker will eventually cease;
4. the steady state rate of growth of the capital stock equals the rate of
income growth, so the K/Y ratio is constant;
5. for a given depreciation rate (δ) the steady state level of output per
worker depends on the savings rate (s) and the population growth rate
(n). A higher rate of saving will increase y*, a higher population growth
rate will reduce y*;
6. the impact of an increase in the savings (investment) rate on the growth
of output per worker is temporary. An economy experiences a period of
higher growth as the new steady state is approached. A higher rate of
saving has no effect on the long-run sustainable rate of growth, although
it will increase the level of output per worker. To Solow this finding was
a ‘real shocker’;
7. the Solow model has particular ‘convergence properties’. In particular,
‘if countries are similar with respect to structural parameters for preferences
and technology, then poor countries tend to grow faster than rich
countries’ (Barro, 1991).
The result in the Solow model that an increase in the saving rate has no
impact on the long-run rate of economic growth contains ‘more than a touch
of irony’ (Cesaratto, 1999). As Hamberg (1971) pointed out, the neo-Keynesian
Harrod–Domar model highlights the importance of increasing the saving rate
to increase long-run growth, while in Keynes’s (1936) General Theory an
increase in the saving rate leads to a fall in output in the short run through its
negative impact on aggregate demand (the so-called ‘paradox of thrift’ effect).
In contrast, the long tradition within classical–neoclassical economics
of highlighting the virtues of thrift come a little unstuck with the Solow
model since it is technological progress, not thrift, that drives long-run growth
of output per worker (see Cesaratto, 1999)!
Following the seminal contributions of Solow (1956, 1957) and Swan (1956),
the neoclassical model became the dominant approach to the analysis of
growth, at least within academia. Between 1956 and 1970 economists refined
‘old growth theory’, better known as the Solow neoclassical model of economic
growth (Solow, 2000, 2002). Building on a neoclassical production
function framework, the Solow model highlights the impact on growth of
saving, population growth and technolgical progress in a closed economy
setting without a government sector. Despite recent developments in endogenous
growth theory, the Solow model remains the essential starting point to
any discussion of economic growth. As Mankiw (1995, 2003) notes, whenever
practical macroeconomists have to answer questions about long-run
growth they usually begin with a simple neoclassical growth model (see also
Abel and Bernanke, 2001; Jones, 2001a; Barro and Sala-i-Martin, 2003).
The key assumptions of the Solow model are: (i) for simplicity it is assumed
that the economy consists of one sector producing one type of
commodity that can be used for either investment or consumption purposes;
(ii) the economy is closed to international transactions and the government
sector is ignored; (iii) all output that is saved is invested; that is, in the Solow
model the absence of a separate investment function implies that Keynesian
difficulties are eliminated since ex ante saving and ex ante investment are
always equivalent; (iv) since the model is concerned with the long run there
are no Keynesian stability problems; that is, the assumptions of full price
flexibility and monetary neutrality apply and the economy is always producing
its potential (natural) level of total output; (v) Solow abandons the
Harrod–Domar assumptions of a fixed capital–output ratio (K/Y) and fixed
capital–labour ratio (K/L); (vi) the rate of technological progress, population
growth and the depreciation rate of the capital stock are all determined
exogenously.
Given these assumptions we can concentrate on developing the three key
relationships in the Solow model, namely, the production function, the consumption
function and the capital accumulation process.
The production function
The Solow growth model is built around the neoclassical aggregate production
function (11.16) and focuses on the proximate causes of growth:
Y = AtF(K, L) (11.16)
where Y is real output, K is capital, L is the labour input and At is a measure of
technology (that is, the way that inputs to the production function can be
transformed into output) which is exogenous and taken simply to depend on
time. Sometimes, At is called ‘total factor productivity’. It is important to be
clear about what the assumption of exogenous technology means in the
Solow model. In the neoclassical theory of growth, technology is assumed to
be a public good. Applied to the world economy this means that every
country is assumed to share the same stock of knowledge which is freely
available; that is, all countries have access to the same production function.
In his defence of the neoclassical assumption of treating technology as if it
were a public good, Mankiw (1995) puts his case as follows:
The production function should not be viewed literally as a description of a
specific production process, but as a mapping from quantities of inputs into a
quantity of output. To say that different countries have the same production
function is merely to say that if they had the same inputs, they would produce the
same output. Different countries with different levels of inputs need not rely on
exactly the same processes for producing goods and services. When a country
doubles its capital stock, it does not give each worker twice as many shovels.
Instead, it replaces shovels with bulldozers. For the purposes of modelling economic
growth, this change should be viewed as a movement along the same
production function, rather than a shift to a completely new production function.
As we shall see later (section 11.15), many economists disagree with this
approach and insist that there are significant technology gaps between nations
(see Fagerberg, 1994; P. Romer, 1995). However, to progress with our
examination of the Solow model we will continue to treat technology as a
public good.
For simplicity, let us begin by first assuming a situation where there is no
technological progress. Making this assumption of a given state of technology
will allow us to concentrate on the relationship between output per
worker and capital per worker. We can therefore rewrite (11.16) as:
Y = F(K, L) (11.17)
The aggregate production function given by (11.17) is assumed to be ‘well
behaved’; that is, it satisfies the following three conditions (see Inada,
1963; D. Romer, 2001; Barro and Sala-i-Martin, 2003; Mankiw, 2003).
First, for all values of K > 0 and L > 0, F(·) exhibits positive but diminishing
marginal returns with respect to both capital and labour; that is, ∂F/∂K
> 0, ∂2F/∂K2 < 0, ∂F/∂L > 0, and ∂2F/∂L2 < 0. Second, the production
function exhibits constant returns to scale such that F (λK, λL) = λY; that
is, raising inputs by λ will also increase aggregate output by λ. Letting λ
=1/L yields Y/L = F (K/L). This assumption allows (11.17) to be written
down in intensive form as (11.18), where y = output per worker (Y/L) and k
= capital per worker (K/L):
y = f (k), where f ′(k) > 0, and f ′′(k) < 0 for all k (11.18)
Equation (11.18) states that output per worker is a positive function of the
capital–labour ratio and exhibits diminishing returns. The key assumption of
constant returns to scale implies that the economy is sufficiently large that
any Smithian gains from further division of labour and specialization have
already been exhausted, so that the size of the economy, in terms of the
labour force, has no influence on output per worker. Third, as the capital–
labour ratio approaches infinity (k→∞) the marginal product of capital (MPK)
The renaissance of economic growth research 605
Figure 11.3 The neoclassical aggregate production function
y
k
y = f (k)
approaches zero; as the capital–labour ratio approaches zero the marginal
product of capital tends towards infinity (MPK→∞).
Figure 11.3 shows an intensive form of the neoclassical aggregate production
function that satisfies the above conditions. As the diagram illustrates,
for a given technology, any country that increases its capital–labour ratio
(more equipment per worker) will have a higher output per worker. However,
because of diminishing returns, the impact on output per worker resulting
from capital accumulation per worker (capital deepening) will continuously
decline. Thus for a given increase in k, the impact on y will be much greater
where capital is relatively scarce than in economies where capital is relatively
abundant. That is, the accumulation of capital should have a much more
dramatic impact on labour productivity in developing countries compared to
developed countries.
The slope of the production function measures the marginal product of
capital, where MPK = f(k + 1) – f(k). In the Solow model the MPK should be
much higher in developing economies compared to developed economies. In
an open economy setting with no restrictions on capital mobility, we should
therefore expect to see, ceteris paribus, capital flowing from rich to poor
countries, attracted by higher potential returns, thereby accelerating the process
of capital accumulation.
The consumption function
Since output per worker depends positively on capital per worker, we need to
understand how the capital–labour ratio evolves over time. To examine the
process of capital accumulation we first need to specify the determination of
saving. In a closed economy aggregate output = aggregate income and comprises
two components, namely, consumption (C) and investment (I) = Savings
(S). Therefore we can write equation (11.19) for income as:
Y = C + I (11.19)
or equivalently Y = C + S
Here S = sY is a simple savings function where s is the fraction of income
saved and 1 > s > 0. We can rewrite (11.19) as (11.20):
Y = C + sY (11.20)
Given the assumption of a closed economy, private domestic saving (sY) must
equal domestic investment (I).
The capital accumulation process
A country’s capital stock (Kt) at a point in time consists of plant, machinery
and infrastructure. Each year a proportion of the capital stock wears out. The
parameter δ represents this process of depreciation. Countering this tendency
for the capital stock to decline is a flow of investment spending each year (It)
that adds to the capital stock. Therefore, given these two forces, we can write
an equation for the evolution of the capital stock of the following form:
Kt+1 = It + (1− δ)Kt = sYt + Kt − δKt (11.21)
Rewriting (11.21) in per worker terms yields equation (11.22):
Kt+1/L = sYt /L + Kt /L − δKt /L (11.22)
Deducting Kt /L from both sides of (11.22) gives us (11.23):
Kt+1/L − Kt /L = sYt /L − δKt /L (11.23)
In the neoclassical theory of growth the accumulation of capital evolves
according to (11.24), which is the fundamental differential equation of the
Solow model:
k˙ = sf (k) − δk (11.24)
where ˙ k = Kt+1 /L – Kt /L is the change of the capital input per worker, and
sf(k) = sy = sYt /L is saving (investment) per worker. The δk= δKt /L term
represents the ‘investment requirements’ per worker in order to keep the
capital–labour ratio constant. The steady-state condition in the Solow model
is given in equation (11.25):
sf (k* ) − δk* = 0 (11.25)
Thus, in the steady state sf(k*) = δk*; that is, investment per worker is just
sufficient to cover depreciation per worker, leaving capital per worker constant.
Extending the model to allow for growth of the labour force is relatively
straightforward. In the Solow model it is assumed that the participation rate is
constant, so that the labour force grows at a constant proportionate rate equal
to the exogenously determined rate of growth of population = n. Because k =
K/L, population growth, by increasing the supply of labour, will reduce k.
Therefore population growth has the same impact on k as depreciation. We
need to modify (11.24) to reflect the influence of population growth. The
fundamental differential equation now becomes:
k˙ = sf (k) − (n + δ)k (11.26)
We can think of the expression (n + δ)k as the ‘required’ or ‘break-even’
investment necessary to keep the capital stock per unit of labour (k) constant.
In order to prevent k from falling, some investment is required to offset
depreciation. This is the (δ)k term in (11.26). Some investment is also required
because the quantity of labour is growing at a rate = n. This is the (n)k
term in (11.26). Hence the capital stock must grow at rate (n + δ) just to hold
k steady. When investment per unit of labour is greater than required for
break-even investment, then k will be rising and in this case the economy is
experiencing ‘capital deepening’. Given the structure of the Solow model the
economy will, in time, approach a steady state where actual investment per
worker, sf(k), equals break-even investment per worker, (n + δ)k. In the
steady state the change in capital per worker ˙ k = 0, although the economy
continues to experience ‘capital widening’, the extension of existing capital
per worker to additional workers. Using * to indicate steady-state values, we
can define the steady state as (11.27):
sf (k* ) = (n + δ)k* (11.27)
Figure 11.4 captures the essential features of the Solow model outlined by
equations (11.18) to (11.27). In the top panel of Figure 11.4 the curve f(k)
graphs a well-behaved intensive production function; sf(k) shows the level of
savings per worker at different levels of the capital–labour ratio (k); the linear
relationship (n + δ)k shows that break-even investment is proportional to k.
At the capital–labour ratio k1, savings (investment) per worker (b) exceed
required investment (c) and so the economy experiences capital deepening
and k rises. At k1 consumption per worker is indicated by d – b and output per
worker is y1. At k2, because (n + δ)k > sf(k) the capital–labour ratio falls,
capital becomes ‘shallower’ (Jones, 1975). The steady state balanced growth
path occurs at k*, where investment per worker equals break-even investment.
Output per worker is y* and consumption per worker is e – a. In the bottom
panel of Figure 11.4 the relationship between ˙ k (the change of the capital–
labour ratio) and k is shown with a phase diagram. When ˙ k > 0, k is rising;
when ˙ k < 0, k is falling.
In the steady state equilibrium, shown as point a in the top panel of Figure
11.4, output per worker (y*) and capital per worker (k*) are constant. However,
although there is no intensive growth in the steady state, there is extensive
growth because population (and hence the labour input = L) is growing at a
rate of n per cent per annum. Thus, in order for y* = Y/L and k* = K/L to
remain constant, both Y and K must also grow at the same rate as population.
It can be seen from Figure 11.4 that the steady state level of output per
worker will increase (ceteris paribus) if the rate of population growth and/or
the depreciation rate are reduced (a downward pivot of the (n + δ)k function),
and vice versa. The steady state level of output per worker will also increase
(ceteris paribus) if the savings rate increases (an upward shift of the sf(k)
function), and vice versa. Of particular importance is the prediction from the
Solow model that an increase in the savings ratio cannot permanently increase
the long-run rate of growth. A higher savings ratio does temporarily
increase the growth rate during the period of transitional dynamics to the new
steady state and it also permanently increases the level of output per worker.
Of course the period of transitional dynamics may be a long historical time
period and level effects are important and should not be undervalued (see
Solow, 2000; Temple, 2003).
So far we have assumed zero technological progress. Given the fact that
output per worker has shown a continuous tendency to increase, at least since
the onset of the Industrial Revolution in the now developed economies, a
model that predicts a constant steady state output per worker is clearly
unsatisfactory. A surprising conclusion of the neoclassical growth model is
that without technological progress the ability of an economy to raise output
per worker via capital accumulation is limited by the interaction of diminishing
returns, the willingness of people to save, the rate of population growth,
and the rate of depreciation of the capital stock. In order to explain continuous
growth of output per worker in the long run the Solow model must
incorporate the influence of sustained technological progress.
The production function (11.16), in its Cobb–Douglas form, can be written
as (11.28):
Y = AtKαL1−α (11.28)
where α and 1 – α are weights reflecting the share of capital and labour in the
national income. Assuming constant returns to scale, output per worker (Y/L)
is not affected by the scale of output, and, for a given technology, At0, output
per worker is positively related to the capital–labour ratio (K/L). We can
therefore rewrite the production function equation (11.28) in terms of output
per worker as shown by equation (11.29):
Y/L = A(t )(K/L) = A(t )K L − /L = A(t )(K/L)
0 0
1
0
α α α (11.29)
Letting y = Y/L and k = K/L, we finally arrive at the ‘intensive form’ of the
aggregate production function shown in equation (11.30):
y = A(t0 )k
α (11.30)
For a given technology, equation (11.30) tells us that increasing the amount
of capital per worker (capital deepening) will lead to an increase in output per
worker. The impact of exogenous technological progress is illustrated in
Figure 11.5 by a shift of the production function between two time periods (t0
⇒t1) from A(t0)kα to A(t1)kα, raising output per worker from ya to yb for a
given capital–labour ratio of ka. Continuous upward shifts of the production
function, induced by an exogenously determined growth of knowledge, provide
the only mechanism for ‘explaining’ steady state growth of output per
worker in the neoclassical model.
Therefore, although it was not Solow’s original intention, it was his neoclassical
theory of growth that brought technological progress to prominence
as a major explanatory factor in the analysis of economic growth. But, somewhat
paradoxically, in Solow’s theory technological progress is exogenous,
that is, not explained by the model! Solow admits that he made technological
progress exogenous in his model in order to simplify it and also because he
did not ‘pretend to understand’ it (see Solow interview at the end of this
chapter) and, as Abramovitz (1956) observed, the Solow residual turned out
to be ‘a measure of our ignorance’ (see also Abramovitz, 1999). While Barro
and Sala-i-Martin (1995) conclude that this was ‘an obviously unsatisfactory
situation’, David Romer (1996) comments that the Solow model ‘takes as
given the behaviour of the variable that it identifies as the main driving force
of growth’. Furthermore, although the Solow model attributes no role to
capital accumulation in achieving long-run sustainable growth, it should be
noted that productivity growth may not be independent of capital accumulation
if technical progress is embodied in new capital equipment. Unlike
disembodied technical progress, which can raise the productivity of the existing
inputs, embodied technical progress does not benefit older capital
equipment. It should also be noted that DeLong and Summers (1991, 1992)
find a strong association between equipment investment and economic growth
in the period 1960–85 for a sample of over 60 countries.
Remarkably, while economists have long recognized the crucial importance
of technological change as a major source of dynamism in capitalist
economies (especially Karl Marx and Joseph Schumpeter), the analysis of
technological change and innovation by economists has, until recently, been
an area of relative neglect (see Freeman, 1994; Baumol, 2002).
Leaving aside these controversies for the moment, it is important to note
that the Solow model allows us to make several important predictions about
the growth process (see Mankiw, 1995, 2003; Solow, 2002):
1. in the long run an economy will gradually approach a steady state equilibrium
with y* and k* independent of initial conditions;
2. the steady state balanced rate of growth of aggregate output depends on
the rate of population growth (n) and the rate of technological progress
(A);
3. in the steady state balanced growth path the rate of growth of output per
worker depends solely on the rate of technological progress. As illustrated
in Figure 11.5, without technological progress the growth of output
per worker will eventually cease;
4. the steady state rate of growth of the capital stock equals the rate of
income growth, so the K/Y ratio is constant;
5. for a given depreciation rate (δ) the steady state level of output per
worker depends on the savings rate (s) and the population growth rate
(n). A higher rate of saving will increase y*, a higher population growth
rate will reduce y*;
6. the impact of an increase in the savings (investment) rate on the growth
of output per worker is temporary. An economy experiences a period of
higher growth as the new steady state is approached. A higher rate of
saving has no effect on the long-run sustainable rate of growth, although
it will increase the level of output per worker. To Solow this finding was
a ‘real shocker’;
7. the Solow model has particular ‘convergence properties’. In particular,
‘if countries are similar with respect to structural parameters for preferences
and technology, then poor countries tend to grow faster than rich
countries’ (Barro, 1991).
The result in the Solow model that an increase in the saving rate has no
impact on the long-run rate of economic growth contains ‘more than a touch
of irony’ (Cesaratto, 1999). As Hamberg (1971) pointed out, the neo-Keynesian
Harrod–Domar model highlights the importance of increasing the saving rate
to increase long-run growth, while in Keynes’s (1936) General Theory an
increase in the saving rate leads to a fall in output in the short run through its
negative impact on aggregate demand (the so-called ‘paradox of thrift’ effect).
In contrast, the long tradition within classical–neoclassical economics
of highlighting the virtues of thrift come a little unstuck with the Solow
model since it is technological progress, not thrift, that drives long-run growth
of output per worker (see Cesaratto, 1999)!
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