Beyond the Solow Model
Although the lack of a theory of technological change is a clear weakness of
the basic neoclassical growth model, Mankiw (1995) argues that many general
predictions from the theory are ‘broadly consistent with experience’. For
example, cross-country data indicate a strong negative correlation between
population growth and income per capita and a strong positive correlation
between income per capita and savings/investment rates (Jones, 2001a). As
predicted by the model, rates of growth in the rich OECD economies are
relatively low while rapid growth rates have been observed in countries
moving from an initial position of relatively low income per capita and low
capital intensity. There is also strong evidence of convergence among relatively
homogeneous economies such as the OECD and between regions and
states within the USA, Europe and Japan (Baumol, 1986; Barro and Sala-i-
Martin, 1995). In larger, more diverse data sets there is little evidence of the
expected negative relationship between growth rates and some initial (for
example 1960) level of income per capita, that is, absolute convergence (P.
Romer, 1986, 1989; DeLong, 1988). However, ‘the central idea of conditional
convergence receives strong support from the data’ (Barro, 1991, 1997)
and has considerable explanatory power for both countries and regions. The
growth accounting research of Alwyn Young (1992, 1994, 1995) has shown
that the rapid growth of the Asian Tiger economies is easily explicable and
can be attributed mainly to rapid accumulation of factor inputs rather than
unusually high total factor productivity growth. As Paul Krugman (1994b)
argues, an implication of this research is that this rapid growth can therefore
be expected to slow down considerably in the future, as it has already done in
Japan. The Solow model has also been used to provide a plausible ‘reconstruction’
account of the ‘miracles’ of Japanese and German post-1945 growth,
and also the relatively good growth performance of France and Italy, in terms
of the transitional dynamics towards a high income per capita steady state. It
seems plausible that these economies grew rapidly in the post-war period
because they were ‘reconstructing’ their capital stock following the destruction
resulting from the Second World War.
However, there are a number of important deficiencies and puzzles which
the Solow model finds difficult to overcome and explain. First, in the Solow
model, while economic policy can permanently influence the level of per
capita output (for example by raising the savings ratio via tax inducements),
it cannot alter the path of long-run growth. Growth rates can only be increased
temporarily during the transitional dynamics en route to the new
steady state. Cross-country growth differentials are also explained in terms of
the transitional dynamics which allow countries to grow faster than their
long-run sustainable growth rates. Sustained growth in the Solow model is
only possible if there is technological progress, since without it per capita
income growth will eventually cease due to the impact of diminishing returns
to capital accumulation. Given that per capita incomes have been rising for
over 100 years in a large number of countries, and growth rates have displayed
no overall tendency to decline, the role of technological progress in
the Solow model in explaining sustainable growth becomes crucial. But
herein lies the obvious shortcoming of the neoclassical model since ‘the
long-run per capita growth rate is determined entirely by an element – the
rate of technological progress – that is outside the model … Thus we end up
with a model of growth that explains everything but long-run growth, an
obviously unsatisfactory situation’ (Barro and Sala-i-Martin, 1995). Furthermore,
as P. Romer (1989) highlights, in terms of policy advice for long-term
growth the neoclassical model has little to offer!
A second problem relates to the evidence, which clearly shows that income
per capita differentials across the world are much greater than predicted by
the model. Differences across countries in capital intensities are too small to
account for the observed disparities in real incomes. Using a Cobb–Douglas
production function framework it is possible to allocate differences in the
level of per capita incomes between countries to variations in levels of total
factor productivity growth and the accumulation of factor inputs. In particular
it is possible to estimate how much of the income disparities witnessed
between rich and poor countries can be attributed to different capital intensities
since total factor productivity is common across all countries. Substituting
from equation (11.34) to equation (11.26) gives equation (11.38):
k˙ = skα − (n + δ)k (11.38)
Setting this equation equal to zero (the steady state condition) and substituting
into the production function yields (11.39):
y* = [s/(n + δ)]α/(1−α) (11.39)
Equation (11.39) is now in a form that enables a solution to be found for the
steady state output per worker (y*). As Jones (2001a) highlights, we can see
from equation (11.39) why some countries are so rich and some are so poor.
Assuming exogenous technology and a similar value for the capital exponent
(α), countries that sustain high rates of saving, and low rates of population
growth and depreciation, will be rich. According to the neoclassical growth
model the high-income economies have achieved their high living standards
because they have accumulated large per worker stocks of capital. However,
although the model correctly predicts the directions of the effects of saving
and population growth on output per worker, it does not correctly predict the
magnitudes. As Mankiw et al. (1992) and Mankiw (1995) argue, the gaps in
output per worker (living standards) between rich and poor countries are
much larger than plausible estimates of savings rates and population growth
predict using equation (11.39). The crux of the problem is that with α = 1/3
there are sharply diminishing returns to capital. This implies that a tenfold
gap in output per worker between the USA and India would require a thousandfold
difference in the capital–labour ratios between these countries! (It
should be noted that this result is highly sensitive to the choice of α = 1/3 for
the share of capital in GDP.)
A third problem with the Solow model is that given a common production
function (that is, exogenous technology) the marginal product of capital
should be much higher in poor countries than in rich countries. Given the
parameters of the Solow model, the observed tenfold differential in output
per worker between rich and poor countries implies a hundredfold difference
in the marginal product of capital if output gaps are entirely due to variations
in capital intensities. Such differentials in the rate of return to capital are
simply not observed between rich and poor countries. As David Romer
(1996) observes, such differences in rates of return ‘would swamp such
considerations as capital market imperfections, government tax policies, fear
of expropriation and so on and we would observe immense flows of capital
from rich to poor countries. We do not see such flows.’ But the rate of return
to capital in poor countries is less than expected and the anticipated massive
flows of capital from rich to poor countries have not been observed across
poor countries as a whole (Lucas, 1990b).
A fourth difficulty relates to the rate of convergence, which is only about
half that predicted by the model. The economy’s initial conditions influence
the outcome for much longer than the model says it should (Mankiw, 1995).
In conclusion, it appears that within the Solow growth framework, physical
capital accumulation alone cannot account for either continuous growth of
per capita income over long periods of time or the enormous geographical
disparities in living standards that we observe. In terms of Figure 11.3, the
data on output per worker (or income per capita) that we actually observe
across the world reveal much greater disparities than those predicted by the
Solow model based on differences in capital per worker.
The new growth models emerging after 1986 depart from the Solow model
in three main ways. One group of models generates continuous growth by
abandoning the assumption of diminishing returns to capital accumulation.
To achieve this, Paul Romer (1986) introduced positive externalities from
capital accumulation so that the creation of economy-wide knowledge emerges
as a by-product of the investment activity of individual firms, a case of
‘learning by investing’ (Barro and Sala-i-Martin, 2003). A second approach
models the accumulation of knowledge as the outcome of purposeful acts by
entrepreneurs seeking to maximize private profits; that is, technological
progress is endogenized (P. Romer, 1990). A third class of model claims that
the role of capital is much more important than is suggested by the α term in
the conventional Cobb–Douglas production function shown in equations
(11.28)–(11.30). In their ‘augmented’ Solow model, Mankiw et al. (1992)
broaden the concept of capital to include ‘human capital’. The first two
classes of model constitute the core of endogenous growth theory whereas
the Mankiw, Romer and Weil (MRW) model constitutes what Klenow and
Rodriguez-Clare (1997a, 1997b) call a ‘neoclassical revival’. The central
proposition of endogenous growth theory is that broad capital accumulation
(physical and human capital) does not experience diminishing returns. The
growth process is driven by the accumulation of broad capital together with
the production of new knowledge created through research and development.
Although the lack of a theory of technological change is a clear weakness of
the basic neoclassical growth model, Mankiw (1995) argues that many general
predictions from the theory are ‘broadly consistent with experience’. For
example, cross-country data indicate a strong negative correlation between
population growth and income per capita and a strong positive correlation
between income per capita and savings/investment rates (Jones, 2001a). As
predicted by the model, rates of growth in the rich OECD economies are
relatively low while rapid growth rates have been observed in countries
moving from an initial position of relatively low income per capita and low
capital intensity. There is also strong evidence of convergence among relatively
homogeneous economies such as the OECD and between regions and
states within the USA, Europe and Japan (Baumol, 1986; Barro and Sala-i-
Martin, 1995). In larger, more diverse data sets there is little evidence of the
expected negative relationship between growth rates and some initial (for
example 1960) level of income per capita, that is, absolute convergence (P.
Romer, 1986, 1989; DeLong, 1988). However, ‘the central idea of conditional
convergence receives strong support from the data’ (Barro, 1991, 1997)
and has considerable explanatory power for both countries and regions. The
growth accounting research of Alwyn Young (1992, 1994, 1995) has shown
that the rapid growth of the Asian Tiger economies is easily explicable and
can be attributed mainly to rapid accumulation of factor inputs rather than
unusually high total factor productivity growth. As Paul Krugman (1994b)
argues, an implication of this research is that this rapid growth can therefore
be expected to slow down considerably in the future, as it has already done in
Japan. The Solow model has also been used to provide a plausible ‘reconstruction’
account of the ‘miracles’ of Japanese and German post-1945 growth,
and also the relatively good growth performance of France and Italy, in terms
of the transitional dynamics towards a high income per capita steady state. It
seems plausible that these economies grew rapidly in the post-war period
because they were ‘reconstructing’ their capital stock following the destruction
resulting from the Second World War.
However, there are a number of important deficiencies and puzzles which
the Solow model finds difficult to overcome and explain. First, in the Solow
model, while economic policy can permanently influence the level of per
capita output (for example by raising the savings ratio via tax inducements),
it cannot alter the path of long-run growth. Growth rates can only be increased
temporarily during the transitional dynamics en route to the new
steady state. Cross-country growth differentials are also explained in terms of
the transitional dynamics which allow countries to grow faster than their
long-run sustainable growth rates. Sustained growth in the Solow model is
only possible if there is technological progress, since without it per capita
income growth will eventually cease due to the impact of diminishing returns
to capital accumulation. Given that per capita incomes have been rising for
over 100 years in a large number of countries, and growth rates have displayed
no overall tendency to decline, the role of technological progress in
the Solow model in explaining sustainable growth becomes crucial. But
herein lies the obvious shortcoming of the neoclassical model since ‘the
long-run per capita growth rate is determined entirely by an element – the
rate of technological progress – that is outside the model … Thus we end up
with a model of growth that explains everything but long-run growth, an
obviously unsatisfactory situation’ (Barro and Sala-i-Martin, 1995). Furthermore,
as P. Romer (1989) highlights, in terms of policy advice for long-term
growth the neoclassical model has little to offer!
A second problem relates to the evidence, which clearly shows that income
per capita differentials across the world are much greater than predicted by
the model. Differences across countries in capital intensities are too small to
account for the observed disparities in real incomes. Using a Cobb–Douglas
production function framework it is possible to allocate differences in the
level of per capita incomes between countries to variations in levels of total
factor productivity growth and the accumulation of factor inputs. In particular
it is possible to estimate how much of the income disparities witnessed
between rich and poor countries can be attributed to different capital intensities
since total factor productivity is common across all countries. Substituting
from equation (11.34) to equation (11.26) gives equation (11.38):
k˙ = skα − (n + δ)k (11.38)
Setting this equation equal to zero (the steady state condition) and substituting
into the production function yields (11.39):
y* = [s/(n + δ)]α/(1−α) (11.39)
Equation (11.39) is now in a form that enables a solution to be found for the
steady state output per worker (y*). As Jones (2001a) highlights, we can see
from equation (11.39) why some countries are so rich and some are so poor.
Assuming exogenous technology and a similar value for the capital exponent
(α), countries that sustain high rates of saving, and low rates of population
growth and depreciation, will be rich. According to the neoclassical growth
model the high-income economies have achieved their high living standards
because they have accumulated large per worker stocks of capital. However,
although the model correctly predicts the directions of the effects of saving
and population growth on output per worker, it does not correctly predict the
magnitudes. As Mankiw et al. (1992) and Mankiw (1995) argue, the gaps in
output per worker (living standards) between rich and poor countries are
much larger than plausible estimates of savings rates and population growth
predict using equation (11.39). The crux of the problem is that with α = 1/3
there are sharply diminishing returns to capital. This implies that a tenfold
gap in output per worker between the USA and India would require a thousandfold
difference in the capital–labour ratios between these countries! (It
should be noted that this result is highly sensitive to the choice of α = 1/3 for
the share of capital in GDP.)
A third problem with the Solow model is that given a common production
function (that is, exogenous technology) the marginal product of capital
should be much higher in poor countries than in rich countries. Given the
parameters of the Solow model, the observed tenfold differential in output
per worker between rich and poor countries implies a hundredfold difference
in the marginal product of capital if output gaps are entirely due to variations
in capital intensities. Such differentials in the rate of return to capital are
simply not observed between rich and poor countries. As David Romer
(1996) observes, such differences in rates of return ‘would swamp such
considerations as capital market imperfections, government tax policies, fear
of expropriation and so on and we would observe immense flows of capital
from rich to poor countries. We do not see such flows.’ But the rate of return
to capital in poor countries is less than expected and the anticipated massive
flows of capital from rich to poor countries have not been observed across
poor countries as a whole (Lucas, 1990b).
A fourth difficulty relates to the rate of convergence, which is only about
half that predicted by the model. The economy’s initial conditions influence
the outcome for much longer than the model says it should (Mankiw, 1995).
In conclusion, it appears that within the Solow growth framework, physical
capital accumulation alone cannot account for either continuous growth of
per capita income over long periods of time or the enormous geographical
disparities in living standards that we observe. In terms of Figure 11.3, the
data on output per worker (or income per capita) that we actually observe
across the world reveal much greater disparities than those predicted by the
Solow model based on differences in capital per worker.
The new growth models emerging after 1986 depart from the Solow model
in three main ways. One group of models generates continuous growth by
abandoning the assumption of diminishing returns to capital accumulation.
To achieve this, Paul Romer (1986) introduced positive externalities from
capital accumulation so that the creation of economy-wide knowledge emerges
as a by-product of the investment activity of individual firms, a case of
‘learning by investing’ (Barro and Sala-i-Martin, 2003). A second approach
models the accumulation of knowledge as the outcome of purposeful acts by
entrepreneurs seeking to maximize private profits; that is, technological
progress is endogenized (P. Romer, 1990). A third class of model claims that
the role of capital is much more important than is suggested by the α term in
the conventional Cobb–Douglas production function shown in equations
(11.28)–(11.30). In their ‘augmented’ Solow model, Mankiw et al. (1992)
broaden the concept of capital to include ‘human capital’. The first two
classes of model constitute the core of endogenous growth theory whereas
the Mankiw, Romer and Weil (MRW) model constitutes what Klenow and
Rodriguez-Clare (1997a, 1997b) call a ‘neoclassical revival’. The central
proposition of endogenous growth theory is that broad capital accumulation
(physical and human capital) does not experience diminishing returns. The
growth process is driven by the accumulation of broad capital together with
the production of new knowledge created through research and development.
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