TECHNOLOGICAL CHANGE AND EQUILIBRIUM GRO
TECHNOLOGICAL
CHANGE AND E Q U I L I B R I U M GROWTH
I N T H E HARROD-DOMAR MODEL*
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There have been several attempts to resolve ‘HARROD’S
problem”,
anathat a growing economy is inherently unstable, In HARROD’S
lytical framework, some economists have said, the economy is always
poised on a knife’s edge within a hair’s breadth ofplunginginto either
secular stagnation or secular inflation by the divergence of the natural and warranted rates of growth. It is this condition which has
attracted attempts at modification. I n general, there have been two
types of modification. On the one hand, there is the substitution of
neoclassical conditions with variable technological coefficients for
HARROD’S
fixed coefficientsa. On the other hand, KALDOR’S
modelS
provides an adjustment mechanism which operates through variations in income distribution and the saving ratio. Some models have
incorporated the two4. It will be sufficient to restrict the discussion
to the neoclassical growth models, since it is these models which
introduce technological modifications to resolve HARROD’S
problem.
I n the first section, a relatively neglected aspect, at least in as far
as it impinges on the neoclassical modifications O f HARROD’S
growth
z
zyxw
* This article is, in part, an outgrowth of the author’s ‘Comment on R.BICA~ 1 6 %The
: Threshold of Economic Growth’, Kyklos, Vol.XV1, Fasc.2. In part,
it is adapted from the author’s doctoral dissertation completed at Rutgers University. The author is grateful to K. K. KURIHARA,
M. GIDEONSE,
W. C. BAGLEY,
S. KLEIN,and S. RATNER
of Rutgers University and A. CARLIPand P. VUKASIN
of
Harpur College for their suggestions and criticisms.
1. This is the term used by H.A.J. GREENto describe the possible and probable divergence of HARROD’S
natural rate of growth from the warranted rate of
growth. See his ‘Growth Models, Capital and Stability’, Economic Journal, Vol. 70,
pp. 57-73.
2. See, for example, R. SOLOW,
‘A Contribution to the Theory of Economic
Growth’, Quarterb Journal of Economics, Vol. 70, pp. 65-94; H. PILVIN,‘Full Capacity vs. Full Employment Growth’, Quarterly Journal of Economics, Vol. 67,
pp. 545-552 ; and J. TOBIN,‘A Dynamic Aggregate Model’, Journal of Political
Economy, Vol. 63, pp. 103-1 15.
3. See his‘AModelofEconomicGrowth’,EconomicJournal,Vol.67,pp. 591-624.
4. For example,seeJ. ROBINSON’S,
The Accumulationof Capital, 1956,pp. 404-406.
zyxw
zyxwvu
207
J. E. LA T O U R E T T E
zyxw
model, will be discussed. This aspect is HARROD’S
assumptions in
respect to technological change, especially neutral technological
change, and the implications of his assumptions for aggregate production functions. The modifications mentioned above fail to resolve
HARROD’S
problem and result from a failure to understand HARROD’S
technological assumptions and to perceive the nature ofhis dynamics.
The author is convinced that not only is the following interpretation
O f HARROD’S
technological assumptions consistent with his intentions,
but also that it is the appropriate one for a theory of economic growth.
I n developing this interpretation, some statistical evidence which
tends to substantiate this conviction will be cited.
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z
zyx
zyxwv
zy
zy
I. Technological Change in the Harrod-Domar Model:
A Suggested Interpretation
To some economists, the production function assumed in the HARROD
model appears to be highly specific. There is, in other words, no possibility for factor substitution. That is, the isoproduct curves are right
angles whose vertices lie along a straight line which passes through
the origin. This view may be represented by a diagram. Following
HARROD,
a two factor aggregate production function is employed.
Y= F ( K , N )
(1)
where Y represents real aggregate income or output and K and N t h e
physical inputs of capital and labor, respectively. From this aggregate
production function, there may be derived what will be referred to
as an aggregate productivity function by dividing through by N.
Therefore, we have :
This functional relationship is to be interpreted to imply that the
productivity of labor ( Y / N )is a function of ‘roundaboutness’, where
roundaboutness refers to the value of the capital-labor ratio ( K / N ) .
In Figure 1, the productivity of labor is measured on they axis,
while the degree of roundaboutness is measured on the x axis. To
represent HARROD’S
highly specific aggregate productivity function,
we draw the rayfo(K/N, 1) with a constant slope, j3. It is a charac208
zy
zyx
zy
zyxwvut
T E C H N O L O G I C A L C H A N G E A N D EQUILIBRIUM GROWTH
teristic of this function that, as various amounts of capital inputs per
worker are employed, the capital-output ratio ( K / Y )remains constant. This is explained by the fact that the slope ofthe ray, /I, is equal
,
reduces T/K, the productivity of capital.
to ( T / N ) / ( K / N )which
Since the productivity of capital is the reciprocal of the capitaloutput ratio, the capital-output ratio is constant along the aggregate
productivity functionfo(K/N, 1).
zyxw
Figure 1
The construction ofFigure 1 is such that any linear function drawn
from the origin represents a constant capital-output ratio. Therefore,
it is arbitrary as to which of the almost infinite number of functions
we select for discussion. The interpretation developed below applies
to any one of these functions. We might think of /I as 0.33. In this
instance, our capital-output ratio is 3, which represents approximately the case of an advanced economy such as the United States.
To achieve more flexibility in the HARROD
model and, in effect,
to attempt to resolve the problem of the potential disequilibrium due
to the divergence between the warranted and natural rates, a 'normal' production function which allows for factor substitution is introduced in the neoclassical models. In this case, the aggregate productivity function would appear asfl(X/N, 1) in Figure 2, where the
productivity of labor (Z"/N)is assumed to be an increasing function
of roundaboutness (the capital-labor ratio), but at a decreasing rate.
zy
209
zyxw
zyxwvu
zyxwv
J. E. LA T O U R E T T E
With functionf~(X/N,I), as capital is substituted for labor, the productivity of capital falls and, therefore, the capital-output ratio rises.
This may be demonstrated by drawing a ray to each successive combination of KIN and TIN long the aggregate productivity function.
Figure 2
The slope of each successive ray and, therefore, the productivity of
capital becomes smaller and smaller. This normal productivity function implies that it is possible for variations in the capital-labor ratio
and, therefore, for variations in the capital-output ratio to bring into
equality the warranted and natural rates. It is to this question, of the
introduction of an adjustment mechanism, that the discussion now
turns.
To examine the neoclassical adjustment mechanism, let us look
first at HARROD’S
condition of secular inflation. I n this case, when
the actual
the natural rate (Gn)exceeds the warranted rate (Gw),
rate (G) lies above the warranted rate such that the required capitaloutput ratio (Cr) is greater than the actual or expost capital-output
ratio (C). In HARROD’S
model, there is a continuous movement away
from the equilibrium rate because the ‘required’ investment ratio
(Cr) would always exceed the actual (expost) investment ratio (C),
creating short capacity and stimulating investment. However, with
a given saving ratio and a given equilibrium rate of interest, ex ante
investment will exceed ex ante saving. Hence, in the neoclassical
210
zyxwv
zy
z
TECHNOLOGICAL CHANGE AND EQUILIBRIUM GROWTH
zy
scheme of the world, the rate of interest is driven up. As the rate of
interest rises, entrepreneurs will substitute labor for capital. There is
a movement back down the normal aggregate productivity function
and the capital-labor ratio falls (see Figure 2). As the discussion in
the preceding paragraph implies, a fall in the required capital-output
ratio (Cr) accompanies the fall in the capital-labor ratio. This causes
Cr to tend to equal C. Finally, the fall in Cr raises the warranted rate
(G,) bringing it into equality with the natural rate (Gn)5. This is the
neoclassical ‘resolution’ of HARROD’S
condition of secular inflation.
On the other hand, if G, > Gn, G, > G and C > Cr. Investment
is discouraged because of excess capacity and there is secular stagnation. With a given saving ratio, ex ante saving exceeds ex ante investment and, in the neoclassical plan, the rate of interest falls. Entrepreneurs will substitute capital for labor. There is a movement outward along the normal productivity function and the capital-labor
ratio rises (see Figure 2). This is accompanied by a rise in the required
capital-output ratio (Cr),so that C,. tends to equal C. Therefore, when
C,. = C, G, = G, = G. Again, it is observed that in the neoclassical
world there is no knife edge.
It should be evident that a radical alteration of the HARROD
model
has occurred at the hands of the neoclassical growth theorists. In
HARROD’S
world, the secular conditions of G, > G, and G, > G ,
are the result of a difference between the community’s saving desires
which determine, with a given capital-output ratio, the warranted
rate and the ratio of saving to output required to have Gn equal G,.
That is, HARROD’S
problem depends on the saving ratio being too
high or too low to have Gn = Gw, for he writes:
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zyxwv
zyxw
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Gn Cr = or #s6
(3)
In the long run, he does not allow for variations in the capital-output
ratio. I t is assumed to be constant because of neutral technological
advance. More of this later, but let us proceed to the interpretation.
More fundamental than the alteration of the HARROD
model is the
misunderstanding of HARROD’S
discussion of the general nature of
dynamics and, in particular, the implications of his correlative as-
zyxw
zyxwvu
5 . The fall in Cr raises G , because Gw= S/Cr. Note that the saving ratio (s)
is assumed to be constant in the neoclassical models.
6. Towards a Dynamic Economics, 1948, p. 87.
211
zyxw
J. E. LA TOURETTE
zy
sumptions of a constant capital-output ratio and a neutral technological advance for a theory of economic growth. The confusion over
model lies
the type of production function employed in the HARROD
within this area. To develop the interpretation, HARROD'S
remarks
on neutral technological advance are quoted at length' :
CHANGE AND E Q U I L I B R I U M GROWTH
I N T H E HARROD-DOMAR MODEL*
zyx
There have been several attempts to resolve ‘HARROD’S
problem”,
anathat a growing economy is inherently unstable, In HARROD’S
lytical framework, some economists have said, the economy is always
poised on a knife’s edge within a hair’s breadth ofplunginginto either
secular stagnation or secular inflation by the divergence of the natural and warranted rates of growth. It is this condition which has
attracted attempts at modification. I n general, there have been two
types of modification. On the one hand, there is the substitution of
neoclassical conditions with variable technological coefficients for
HARROD’S
fixed coefficientsa. On the other hand, KALDOR’S
modelS
provides an adjustment mechanism which operates through variations in income distribution and the saving ratio. Some models have
incorporated the two4. It will be sufficient to restrict the discussion
to the neoclassical growth models, since it is these models which
introduce technological modifications to resolve HARROD’S
problem.
I n the first section, a relatively neglected aspect, at least in as far
as it impinges on the neoclassical modifications O f HARROD’S
growth
z
zyxw
* This article is, in part, an outgrowth of the author’s ‘Comment on R.BICA~ 1 6 %The
: Threshold of Economic Growth’, Kyklos, Vol.XV1, Fasc.2. In part,
it is adapted from the author’s doctoral dissertation completed at Rutgers University. The author is grateful to K. K. KURIHARA,
M. GIDEONSE,
W. C. BAGLEY,
S. KLEIN,and S. RATNER
of Rutgers University and A. CARLIPand P. VUKASIN
of
Harpur College for their suggestions and criticisms.
1. This is the term used by H.A.J. GREENto describe the possible and probable divergence of HARROD’S
natural rate of growth from the warranted rate of
growth. See his ‘Growth Models, Capital and Stability’, Economic Journal, Vol. 70,
pp. 57-73.
2. See, for example, R. SOLOW,
‘A Contribution to the Theory of Economic
Growth’, Quarterb Journal of Economics, Vol. 70, pp. 65-94; H. PILVIN,‘Full Capacity vs. Full Employment Growth’, Quarterly Journal of Economics, Vol. 67,
pp. 545-552 ; and J. TOBIN,‘A Dynamic Aggregate Model’, Journal of Political
Economy, Vol. 63, pp. 103-1 15.
3. See his‘AModelofEconomicGrowth’,EconomicJournal,Vol.67,pp. 591-624.
4. For example,seeJ. ROBINSON’S,
The Accumulationof Capital, 1956,pp. 404-406.
zyxw
zyxwvu
207
J. E. LA T O U R E T T E
zyxw
model, will be discussed. This aspect is HARROD’S
assumptions in
respect to technological change, especially neutral technological
change, and the implications of his assumptions for aggregate production functions. The modifications mentioned above fail to resolve
HARROD’S
problem and result from a failure to understand HARROD’S
technological assumptions and to perceive the nature ofhis dynamics.
The author is convinced that not only is the following interpretation
O f HARROD’S
technological assumptions consistent with his intentions,
but also that it is the appropriate one for a theory of economic growth.
I n developing this interpretation, some statistical evidence which
tends to substantiate this conviction will be cited.
zyx
zyx
z
zyx
zyxwv
zy
zy
I. Technological Change in the Harrod-Domar Model:
A Suggested Interpretation
To some economists, the production function assumed in the HARROD
model appears to be highly specific. There is, in other words, no possibility for factor substitution. That is, the isoproduct curves are right
angles whose vertices lie along a straight line which passes through
the origin. This view may be represented by a diagram. Following
HARROD,
a two factor aggregate production function is employed.
Y= F ( K , N )
(1)
where Y represents real aggregate income or output and K and N t h e
physical inputs of capital and labor, respectively. From this aggregate
production function, there may be derived what will be referred to
as an aggregate productivity function by dividing through by N.
Therefore, we have :
This functional relationship is to be interpreted to imply that the
productivity of labor ( Y / N )is a function of ‘roundaboutness’, where
roundaboutness refers to the value of the capital-labor ratio ( K / N ) .
In Figure 1, the productivity of labor is measured on they axis,
while the degree of roundaboutness is measured on the x axis. To
represent HARROD’S
highly specific aggregate productivity function,
we draw the rayfo(K/N, 1) with a constant slope, j3. It is a charac208
zy
zyx
zy
zyxwvut
T E C H N O L O G I C A L C H A N G E A N D EQUILIBRIUM GROWTH
teristic of this function that, as various amounts of capital inputs per
worker are employed, the capital-output ratio ( K / Y )remains constant. This is explained by the fact that the slope ofthe ray, /I, is equal
,
reduces T/K, the productivity of capital.
to ( T / N ) / ( K / N )which
Since the productivity of capital is the reciprocal of the capitaloutput ratio, the capital-output ratio is constant along the aggregate
productivity functionfo(K/N, 1).
zyxw
Figure 1
The construction ofFigure 1 is such that any linear function drawn
from the origin represents a constant capital-output ratio. Therefore,
it is arbitrary as to which of the almost infinite number of functions
we select for discussion. The interpretation developed below applies
to any one of these functions. We might think of /I as 0.33. In this
instance, our capital-output ratio is 3, which represents approximately the case of an advanced economy such as the United States.
To achieve more flexibility in the HARROD
model and, in effect,
to attempt to resolve the problem of the potential disequilibrium due
to the divergence between the warranted and natural rates, a 'normal' production function which allows for factor substitution is introduced in the neoclassical models. In this case, the aggregate productivity function would appear asfl(X/N, 1) in Figure 2, where the
productivity of labor (Z"/N)is assumed to be an increasing function
of roundaboutness (the capital-labor ratio), but at a decreasing rate.
zy
209
zyxw
zyxwvu
zyxwv
J. E. LA T O U R E T T E
With functionf~(X/N,I), as capital is substituted for labor, the productivity of capital falls and, therefore, the capital-output ratio rises.
This may be demonstrated by drawing a ray to each successive combination of KIN and TIN long the aggregate productivity function.
Figure 2
The slope of each successive ray and, therefore, the productivity of
capital becomes smaller and smaller. This normal productivity function implies that it is possible for variations in the capital-labor ratio
and, therefore, for variations in the capital-output ratio to bring into
equality the warranted and natural rates. It is to this question, of the
introduction of an adjustment mechanism, that the discussion now
turns.
To examine the neoclassical adjustment mechanism, let us look
first at HARROD’S
condition of secular inflation. I n this case, when
the actual
the natural rate (Gn)exceeds the warranted rate (Gw),
rate (G) lies above the warranted rate such that the required capitaloutput ratio (Cr) is greater than the actual or expost capital-output
ratio (C). In HARROD’S
model, there is a continuous movement away
from the equilibrium rate because the ‘required’ investment ratio
(Cr) would always exceed the actual (expost) investment ratio (C),
creating short capacity and stimulating investment. However, with
a given saving ratio and a given equilibrium rate of interest, ex ante
investment will exceed ex ante saving. Hence, in the neoclassical
210
zyxwv
zy
z
TECHNOLOGICAL CHANGE AND EQUILIBRIUM GROWTH
zy
scheme of the world, the rate of interest is driven up. As the rate of
interest rises, entrepreneurs will substitute labor for capital. There is
a movement back down the normal aggregate productivity function
and the capital-labor ratio falls (see Figure 2). As the discussion in
the preceding paragraph implies, a fall in the required capital-output
ratio (Cr) accompanies the fall in the capital-labor ratio. This causes
Cr to tend to equal C. Finally, the fall in Cr raises the warranted rate
(G,) bringing it into equality with the natural rate (Gn)5. This is the
neoclassical ‘resolution’ of HARROD’S
condition of secular inflation.
On the other hand, if G, > Gn, G, > G and C > Cr. Investment
is discouraged because of excess capacity and there is secular stagnation. With a given saving ratio, ex ante saving exceeds ex ante investment and, in the neoclassical plan, the rate of interest falls. Entrepreneurs will substitute capital for labor. There is a movement outward along the normal productivity function and the capital-labor
ratio rises (see Figure 2). This is accompanied by a rise in the required
capital-output ratio (Cr),so that C,. tends to equal C. Therefore, when
C,. = C, G, = G, = G. Again, it is observed that in the neoclassical
world there is no knife edge.
It should be evident that a radical alteration of the HARROD
model
has occurred at the hands of the neoclassical growth theorists. In
HARROD’S
world, the secular conditions of G, > G, and G, > G ,
are the result of a difference between the community’s saving desires
which determine, with a given capital-output ratio, the warranted
rate and the ratio of saving to output required to have Gn equal G,.
That is, HARROD’S
problem depends on the saving ratio being too
high or too low to have Gn = Gw, for he writes:
zyxwvu
zyxwv
zyxw
zy
zyxw
Gn Cr = or #s6
(3)
In the long run, he does not allow for variations in the capital-output
ratio. I t is assumed to be constant because of neutral technological
advance. More of this later, but let us proceed to the interpretation.
More fundamental than the alteration of the HARROD
model is the
misunderstanding of HARROD’S
discussion of the general nature of
dynamics and, in particular, the implications of his correlative as-
zyxw
zyxwvu
5 . The fall in Cr raises G , because Gw= S/Cr. Note that the saving ratio (s)
is assumed to be constant in the neoclassical models.
6. Towards a Dynamic Economics, 1948, p. 87.
211
zyxw
J. E. LA TOURETTE
zy
sumptions of a constant capital-output ratio and a neutral technological advance for a theory of economic growth. The confusion over
model lies
the type of production function employed in the HARROD
within this area. To develop the interpretation, HARROD'S
remarks
on neutral technological advance are quoted at length' :