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11.10 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)

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.

Figure 11.4 The Solow growth model

0

k ·

> 0

0

k ·

< 0

y1

y*

y = Y/L

d

b

c

a

k1 k* k2 k = K/L

f (k)

(n + δ)k

sf (k)

e

k* k = K/L

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)

Figure 11.5 Technological progress

y = Y/L

k = K/L

yb

ya

ka

A(t1)kα

A(t0)kα

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)!