ka k = K/L

Figure 11.5 Technological progress

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)k± to A(t1)k±, 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, pro-

vide 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 neo-

classical theory of growth that brought technological progress to prominence

as a major explanatory factor in the analysis of economic growth. But, some-

what 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

The renaissance of economic growth research 611

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 identi¬es 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 accumula-

tion if technical progress is embodied in new capital equipment. Unlike

disembodied technical progress, which can raise the productivity of the exist-

ing inputs, embodied technical progress does not bene¬t older capital

equipment. It should also be noted that DeLong and Summers (1991, 1992)

¬nd 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 impor-

tance 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 equi-

librium 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 illus-

trated 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 ¬nding was

a ˜real shocker™;

612 Modern macroeconomics

7. the Solow model has particular ˜convergence properties™. In particular,

˜if countries are similar with respect to structural parameters for prefer-

ences 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™ ef-

fect). 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)!

11.11 Accounting for the Sources of Economic Growth

Economists not only need a theoretical framework for understanding the

causes of growth; they also require a simple method of calculating the rela-

tive importance of capital, labour and technology in the growth experience of

actual economies. The established framework, following Solow™s (1957) semi-

nal contribution, is called ˜growth accounting™ (see Abel and Bernanke, 2001.

Some economists remain highly sceptical about the whole methodology and

theoretical basis of growth accounting, for example Nelson, 1973). As far as

the proximate causes of growth are concerned we can see by referring back to

equation (11.28) that increases in total GDP (Y) come from the combined

weighted impact of capital accumulation, labour supply growth and techno-

logical progress. Economists can measure changes in the amount of capital

and labour that occur in an economy over time, but changes in technology

(total factor productivity = TFP) are not directly observable. However, it is

possible to measure changes in TFP as a ˜residual™ after taking into account

the contributions to growth made by changes in the capital and labour inputs.

Solow™s (1957) technique was to de¬ne technological change as changes in

aggregate output minus the sum of the weighted contributions of the labour

and capital inputs. In short, the Solow residual measures that part of a change

in aggregate output which cannot be explained by changes in the measurable

quantities of capital and labour inputs. The derivation of the Solow residual

can be shown as follows. The aggregate production function in equation

(11.28) shows that output (Y) is dependent on the inputs of capital (K), labour

(L) and the currently available technology (A), which acts as an index of total

The renaissance of economic growth research 613

factor productivity. Output will change if A, K or L change. In equation

(11.28) the exponent on the capital shock ± measures the elasticity of output

with respect to capital and the exponent on the labour input (1 “ ±) measures

the elasticity of output with respect to labour. The weights ± and 1 “ ± are

estimated from national income statistics and re¬‚ect the income shares of

capital and labour respectively. Since these weights sum to unity, this indi-

cates that (11.28) is a constant returns to scale production function. Hence an

equal percentage increase in both factor inputs (K and L) will increase Y by

the same percentage. Since the growth rate of the product of the inputs will

be the growth rate of A plus the growth rate of K± plus the growth rate of L1“±,

equation (11.28) can be rewritten as (11.31), which is the basic growth

accounting equation used in numerous empirical studies of the sources of

economic growth (see Maddison, 1972, 1987; Denison, 1985; Young, 1995,

Crafts, 2000; Jorgenson, 2001).

∆Y /Y = ∆A/ A + ±∆K / K + (1 ’ ± )∆L/ L (11.31)

Equation (11.31) is simply the Cobb“Douglas production function written in

a form representing rates of change. It shows that the growth of aggregate

output (∆Y/Y) depends on the contribution of changes in total factor produc-

tivity (∆A/A), changes in the weighted contribution of capital, ±∆K/K, and

changes in the weighted contribution of labour (1 “ ±)∆L/L. By rearranging

equation (11.28) we can represent the productivity index (TFP) which we

need to measure as equation (11.32):

TFP = A = Y / K ± L1’± (11.32)

As already noted, because there is no direct way of measuring TFP it has to

be estimated as a residual. By writing down equation (11.32) in terms of rates

of change we can obtain an equation from which the growth of TFP (techno-

logical change) can be estimated as a residual. This is shown in equation

(11.33):

∆A/ A = ∆Y /Y ’ [±∆K / K + (1 ’ ± )∆L/ L] (11.33)