sf ( k * ) ’ δk * = 0 (11.25)

Thus, in the steady state sf(k*) = δk*; that is, investment per worker is just

suf¬cient to cover depreciation per worker, leaving capital per worker con-

stant.

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 re¬‚ect the in¬‚uence 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 re-

quired 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 de¬ne 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)

608 Modern macroeconomics

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. How-

ever, 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.

f(k)

y = Y/L

e

(n + δ)k

y*

d

y1

a sf(k)

b

c

k1 k* k2

0 k = K/L

·

k>0

k = K/L

k*

0

·

k<0

Figure 11.4 The Solow growth model

The renaissance of economic growth research 609

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 in-

crease 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 diminish-

ing 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 continu-

ous growth of output per worker in the long run the Solow model must

incorporate the in¬‚uence of sustained technological progress.

The production function (11.16), in its Cobb“Douglas form, can be written

as (11.28):

Y = At K ± L1’± (11.28)

where ± and 1 “ ± are weights re¬‚ecting 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(t0 )( K / L) = A(t0 ) K ± L1’± / L = A(t0 )( K / L)± (11.29)

Letting y = Y/L and k = K/L, we ¬nally arrive at the ˜intensive form™ of the

aggregate production function shown in equation (11.30):

y = A(t0 )k ± (11.30)

610 Modern macroeconomics

y = Y/L

A(t1)k ±

yb

A(t0)k ±

ya