1. No category

Why Is Systematically Observed as Smaller than 1?
And Why Is It of Fundamental Importance?
Olivier de La Grandville1
Stanford University, Department of Management Science and Engineering
475 Via Ortega, Stanford, CA 94305; email: [email protected]
April 24, 2012
Abstract. We show that if an economy is driven by a CES production function with capital-augmenting progress, an elasticity
of substitution higher than 1 is incompatible with competitive
equilibrium. Only an elasticity lower than 1 can be associated
with competitive equilibrium and the current and future rewards
society may derive from it. We also show that there always exists
a threshold value of the elasticity of substitution precluding competitive equilibrium whatever the types of technological progress
prevailing in the economy.
JEL code numbers E20, E25, O40, O41.
1. Introduction
Throughout the ages, one of the constant aims of any society has been to
resort to capital to alleviate labor. Intimately linked to technical progress,
this aim turned out to be the main engine of economic growth, de…ned as a
steady increase of real income per person. The substitution process between
capital and labor was analysed very early by Hicks (1932) and Allen (1938)
as the increasing relationship between the ratio rental rate of capital/ wage
rate (denoted q=w), on the one hand, and the capital/labor ratio (r = K=L)
on the other; the sensitivity of this relationship was to be measured by the
so-called elasticity of substitution ( ), i.e. the elasticity of K=L with respect
to q=w. Both authors showed that if the production function Y = F (K; L)
is homogeneous of degree one, could be expressed as = FK FL =rF FKL :
Even considering as a constant, it would have taken a lot of inventiveness
to determine from this second-order partial di¤erential equation the implied
analytical form of F (K; L).
1
The extremely useful comments and suggestions by Ken Arrow, Bjarne Jensen, David
de La Croix, Ulla Lehmijooki, Miguel Leon-Ledesma, Bernardo Maggi, Pietro Peretto and
Robert Solow are gratefully acknowledged.
1
It is only about a quarter of a century later that a surprising, non-intuitive
discovery was made; it would have two very signi…cant outcomes: it would
…rst provide an e¢ cient way of measuring ;and, as importantly, it would
be key to deriving the solution to the above-mentioned equation. Indeed,
in 1961 Arrow, Chenery, Minhas and Solow showed that if the production
process is homogeneous of degree one, the elasticity of substitution is equal
to the elasticity of income per person with respect to the wage rate. They
then tried to …nd the best possible …t between the wage rate and income per
person (y). Settling for a concave, power function of the type y = aw ; and
using the fact that w = y(r) ry 0 (r); they integrated the resulting …rst order,
ordinary di¤erential equation to obtain the celebrated constant elasticity of
substitution production function.
We have emphasized the word "concave": indeed, there was overwhelming
evidence that was smaller than one. In the half-century that followed, this
fact was con…rmed time and again (for the many ways to measure and
for numbers; see Antras (2004), Sato (2006), Klump, McAdam and Willman
(2007), Chirinko (2008), León-Ledesma, McAdam and Willman (2010)).
In this paper we will answer the following question: why is systematically observed as smaller than 1? We will show that there is a very powerful
reason behind this fact: if technological progress is capital-augmenting, then
> 1 is incompatible with competitive equilibrium.
This property will no doubt come as a surprise. Indeed, we have become
accustomed to the well documented link between a higher elasticity of substitution and enhanced growth –in 1960 already Pitchford had shown that
high enough a value of could generate permanent growth of income per
person even in the absence of technological progress. (In the same vein, and
for the role of in the growth process, see for instance La Grandville (1989,
2011), Yuhn (1991), Klump and La Grandville (2000), Klump and Preissler
(2000), La Grandville and Solow (2006), Thanh and Minh (2008)).
Above all, this …nding brings signi…cant – and we may add – comforting news: it means that competitive equilibrium and its numerous rewards
require an increasing share of labor in total income. We will show that
a diminishing share is de…nitely incompatible with competitive equilibrium,
while a constant share, compatible as it may be, would require an exceedingly
high optimal savings rate.
We will proceed as follows. In Section 2 we state our hypotheses and
show what competitive equilibrium implies in terms of welfare for society in
an aggregated model. We then demonstrate that competitive equilibrium
2
is not sustainable if
> 1 when technical progress is capital-augmenting
(Section 3), and that, whatever the types of technological progress prevailing
in the economy, there always exists a threshold value of the elasticity of
substitution precluding at any time competitive equilibrium (Section 4). In
Section 5 we explain these surprising results. The conclusion will stress their
importance.
2. Competitive equilibrium and its optimality.
2.1 Hypotheses.
Suppose the production process is driven by a CES function with capital- and
labor-augmenting technical progress represented by the increasing functions
GK (t) and GL (t) with the following general properties: they are positive,
increasing and unbounded; their growth rates, denoted gK (t) and gL (t); may
be increasing or decreasing over various time intervals. Initial
R t conditions are
set as GK (0) = GL (0) = 1: We thus have GK (t) = exp( 0 gK ( )d ) and
Rt
GL (t) = exp( 0 gL ( )d ):
The production function, supposed to be linearly homogeneous, is then
Yt = F (Kt GK (t); Lt GL (t)) = f [Kt GK (t)]p + (1
)[Lt GL (t)]p g1=p ; p 6= 0
(1)
where Yt represents, in index form, net national income (net of capital depreciation). To simplify notation, Yt ; Kt and Lt stand for the indices Yt =Y0 ;
Kt =K0 and Lt =L0 ; respectively: The order of this general mean of Kt GK (t)
and Lt GL (t) is p, an increasing function of the elasticity of substitution ;
an e¢ ciency parameter; p = 1 1= 2 :
In the case p = 0 or = 1; the production function is the mean of order
0, i.e. the geometric mean
Yt = [Kt GK (t)] [Lt GL (t)]1
2
:
(2)
Income per person, denoted yt , will be a general mean of order p = 1 1= as well:
yt = f [rt GK (t)]p +(1 )[GL (t)]p g1=p ; p 6= 0: The reason why is an e¢ ciency parameter
stems from the fundamental property of the general mean, as an increasing function of its
order. A proof of this property is in Hardy, Littlewood and Pólya (1952). This proof is
admittedly di¢ cult; an easier one can be found in La Grandville and Solow (Appendix of
chapter 5, in La Grandville, 2009). The property has an immediate geometric interpre1=p
tation: if p increases, the general mean M (p) = [ xp1 + (1
)xp2 ]
of two numbers x1
and x2 is a surface in (M; x1; x2 ) space that opens up around the ray M = x1 = x2 : This
property will prove to be very useful in Section 5.
3
If competitive equilibrium applies, the marginal productivity of capital
must equal the real rate of interest, represented by the exogeneous function
of time i(t):This rate, incorporating a risk premium, should be considered as
society’s long term rate of preference for the present. We then must have
@F
(Kt GK (t); Lt GL (t)) = i(t):
(3)
@K
What could be the magnitude of i(t) and its evolution? In a …rst approach, we
could reasonably argue that, throughout history, it has probably experienced
a small but signi…cant downward trend. Many factors may have contributed
to that trend; perhaps the most important is a slow decrease of fear among
societies. Pierre Gaxotte famously wrote that "the man of the Middle-Ages
does not know of time and numbers", and in those dark days the recurring
plagues, famine and wars would certainly have induced a fear for the future
de…nitely higher than today. On the other hand, we know too well of the fractal nature that characterises human evolution in historic times, considered
of course not only from the economic point of view, but from the far more
broad, and important, perspective of the evolution of ideas, characterised
by the sudden appearance of ideologies carrying disastrous consequences –
so disastrous in fact that it is hard to decide whether the 14th century was
worse for Europe than the 20th century turned out to be for the world.
We still can put bounds on i(t). We can safely say it is smaller than ,
the share of capital in any base year we might consider (FK (0) K0 =Y0 = ):
Indeed, together with (3), the inequality i(0) < implies FK (0) = Y0 =K0 <
, and therfore Y0 < K0 ; which is quite justi…able.
L(t) is supposed to be exogeneous, the wage rate being equal to the
marginal productivity of labor. The growth rate of L(t) is n(t); a function
of time which may be Rincreasing or decreasing on some intervals; thus, with
t
L(0) = 1; L(t) = exp( 0 n( )d ):
2.2 Welfare implications
From a welfare point of view, the systems of equations (1),(3) and (2),(3)
are far from trivial. Indeed, if for any of these systems a solution exists
(denoted as the optimal trajectory Kt ), it will ful…l no less than 3 objectives
for society:
1. The trajectory Kt minimizes the cost of producing any amount Yt (this
follows immediately from …rst principles).
4
2. Kt maximizes intertemporally, over a time interval of in…nite length,
the sum of all discounted consumption ‡ows society will receive.
Indeed, suppose that at time 0 society chooses to maximize
Z 1
Z 1
Rt
Rt
i(z)dz
0
W =
Ct e
dt =
[F (Kt GK (t); Lt GL (t)) K_ t ]e 0 i(z)dz dt ;
0
0
(4)
_ t) the integrand of the right-hand side of (4) and applying
denoting B(K; K;
_ t) is concave in
Euler’s equation BK dtd BK_ = 0 yields (3). Since B(K; K;
K and K_ , we can apply Takayama’s theorem to obtain the following result:
competitive equilibrium, expressed in (3), constitutes both a necessary and
a su¢ cient condition for the maximisation of W:
3. Kt maximizes at any point of time the value of society’s activity,
de…ned as consumption plus the rate of increase in the value of the capital
stock at that time.
To demonstrate this, we will rely on Robert Dorfman’s remarkable interpretation of the Pontryagin principle, where he introduced what he called a
modi…ed Hamiltonian, de…ned as the traditional Hamiltonian augmented by
the product of the state variable and the derivative of the adjoint variable.
In our case the Hamiltonian would be
_ t) = B(K; K;
_ t) +
H(K; K;
_
t K;
thus the modi…ed Hamiltonian –which, to honor Professor Dorfman’s memory, we choose to call a Dorfmanian and denote as D –is
_ t) = H(K; K;
_ t) + _ t K =
D(K; K;
[F (Kt GK (t); Lt GL (t))
K_ t ]e
Rt
0
i(z)dz
+
_ + _ t K (5)
tK
where the adjoint variable t , has an all-important signi…cance: it is the
present value of one unit of capital in existence at time t when the optimal
_ t) is equal, in
trajectory Kt is followed, as will be shown. Then D(K; K;
present value, to the sum of the consumption enjoyed by society plus the
_ t) can
rate of increase in the value of its capital stock at time t. D(K; K;
thus be viewed as the present value of society’s activity at time t:
Equating to 0 the gradient of D with respect to the state variable Kt and
the control variable K_ yields
5
@D
@F
=
(Kt GK (t); Lt GL (t))e
@K
@K
and
@D
=
@ K_
e
Rt
0
i(z)dz
+
Rt
0
t
i(z)dz
+ _t = 0
(6)
(7)
= 0:
Di¤erentiating (7) with respect to time and replacing _ t into (6) yields (3)
again, a su¢ cient condition for maximizing D since D is concave in K and
K_ 3 .
Rt
We still have to prove that t ; equal – from (7) – to exp( 0 i(z)dz) is
the present value of one additional unit of capital received at time t if the
economy is on its optimal path. This will be true if and only if at time t the
derivative with respect to K of the integral of the discounted consumption
‡ows received by society over the in…nite interval of time [t; 1) is equal to
1. Let this integral be denoted as I : We have
Z 1
R
I =
[F (K GK ( ); L GL ( )) K_ ]e t i(z)dz d :
(8)
t
The derivative of I with respect to Kt is
Z 1
@I
@
=
[F (K GK ( ); L GL ( )]e
@Kt
@K
t
R
t
i(z)dz
d :
(9)
Since K follows an optimal trajectory, we can replace in the integrand of
the right-hand side of (9) the marginal productivity of capital at any time
by i( ); we thus obtain
Z 1
R
R
@I
=
i( )e t i(z)dz d = e t i(z)dz j1
=1
(10)
t
@Kt
t
Notice the generality of the Dorfmanian: here we have considered investment K_ t as
the control variable; but we could equally well choose consumption Ct as our control. In
that case the Dorfmanian, still equal to the present value of consumption plus the rate of
increase in the value of capital would now be a function of consumption, capital and time;
denoted D ; it is equal to
3
D (K; C; t)
= Ct e
Ct e
Rt
0
Rt
0
_ + _ tK =
i(z)dz
+
tK
i(z)dz
+
t [F (Kt GK (t); Lt GL (t))
Ct ] + _ t K:
Setting the gradient of D (K; C; t) with respect to K; C equal to 0 yields (3) as well.
6
Rt
as was to be proved. Therefore t = e 0 i(z)dz is indeed the present value
of one additional unit of capital received at time t since it is equal to the
present value of all future rewards brought by this capital.
3. The incompatibility of competitive equilibrium with
capital-augmenting progress and > 1:
We will now prove the following theorem.
Theorem 1. Suppose an economy is driven by a constant elasticity
of substitution production function exhibiting capital- and labor-augmenting
technical progress with the following properties: the positive factor-enhancing
functions are increasing and unbounded. If the economy is to sustain a situation of competitive equilibrium, its elasticity of substitution must be smaller
than or equal to 1.
Proof.
Solving systems (1),(3) and (2),(3) yields
Kt =
1
1=p
Lt GL (t)=GK (t)
; p 6= 0;
[i(t) 1
GK (t)1
1]1=p
and
6= 1 ;
(11)
1
1
Kt =
i(t)
GK (t) 1 Lt GL (t); p = 0;
=1
(12)
respectively.
In the latter case ( = 1) a solution obviously always exists. Now consider
the cases 0 < < 1 and > 1: For a positive solution to exist, we must have
GK (t)1
> i(t)1
;
(13)
i.e. GK (t)1 must be larger than the geometric average of i(t) and , where
plays the role of the weight.
If 0 < < 1, this condition will always be ful…lled since GK (t) 1:
If, however, > 1 the right-hand side of the above inequality is larger
than , while the left-hand side tends to 0 with increasing t: Hence there always exists a time t from which system (1),(3) admits no solution. Therefore
7
competitive equilibrium is not sustainable if the elasticity of substitution is
larger than 1:
It is worth examining some examples, because in the case > 1 even when
competitive equilibrium can be achieved during a limited period, serious
problems arise regarding some fundamental characteristics of the economy,
namely its optimal growth rate and savings rate.
We will choose gK (t) and gL (t) as the constants gK = 0:005 and gL = 0:02
– these numbers are very close to those obtained by Sato (2006) for the
U.S. economy over an 80-year period. We will also consider i as a constant,
to which di¤erent values will be assigned in comparative dynamics. With
GK (t) = egK t , the time t is now given by the solution of
)gK t
e(1
= i1
i.e. by
ln + (1
) ln i
:
(1
)gK
t=
As an example, suppose that
to be as low as 15.4 years.
(14)
= 1=3 and i = 0:04: Then t turns
= 1:5,
Consider now what would be the evolution of the economy in the years
preceding t. We will show that even though solutions to system (1),(3)
exist, they are utterly unrealistic for two reasons: …rst, the economy exhibits
explosive behavior; and second the implied savings – or investment – rates
have unrealistically high values.
Suppose that L(t) = exp(nt): From (11), we deduce the growth rate of
the capital stock as
K_ t =Kt = gL + n
gK (1
i1
1
e
gK (1
)t
):
(15)
On the other hand, plugging (11) into (1) leads to the optimal time path of
income per person, denoted yt :
yt = egL t
1=p
1
1
i1
e
gK (1
;
)t
(16)
from which the growth rate of income per person can be determined as
y_ t =yt = gL +
gK
i
8
1 e(1
)gK t
1
:
(17)
Refering to (11) and (16), it can be immediately seen that even if a
solution is available for a …nite time span (before the fateful time t is reached),
the capital stock as well as income per person tend to in…nity in …nite time.
Worse, their growth rates tend to in…nity as well. Indeed, from (15) limt!t
(K_ t =Kt ) = 1, and from (17) limt!t (y_ t =yt ) = 1:
Let us now determine what would be the optimal savings ratio st , equal
to K_ t =F (Kt ; Lt ; t); from (1) and (11) we get
st = ( =i) e
(1
)gK t
: n + gL + gK (
i1
1
e
(1
)gK t
1) ;
(18)
a formula valid for all = 0. We notice that if > 1, st is an increasing function of time, certainly an unwelcome feature in an economy where
both capital and labor are enhanced by technological progress; indeed, one
certainly would expect technical progress to enable society to reduce, not
increase the proportion of its income that should be saved for the future. At
least as problematic is the fact that, already at the initial time, the optimal
saving rates are exceedingly high. At time t = 0, from (18) we have
s0 = ( =i)
n + gL + gK (
1
i1
1) :
(19)
Table 1 gives those initial optimal saving rates and o¤ers ample evidence
that the case > 1 leads to unrealistically high savings rates. Some cells in
Table 4 correspond to savings rates higher than 100%. Others, marked n.a.
for "not available" (cases i = 0:035; = 1:5 and i = 0:04; = 1:5 in the
table) correspond to a situation where is too high for a solution even to
exist at any time, whatever the types and combination of factor-enhancing
functions prevailing in the economy; our theorem 2 in next section will make
those circumstances precise.
9
Table 1. The inordinate initial optimal savings rate s0 corresponding to an elasticity of substitution equal to or higher than 1
(in percent;
= 1=3; gL = 0:02; gK = 0:005; n = 0:01).
1
1.1
1.2
1.3
1.4
1.5
36.1
31.0
27.1
24.1
21.7
48.8
41.1
35.4
31.0
27.6
68.4
56.2
47.4
40.8
35.8
>100
82.1
67.0
56.2
48.1
>100
>100
>100
86.1
70.2
n.a.
n.a.
>100
>100
>100
i
0.03
0.035
0.04
0.045
0.05
A situation where > 1 is de…nitely not compatible with competitive
equilibrium. We may even add that good old case = 1 is highly suspicious,
because the very high implied net savings or investment rates in Table 1,
column 1, ranging from 22% to 36%, were never observed anywhere at any
time (it may be useful to remember here that i represents real interest rates).
Now contrast this inordinate behavior of the economy with a situation
of competitive equilibrium when is in the range most frequently observed
(0.5 to 0.8). From (18) we …rst observe that the optimal savings rate is a
decreasing function of time, a most natural behavior in an economy exhibiting
capital- and labor-augmenting technical progress. Furthermore, as shown in
Table 2, the initial optimal savings rates s0 now take very reasonable values;
for instance when i = 0:04, s0 is between 8.0 and 16.4 percent.
10
Table 2. The reasonable initial optimal savings rates s0
corresponding to values of the elasticity of substitution in the
most observed range: 0:5 < < 0:8
( in percent; = 1=3; gL = 0:02; gK = 0:005; n = 0:01)
0.5
0.55
0.6
0.65
0.7
0.75
0.8
9.3
8.6
8.0
7.6
7.2
10.6
9.7
9.0
8.5
8.0
12.1
11.0
10.2
9.5
8.9
13.8
12.5
11.5
10.6
9.9
15.7
14.2
12.9
11.9
11.1
18.0
16.1
14.5
13.3
12.3
20.6
18.3
16.4
14.9
13.8
i
0.03
0.035
0.04
0.045
0.05
Over time these rates will decrease slowly. Table 3 gives those optimal
savings rates in 30 years.
Table 3. The optimal savings rates in 30 years s30 corresponding
to values of the elasticity of substitution in the most observed
range: 0:5 < < 0:8
( in percent; = 1=3; gL = 0:02; gK = 0:005; n = 0:01)
0.5
0.55
0.6
0.65
0.7
0.75
0.8
8.6
8.0
7.4
7.0
6.7
9.7
8.9
8.3
7.8
7.4
11.0
10.0
9.3
8.7
8.1
12.5
11.3
10.4
9.6
9.0
14.1
12.7
11.6
10.7
9.9
16.0
14.3
13.0
11.9
11.0
18.2
16.1
14.5
13.2
12.1
i
0.03
0.035
0.04
0.045
0.05
Let us now turn to the growth rates implied by those optimal savings
rates. From (17), it can be seen that y_ t =yt is a decreasing function of time,
with a limiting value equal to limt!1 y_ t =yt = gL . This last property may
come as a surprise, because a widely held belief is that the growth rate of
11
income per person will converge toward gL only if technical progress is laboraugmenting (with the exception of the Cobb-Douglas case). The reason why
it can also apply if progress is capital-augmenting with the CES function
rests upon a property of general means; on this see La Grandville (2011).
Table 4 shows the initial optimal growth rates y_ 0 =y0 : Values for higher t
would decrease very slowly toward gL = 2%:
Table 4. The optimal growth rates of income per person y_ 0 =y0
(in percent, = 1=3; gL = 0:02; gK = 0:005; n = 0:01).
0.5
0.55
0.6
0.65
0.7
0.75
0.8
2.03
2.03
2.03
2.03
2.04
2.03
2.04
2.04
2.05
2.05
2.04
2.05
2.05
2.06
2.06
2.05
2.06
2.06
2.07
2.07
2.06
2.07
2.07
2.08
2.08
2.08
2.09
2.09
2.09
2.10
2.10
2.11
2.11
2.12
2.12
i
0.03
0.035
0.04
0.045
0.05
A striking feature is how little dependent on the rate of interest those numbers
are, and how close they are to these observed in the U.S., where the longterm growth rate of real income per person between 1919 and 2008 has been
2.1% (see Johnston and Williamson, 2009). Note …nally that the growth
rates implied by the Cobb-Douglas case, deduced from (17), would be the
constant gL + 1 gK = 2:25; a number signi…cantly higher than those above
and those observed.
4. The threshold value of precluding competitive equilibrium
at any time whatever the types of technological progress prevailing
in the economy.
It is important to realize that there always exists a threshold value of
, denoted , precluding competitive equilibrium at any time, including of
course the initial time t = 0; whatever the types of technological progress
prevailing in the economy. We will now prove the following theorem.
12
Theorem 2. Suppose an economy is driven by a constant elasticity of
substitution production function exhibiting any combination of factor-enhancing
functions. The economy will not be able to be in a situation of competitive
equilibrium at any time if its elasticity of substitution is equal or larger than
= 1=(1 ln = ln i):
Proof.
Suppose …rst that technical progress is labor-augmenting only. If
from (11) the optimal time path of capital is given by
Kt =
1
1=p
Lt GL (t)
; p 6= 0;
[i
1]1=p
1
6= 1 :
6= 1,
(20)
We observe that the nature of the labor-enhancing function will have no
bearing on the existence of a solution. A necessary and su¢ cient condition
for such existence is
i 1
1 > 0;
(21)
or equivalently
<
=
1
1
:
ln = ln i
(22)
This threshold is the same as that preventing competitive equilibrium at any
time if there was capital-augmenting technical progress, jointly with laboraugmenting progress or not. Indeed, equations (21) and (22) for the existence
of a solution apply already at time t = 0 because GK (0) = 1 . Hence, for
any combination of factor enhancing progress functions, a value of equal
to or larger than = 1=(1 ln = ln i) will preclude at any time competitive
equilibrium.
The threshold depends on and i only. In our case ( = 1=3 and
i = 0:04); = 1:52 (rounded). The following table gives for various values
of and i:
13
Table 5. Value of the elasticity of substitution
impairing competitive equilibrium at any time, whatever the
types of technological progress prevailing in the economy.
(gL = 0:02; gK = 0:005; n = 0:01)
i
0.25
0.3
0.¯3
0.03
0.035
0.04
0.045
0.05
1.65
1.52
1.46
1.71
1.56
1.49
1.76
1.60
1.52
1.81
1.63
1.55
1.86
1.67
1.58
4. Explaining the incompatibility of competitive equilibrium
and high values of :
In the preceding Sections, we have purely relied on the mathematical properties of system of equations (1); (3) and its solution (11) to demonstrate
a) the incompatibility of competitive equilibrium and > 1 when technical progress is capital-augmenting (Section 3), and
b) the existence of a threshold value of the elasticity of substitution
precluding competitive equilibrium whatever the types and combination of
technological progress prevailing in the economy (Section 4). We now want
to explain these results.
To do so, it will be helpful to examine the structure of the production
function from a geometric point of view. It will be key to the economic
explanation of the above results.
First denote the augmented variables GK (t)K(t) U (t) and GL (t)L(t)
V (t). Since K and L are index numbers and GK and GK are pure numbers,
U and V are pure numbers as well. The production function then is a mean
of order p of Ut and Vt ; written Mp (Ut ; Vt ) :
Yt = F (Kt GK (t); Lt GL (t)) = Mp (Ut ; Vt ) = [ Utp + (1
)Vtp ]1=p ; p 6= 0: (23)
As
increases from 0 to 1 (equivalently, when p; equal to 1 1= ;
increases from 1 to 1) the mean Mp (Ut ; Vt ); a concave surface in (M; U; V )
space, opens up around the ray from the origin M = U = V: This is just
14
the geometric interpretation of the property of general means as increasing
functions of their order, and of the fact that, independently of its order, the
mean is equal to any of the numbers U or V when the latter are equal. When
the surface opens up around that ray, it does so from the corner of a pyramid
M 1 = min(U; V ); to the plane from the origin M1 (U; V ) = U + (1
)V .
Figures 1a to 1c describe this opening up, for four succesive values of : 0; 1; 2
and 1:
Y
1
U
1
2
1
0
V
Figure 1a. The case = 0; p = 1; the corner of a pyramid
Y = M 1 = min(U; V ):
Y
1
U
1
2
1
0
Figure 1b. The case
V
= 1; p = 0; Y = M0 = U 1=3 V 2=3 :
15
Y
1
U
1
2
1
0
Figure 1c. The case
V
= 2; p = 1=2; Y = M1=2 = [(1=3)U 1=2 + (2=3)V 1=2 ]2 :
Y
1
U
1
2
1
0
Figure 1d. The case
V
= 1; p = 1; Y = M1 = (1=3)U + (2=3)V:
Consider now the all-important –for our purposes –behavior of the marginal productivity of capital at any time t; denoted FK (K; ) when increases. This behavior is fundamentally di¤erent whether U is smaller or
larger than V:
1. Case U < V ; two possibilities must be considered: i) if
1; an
increasing generates a decrease in FK ; ii) if < 1, the sign of the change
in FK will depend upon K and :
16
2. Case U > V ; the marginal productivity of capital always increases,
whatever the values of K and ; formally, if U > V , FK; (K; ) > 0:
The latter is exactly our case. Indeed, it can easily be shown (see Appendix) that at all times competitive equilibrium entails U > V ; therefore the
elasticity of substitution in that case will be an extremely e¢ cient parameter
for two reasons: …rst, it increases production at any point (U; V ); second, it
increases the marginal productivity of capital.
But this is precisely where may become too e¢ cient. When > 1,
already at the initial time a competitive equilibrium may fail to exist, because
the marginal productivity of capital is always larger than =( 1) –this is the
lower bound of FK : For a solution of equation FK = i to exist at the initial
time t = 0, that lower bound must be smaller than i and hence must
be smaller than = 1=(1 ln = ln i) (our equation (22)). And of course, if
1 < < , when time increases, not only the surface increases, but so does its
slope. The excessive e¢ ciency thus generated through time forces capital to
increase very quickly (through exceedingly high savings rates) to levels where
its marginal productivity returns to the level of the rate of interest. This
explains the explosive behavior of an economy in competitive equilibrium
whenever > 1: On the contrary, if < 1; at any time t limK!1 @F=@K = 0,
and therefore the model always yields a solution (and a reasonable one, as
we have seen).
Figure 2a illustrates the case of competitive equilibrium, with = 0:5.
Points A and B represent equilibria at time 0 and after 30 years respectively;
the other parameters are those we used before: = 1=3; gL = 0:02; gK =
0:005; n = 0:01:The optimal trajectory linking A to B generates to an income
per person growing from 1.327 to 2.433, i.e. at a rate of 2.02%; as we have
seen earlier from (17), that rate will slowly decrease toward gL = 2%: The
optimal savings and investment rates are 8% at point A and 7.4% at B
(Tables 2 and 3).
On the other hand, Figure 2b corresponds to the case = 2; where no
equilibrium is possible even at the initial time 0. The lower bound of the
marginal productivity of capital is limK!1 @F=@K = =( 1) = (1=3)2 =
0:1; i.e. nearly 3 times more than the rate of interest i = 0:04:
17
4
Y
3,5
B
*
Equilibrium point
at time t = 30
3
2,5
F(K) at time t = 30
2
1,5
A
*
Equilibrium point a time t = 0
1
F(K) at time t = 0
0,5
0
0
2
4
6
8
10
K
Fig.2a. Case = 0:5: Competitive equilibrium points A and B are linked
by an optimal trajectory with an average growth rate of income per person
of 2.02%; the optimal savings rate is 8% at A; it decreases to 7.4% at B.
Y
5
4,5
4
3,5
3
F(K) at time t = 30
2,5
2
Slope of this straight line:
lower bound of F'(K) = 0.11
1,5
1
Slope of this line:
i = 0.04
F(K) at time 0
0,5
0
0
2
4
6
8
10 K
Fig. 2b. Case = 2: No competitive equilibrium is possible; already
at time t = 0; the lowest possible value of F 0 (K); limK!1 F 0 (K) = 0:11, is
much higher than i = 0:04:
18
5. Conclusions.
The concept of elasticity of substitution should be handled with care. We
hope to have …rmly established the theoretical reason backing the fact that
has been time and again observed as smaller than 1. Furthermore, the
case = 1 has been shown to entail exceedingly high, unobserved, savings
values. Despite being constrained by the fact that it has to be smaller than 1
to ensure competitive equilibrium, remains a marker as well as a powerful
engine of growth. For instance, the integral of future discounted consumption
‡ows is signi…cantly more sensitive to a relative increase in than to a similar
change in gK , as could be easily shown.
Finally, we should stress two important results. First, if competitive equilibrium is established, society will reap all three bene…ts we have described:
at any point in time, the minimisation of its production costs; the maximisation of the value of its activity; and, over a time interval of in…nite length,
the sum of all discounted consumption ‡ows society will receive.
Second, the share of labor will permanently increase over time; it will do
so at a slow pace, but it will do so. For instance, if today = 0:8 and if the
other parameters of the economy we used in this paper apply, the share of
labor will increase from 78.2% to 78.8% in 30 years and 79.3% in 50 years.
Consequently, if the model we have used is a fair description of the production
process, an unescapable conclusion is that the increase of the capital share we
observe today does not …t well with a situation of competitive equilibrium.
This implies that society does not reap the bene…ts it could have otherwise
secured not only for present, but also for future generations.
Appendix.
We need to prove that U > V , for any > 0: If = 1, the proof is
immediate from (12). Consider now
6= 1; p 6= 0: From (11) we have
U = G t Kt =
1
1
[i
1=p
[i
1=p
1
V
GK (t)1
19
1
Lt GL (t)
GK (t)1
1]1=p
; p 6= 0;
1]1=p
=
6= 1 ;
(24)
and therefore, after simpli…cations,
U=V =
( =i)1
1
GK (t)1
1=p
(25)
Two cases must now be dealt with:
1
a) < 1; p < 0 : in the denominator of (24), the term ( =i) GK (t)1
is larger than 1; therefore the number in brackets is smaller than 1; since
1=p < 0, U=V > 1:
1
b) > 1; p > 0 : the term ( =i) GK (t)1 is now smaller than 1; the
number in brackets is larger than 1; since 1=p > 0, U=V > 1 as well.
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