Endogenous Growth: Constant Returns to Capital
Accumulation
During the mid-1980s several economists, most notably Paul Romer (1986,
1987b) and Robert Lucas (1988), sought to construct alternative models of
growth where the long-run growth of income per capita depends on ‘investment’
decisions rather than unexplained technological progress. However, as
Crafts (1996) notes, the term investment in the context of these new models
refers to a broader concept than the physical capital accumulation reported in
the national accounts; research and development (R&D) expenditures and human
capital formation may also be included. ‘The key to endogenous steady
state growth is that there should be constant returns to broad capital accumulation’.
Hence in order to construct a simple theory of endogenous growth, the
long-run tendency for capital to run into diminishing returns needs to be
modified to account for the extraordinary and continuous increases in observed
per capita incomes across the world’s economies. In the early versions of the
new endogenous growth theory the accumulation of capital plays a much
greater role in the growth process than in the traditional neoclassical model. In
many ways the work of Romer revives the earlier seminal contribution of
Arrow (1962) on ‘learning by doing’. Arrow had shown how the productivity
of labour increases with experience, and experience is a function of cumulative
investment expenditures that alter the work environment. That is, a firm’s
accumulation of capital produces external effects on learning. However, as
Blaug (2002) argues, ‘it strains credulity to believe that this could account, not
just for a once-and-for-all increase in output, but also for a constant rate of
increase in total factor productivity year in year out’.
Building on Arrow’s insight, Romer broadened the concept of capital to
include investment in knowledge as well as the accumulation of physical
capital goods. Since the knowledge gained by workers in one firm has public
good characteristics and is at best only partially excludable, then knowledge
spillovers occur such that investment in knowledge (R&D) by one firm increases
the production potential of other firms. No individual firm can
completely internalize the positive impact that their investment in physical
and human capital has on the economy-wide stock of knowledge.
Paul Romer’s 1986 model can be illustrated by modifying the production
function. In equation (11.40) the production function includes technology (A)
as an endogenous input:
Y = F(K, L, A) (11.40)
At the micro level, the output of an individual firm (j) depends on its own
inputs of capital (Kj), labour (Lj) and the economy-wide state of knowledge
(A), as indicated in equation (11.41):
Yj = F(Kj, Lj, A) (11.41)
In this formulation the growth of knowledge (technology) is assumed to
depend on the growth of capital because capital deepening fosters technological
spillovers that raise the marginal productivity of capital across the economy
as a whole. Therefore any increase in aggregate K will improve A and hence
the productivity of all firms. In Romer’s (1986) endogenous growth model
the expansion of aggregate knowledge results from learning externalities
among firms. In effect, the higher the level of the capital stock in an economy,
the more productive each firm will be via a process of ‘learning by doing’. So
while a firm’s production function exhibits constant returns to scale and
diminishing returns to capital accumulation, the aggregate production function
will exhibit increasing, rather than constant, returns to scale.
One of the simplest models of endogenous growth is the AK* model shown
in equation (11.42) below (Rebelo, 1991):
Y = KαHβ = AK* (11.42)
Here A is a constant, K* represents a broad measure of capital (Kα Hβ), and α
+ β = 1. As Crafts (1995) points out, ‘models of this kind put investment
centre stage and see growth as an investment-driven process. There is no role
for the Solow residual.’ Therefore there is a close similarity between the AK
model and the Harrod–Domar model. In both models there are no diminishing
returns and hence no reason for growth to slow down as capital deepening
occurs. If one group of countries has higher average savings rates, lower
depreciation rates and lower capital–output ratios than some other group of
countries, then the first group will grow faster than the second group permanently
and ‘divergence, big time’ will be the rule.
The AK class of endogenous growth models has been subject to heavy
criticism, mainly on account of their key assumption of an absence of diminishing
returns to the capital input. The AK model predicts a permanent increase
in the growth rate following an increase in the investment/GDP ratio of an
economy. However, Jones (1995), in a time series analysis of 15 OECD
countries in the post-1945 period, argues that the AK models are inconsistent
with the empirical evidence. Although the investment/GDP ratios increased
significantly in the 1950–89 period, growth rates of GDP per worker remained
stable or have fallen. This finding has been challenged by McGrattan
(1998). By considering time series evidence from a larger sample of countries
over a longer time period McGrattan finds the main predictions of AK
theory to be confirmed by the data. Using data from Maddison (1995) for the
period 1870–1989, McGratton finds that ‘higher investment rates correspond
to higher growth rates, with the exception of the US economy where there is
little variation in the growth rate of GDP per capita’. Extending the analysis
to cross-sectional data for 125 economies in the period 1960–85 also reveals
‘a definite positive correlation between investment rates and growth rates’.
Accumulation
During the mid-1980s several economists, most notably Paul Romer (1986,
1987b) and Robert Lucas (1988), sought to construct alternative models of
growth where the long-run growth of income per capita depends on ‘investment’
decisions rather than unexplained technological progress. However, as
Crafts (1996) notes, the term investment in the context of these new models
refers to a broader concept than the physical capital accumulation reported in
the national accounts; research and development (R&D) expenditures and human
capital formation may also be included. ‘The key to endogenous steady
state growth is that there should be constant returns to broad capital accumulation’.
Hence in order to construct a simple theory of endogenous growth, the
long-run tendency for capital to run into diminishing returns needs to be
modified to account for the extraordinary and continuous increases in observed
per capita incomes across the world’s economies. In the early versions of the
new endogenous growth theory the accumulation of capital plays a much
greater role in the growth process than in the traditional neoclassical model. In
many ways the work of Romer revives the earlier seminal contribution of
Arrow (1962) on ‘learning by doing’. Arrow had shown how the productivity
of labour increases with experience, and experience is a function of cumulative
investment expenditures that alter the work environment. That is, a firm’s
accumulation of capital produces external effects on learning. However, as
Blaug (2002) argues, ‘it strains credulity to believe that this could account, not
just for a once-and-for-all increase in output, but also for a constant rate of
increase in total factor productivity year in year out’.
Building on Arrow’s insight, Romer broadened the concept of capital to
include investment in knowledge as well as the accumulation of physical
capital goods. Since the knowledge gained by workers in one firm has public
good characteristics and is at best only partially excludable, then knowledge
spillovers occur such that investment in knowledge (R&D) by one firm increases
the production potential of other firms. No individual firm can
completely internalize the positive impact that their investment in physical
and human capital has on the economy-wide stock of knowledge.
Paul Romer’s 1986 model can be illustrated by modifying the production
function. In equation (11.40) the production function includes technology (A)
as an endogenous input:
Y = F(K, L, A) (11.40)
At the micro level, the output of an individual firm (j) depends on its own
inputs of capital (Kj), labour (Lj) and the economy-wide state of knowledge
(A), as indicated in equation (11.41):
Yj = F(Kj, Lj, A) (11.41)
In this formulation the growth of knowledge (technology) is assumed to
depend on the growth of capital because capital deepening fosters technological
spillovers that raise the marginal productivity of capital across the economy
as a whole. Therefore any increase in aggregate K will improve A and hence
the productivity of all firms. In Romer’s (1986) endogenous growth model
the expansion of aggregate knowledge results from learning externalities
among firms. In effect, the higher the level of the capital stock in an economy,
the more productive each firm will be via a process of ‘learning by doing’. So
while a firm’s production function exhibits constant returns to scale and
diminishing returns to capital accumulation, the aggregate production function
will exhibit increasing, rather than constant, returns to scale.
One of the simplest models of endogenous growth is the AK* model shown
in equation (11.42) below (Rebelo, 1991):
Y = KαHβ = AK* (11.42)
Here A is a constant, K* represents a broad measure of capital (Kα Hβ), and α
+ β = 1. As Crafts (1995) points out, ‘models of this kind put investment
centre stage and see growth as an investment-driven process. There is no role
for the Solow residual.’ Therefore there is a close similarity between the AK
model and the Harrod–Domar model. In both models there are no diminishing
returns and hence no reason for growth to slow down as capital deepening
occurs. If one group of countries has higher average savings rates, lower
depreciation rates and lower capital–output ratios than some other group of
countries, then the first group will grow faster than the second group permanently
and ‘divergence, big time’ will be the rule.
The AK class of endogenous growth models has been subject to heavy
criticism, mainly on account of their key assumption of an absence of diminishing
returns to the capital input. The AK model predicts a permanent increase
in the growth rate following an increase in the investment/GDP ratio of an
economy. However, Jones (1995), in a time series analysis of 15 OECD
countries in the post-1945 period, argues that the AK models are inconsistent
with the empirical evidence. Although the investment/GDP ratios increased
significantly in the 1950–89 period, growth rates of GDP per worker remained
stable or have fallen. This finding has been challenged by McGrattan
(1998). By considering time series evidence from a larger sample of countries
over a longer time period McGrattan finds the main predictions of AK
theory to be confirmed by the data. Using data from Maddison (1995) for the
period 1870–1989, McGratton finds that ‘higher investment rates correspond
to higher growth rates, with the exception of the US economy where there is
little variation in the growth rate of GDP per capita’. Extending the analysis
to cross-sectional data for 125 economies in the period 1960–85 also reveals
‘a definite positive correlation between investment rates and growth rates’.
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