PRODUCTION FUNCTION AND NEUTRAL TECHNICAL
PROGRESS IN HICKS’S SENSE (*)
by Ken-ichi Inada (Tokyo Metropolitan University
and The University of New South Wales)
T
Professor Swan [ z ] has neatly proved that for a Golden Age Path
to exist in the Solow-Swan growth model for every value of the saving
ratio in a certain open interval it is necessary that technical progress
be neutral in Harrod’s sense. We easily see from this that if a Golden
Age Path must exist for every value of the saving ratioinanopen
interval, technical progress cannot be neutral in Hicks’s sense except
in the case where Hicks neutrality reduces to Harrod neutrality, i. e.,
in the case where the production function is Cobb-Douglas. This
result is also obtained by Professor Kurz [I! independently from Professor Swan.
As Professor Swan himself admits, technical progress of notHarrod neutral type may be compatible with a Golden Age Path
for some particular value of s . If we consider a general class of types
of technical progress, we can discover in it some non-Harrod neutral
technical progress which is compatible with a Golden Age Path for
a particular value of s . But if we are concerned only with neutral
technical progress in Hicks’s sense, is the statement still true? The
purpose of this note is to study this problem. That is, j f we assume
that a Golden Age Path exists under Hicks neutral technical progress
for some particular value of s , what can we say about the shape of
the production function and the nature of the dynamic path? We
can show that the production function must be Cobb-Douglas in a
domain such that any dynamic path eventually locates. That is,
Professor Kurz’s results still hold in the eventually relevant region.
Moreover, any dynamic path approaches asymptotically the Golden
Age Patli.
2.
Consider the following Solow-Swan-Kurz model.
Ii ( t ) = s Y ( t ) ,
Y ( t ) = ehfF ( K ( t ) , N ( t ) ) ,
(A > 0) ,
N ( t ) = 1%’ (0)ent .
(*) I am obliged to Professor M. C. Kemp for his valuable suggestions.
Neutral technical progress in Hicks’s sense was originally defined as a
once for all technical progress. It is expressed by a shift parameter which locates
(I)
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63 -
We follow Professor Kurds notation and assumptions, That is,
the function F is continuous, twice differentiable, homogeneous of
degree one, concave, and
F ( K (t) , o) = o and I; ( K ( t ) ,
03)
= 03,
a t the front of the production function. That is, the shift of the value of a parameter p in the following expression indicates neutral technical progress in
Hicks’s sense. (See [4]).
Y = p F ( K , N).
But, when we incorporate this technical progress in the growth model, we
have to assume some time shape of the shift of the parameter. Let this time
shape be p ( t ) . Then we get
y ( t ) = I“ 0 ) F ( K ( t ) N 0 ) ).
Under this general time shape of a shift parameter, the neutral technical
progress in Hicks’s sense is compatible with a Golden Age Path for a non-CobbDouglas production function and a particular value of the saving ratio, provided
the time shape is in harmony with them. The time shape in the text has the
special feature that the value of the parameter which expresses the extent of the
technical progress shifts a t a constant rate. Our arguments in the following
are only concerned with this case. Strictly speaking, therefore, we should call
the technical progress assumed here the constant rate neutral technical progress
in Hicks’s sense. This is too long, and other economists, possibly except Professor Swan, have discussed or studied only this case. We therefore follow them,
and skip the word ’constant rate’ in the following.
Professor Solow [ z ]also studied the case of Hicks neutral technical progress.
He, however, only studied the Cobb-Douglas case. In this case the Golden Age
Path exists and any dynamic path approaches it. If he had studied the general
case, he would find that the Golden Age Path does not necessarily exist. For
example, i f instead of Cobb-Douglas function, he employed the production
function (which is employed by himself in the same paper for another occasion)
F ( K , N)= ( a KP
NP)1/,, ,
he would discover that the result obtained for his model without technical progress cannot be extended to his model with Hicks neutral technical progress.
This fact shows the danger of specifying the production function t o some special
form - say Cobb-Douglas function. The result obtained is not necessarily
valid for the cases of general production functions. This fact is also an alarm
against the mode which many growth theories assume the Cobb-Douglas production function.
We can incorporate in our model neutral technical progress in Harrod’s
sense also. That is,
Y ( t ) = ell F ( K ( t ) , ev! N ( t ) ) .
But if we define
evf N ( t ) = Lq( t ) = N (0) eb+v)t ,
the following arguments hold completely. So, our model is viewed as incorporating both Harrod and Hicks neutral technical progress. Our model also covers
the following model which only seems to be more general
Y ( t ) = eh* F (ept K ( t ) , evt N ( t ) ) .
For,
Y ( t ) = eO.+N F ( K ( t ) , e(V-p)t N ( t ) )
= eh’t F ( K ( t ) , eb’t l\r ( 1 ) ) .
Here,
h‘ E h
I” and V - p
V’
(z) From concavity, and F ( K , 0) = o and F ( K , co) = co, we can derive
7
+
+
-
64 -
Our system reduces to a single differential equation
K ( t ) = seLt F ( K ( t ) , N (0)en+) .
Suppose that a Golden Age Path exists for a particular value of s .
Let it be written
K ( t ) = K ( 0 ) e g t , (0 6 t < co) .
Here, both
(0) and g may depend on the fixed value of s , and
(0)and g are constant
N (0). But once s and N (0)are fixed, both
over time. To show explicitly that K ( 0 ) and g may depend on s ,
we write
(0 ; s) and g (s) . Then
g (s)
(0 ;
s)
eg(s)t
= selt F
(E ( 0 ; s)
egcS)*
,N
(0)ent)
.
From the first degree homogeneity of F ,
g (s) = se” F
(1)
seSt
f
x represents the labour-capital ratio. Now relation (I) can be
written as
g (s) = s exr f ( x )
(3)
Write
(4)
A
R
-E
(4 - n
.
p (s).
dF
> 0 . Professor Kurz assumes only the concavity, instead of strict
dN
concavity. Thus, as is seen later the following function is possible
F(K,L)=uL.
Correctly speaking, his assumption is slightly different from Professor
dF(K, N)
--> 0 , but the forSwan’s. For example, the latter assumes dK
mer not.
that
~
Then, from
(2) and
(4)
Then from ( 3 )
Here
Relation (5) shows that the production function is Cobb-Douglas,
provided o < f3 (s) 5 I and x covers from zero to infinity. For, since
f (x)= A (s) x I j @ ) , F ( I < , N ) must take the form
F ( K , N ) = A (s) K*--F(S)I L ' P t S , ,
But, in our case, x docs not cover from zero to infinity. On the
other hand, we can show that o < f3 ( s ) 6 I .3 First we shall show this.
; g (s) . Consider relation (I). The left
Suppose F ( s )< o . Then 'y2 >
hand side is constant. The right hand side increases to infinity. For,
N (0)
e(n-p(s))t
also increases to
eh* increases to infinity, and j
2 (0; s)
infinity when t t m c t s to infinity. We have rcached a contradiction.
Suppose p (s) = ( I . Thcn A - o . We howevcr assumed A > 0 . This
case therefore ncver arises. Suppose p (s) > z . We assumed that
F (R, N ) is c o n a v e . This contmdict the assumption that p (s) > I .
The only remaining possibility is t h a t o < f. (s) 5 I .
Sow, we study the region co\.ercd by x . As shown above,
co, x moves
I.: (s) . 0 , or g (s) =. n . When t moves from o to
n; ( 0 )
from a certain positive \ d u e ___- to zero. This is easily seen
1
i
+-
E
(3)
I f wc
:i\>iirnc
See footnote 2.
(0 ;
s)
t h e 5 t r i c t t o n r n m t \ . o f tlir production lunction, !3# I .
- 66
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from relation ( 2 ) . On a Golden Age Path, the value of x (labour-capital
ratio) constantly decreases and thus x never becomes bigger than its
N (0)
initial value ___. That is, the region covered by x is 0 < x 5
K ( o ; s)
5
(O)
-
.
We therefore could show that I; ( K , N) is a Cobb-
K ( 0 ; s)
Douglas function in the domain of ( K , N ) such that
N
-
N
(0)
K ' K ( o ; s)
F ( K , N ) , however, can take any other functional form in the
domain of ( K , N) such that
N
->
K
N
K
(0)
(0;
s)
Each domain is separated by a half line which goes through the
origin and the slope of which depends on the value of s . The shaded
area in Fig. I shows the domain in which the production function is
Cobb-Douglas. For simplicity, we call this area a Cobb-Douglas
domain in what follows.
K
0
Fig. 1.
I n the above, we assumed the existence of a Golden Age Path
extending from the present (t = 0 ) to the infinitely distant future
(t =
co) . If, instead, we assume the existence r.f a Golden -4ge
Path extending from the infinitely distant past (t = - co) to the ina),
we can show that the production
finitely distant future (t =
function is Cobb-Douglas for any ( K , N ) > o . In this case, if t moves
+
+
from - cc to 1- C C , then the valutb of A , tlefiiicd in
+ ac: t o zero. Thus,
i (,I) == ,i (s)
for ( t ... ,v L TC .
( 2 ) , moves
from
.vlji*)
This s h o w that 1; ( K , L ) is a Col>b-l)ouglas fuiiction for any
( K , A')
;. 0 .
So\\, back to the case of a Golden Age Path extending from the
present to the infinitely distant future. We mentioned that g and
K (0)may depend on c . But 11 e will s h o ~ vthat g does not depend on s
and K ( 0 ; s) is incrclasing in s .
Suppose that there exist two (;olden Age l'aths, which correspond,
respecti\-ely, to two x.alus of s , say s 1 and s2 , with s1 ;s z . Then, for
each value of s , wc can carry out the above reasoning and get g (sl),
g (sz) , K (o ; sl) and K (o ; s,) . Here AY(01 i.; assumed to be the same
for the two Golden Age Paths. W e obtain also two Cobb-Douglas
domains. At this stage, we don't know \vhich domain covers the
other domain. But we know one t1om;iin i\ a1n.a~-sincluded in anothrr domain. For the dividing linv hetween the Cobb-l)ouglas domain
and thc- possi1)ly notI-('obt)-I)oiih.la.; clomaiii is a straight half line
which goes through the origin, and thus one dividing half line, corresponding to one \ d u e of s, must locntc above 01- coincide with the other
dividing half lint,. Kow the prodnction function has the form
A (s,)
'
-P
1%)
sP
(SLl
,
or
,4 (sJ
I
--ISM
,yPoa)
in each of two Cobb-huglas domains. I n thc common area of the two
domains, this is also valid. On the other hand, thc function I; ( K , N )
must be the same in the common arca. Thxt is,
for any (I<, -77) in the common area. From this,
13
(Sl) =
.4 (s,)
,
;Ind
p (Sl)
- :1, (s,)
.
This shows that A (s) and [I, (s) (lo not depend on s . Then, from (41,
R (s) does not depend on s .
Sext, from above
=
A
=
constant (indc1)cndent on tlic. \.aluc. of s) .
6s
-
(O)
I t follows that
i?
(0;
-
must be decreasing in s . This shows
s)
that the slope of the dividing line must be decreasing in s and that
K ( 0 ; s) is increasing in s .
Suppose that s is fixed again, and that a Golden Age Path exists
for this particular value of s . TVe study the dynamic features of the
path which starts from any initial point.
Fist, suppose that a dynamic path starts from a point on the
dividing line. I n this case the path is represented by
K ( t ) = I?( 0 ) ear ,
N ( t ) = N (0) ent ,
Y ( t ) = Y ( 0 ) cot.
N (4
This is a Golden Age Path and ___ monotonically decreases over
K
(t)
time. This means that the dynamic path 1ocz.tes in the Cobb-Douglas
domain permanently.
Second, suppose that the dynamic path starts from a point in
the Cobb-Douglas domain. Then as long as the path is in the domain,
it is described by the differential equation
I;; = s
Here
eki
K" ( N ( 0 )
erlt)p
.
cc G 1-;2.
Integrating this differential equation, we get
(6)
K ( t ) = (p s A ( N (o))p--
I
n$
+A
e(*p+k)t+
C
$)$ .
Here, the value of C depends on the initial value of K . I n this case
the initial point is in the Cobb-Douglas domain. Then, we easily
see that C > o . We will show that
N (t)
A 7 (0)
<for any t > 0 .
K
We calculate
in relation (6)
(0).
z
(7)
(0)
This is easily obtained if we put C
(0)
Thus,
z
(4
=
I
p s A ( N (0))Ij
I
@+h.
I-+
=0
and t
== 0
On the other hand
From (7) and (S), we see that
This shows that the dynamic Imth \ t a y i t i thc ('obb-Douglas domain,
and never goes out o f it. From (O), W Y WY* that thc. movcmcnt of I< (f)
is asymptotically clcscrilxxl by
The growth rate g is
92
+
?L
Finally, consider a dynamic 1mth starting from a point in the
possibly non-('obb-l)ouh.las domain. \Ve shall how that the path
sooner or later enters thc Cobb-Tlouglas domain and stnvs there permanently. Slippose that the p i t h Stay$ in the possibly non-Cob1I Douglas domain 1)ermanently. That is, supposc that
From this,
This shous that t h e growth mtc of K i i alwavs larger than a value
whiclI increases exponentidl>. 'I hus it will sooner or later csceed the
A- ( t )
growth rate of h' which is equal to a constant n. Then -- - - miist
h-( t )
tlccreaw to zcro evc.ntuallj7. Thi.. is a contradiction. 3 e x t we $how
that oncc t l i t b l ~ ~ t vntc*r,
li
tlic. ( ' o l ~ I ~ - l ) o i i donlain
~ l a ~ it stays there
perniancntli-. 5iilq1ow that t h c ]);it11 hits the dividing line a t a time
- 70
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point I' . Since F ( K , N) is twice differentiable, it is continuous.
This shows that the switch from the possibly non-Cobb-Uouglas
domain to the Cobb-Douglas domain is carried out smoothly. The
differential equation just after time point i' will be expressed by
Here K ( t ' ) =- K ( 0 ) enh'. 115 long as the path remains in the CobbDouglas domain, the path is subjected to this differential equation.
Solving explicitly thi5 differential equation, we get the spme form
as ( 5 ) . We can show, in a manner similar to but slightly different from
that used above, that the dynamic path never goes out of the CobbDouglas domain. Hence, the path asymptcticdy grows a t a constant
rate n
+ -h.
P
We could show that any dynamic path always enters the CobbDouglas domain. In other words, if there exists a Golden Age Path
for a particular value of thc saving ratio, the production function
must be Cobb-Douglas in a n evrxntnally relevant domain, provided
that the technical progreby is neutr,il in Hicks's sense. We have also
shown that any dynamic p t h approachcs asymptotically a Golden
Age Path. Here the value of s is fixed. But by reasoning similar to
that used above, wc can prove that if there exists a Golden Age Path
fcr a particular value of s , then any dvnamic path (even that which
corresponds to any diffcrent value of s ) asymptotically grows at a
constant rate. In particular, for a larger value of s than the particular
one, there exists a Golden -4ge Path extending from the present to
the infinitely distant future ( 4 ) .
I n the previous section, we have shown that if there exists a
Golden Age Path for a particular value of s, then all dynamic paths
eventually grow at ;I constant rate. I n this section, we shall assume
a weaker condition than that assumcd in the previous section. That is,
assume that there exists a 7 2 1 ~ sGolden
~
Age Path for a particular value
of s . Here a gztasi Golden Age Path is defined as a path such that
K ( t ) approaches asymptotically K* e g t . I n the previous section we
have shown that any dynamic path is a quasi Golden ,4ge Path if
( 4 ) \Ye have assumed that A > o . The almost same discussion as above
~ z p< 1. < o . This is not an indeveloped is carried out for the case where
teresting case, but is not unconceivable. Siippose the beginning of a glacial
period. The temperaturc tieclines constantly arid the output, especially, the
agricultural product declines constantly even with the same capital and labour
input. Golden Age during a Glacial Epoch ! Such a situation may be expressed
by a model in which A i o . B u t still I would like t o exclude this CRSC. For,
the p e r capita output declines constantly on a Golden Age Path. The situation,
I believe, is not suitable t o be called by the name.
j1 --
esists a (;olclen Age Path for ;L ~)articul:~r
yalue of s . We don't
in this section assume the existeiicc of a Golden ilge Path. Then what
can wc say about the sliape of the production function and tlic nature
of the dynamic path? By a mctliotl almost the same as used above
we can sho\v that the production fuiiction must bc asymptotically
t1lc.i-e
Or eqoivalently,
:l* xlj \\-hen x tciids to zero.
i (4
*
,,j ,I$
approaches
one wlicn x tends to zero. Moreover, any dynamic path is a qumi
Golden Xgc. Path, that is, i t ;tppro;ichr's asymptotic;Llly a path expressed b y k'* e g b . The class of as~-nii~tntic.;Lll?,
('ol)b-T)oiiglns functions is
rather \vide. But it dotxs not cover 1 1 1 ~ .! ~ i i t i rposGble
~~
class of prodtiction functions. For cx;uiii~le,tl!c folloiving frinction is not asgmptoticallv c'o 17 b- lkmgl :I 5.
./ (s)
log (I
7x) ( I
-:-
'1'
~ - =.Y
x
j' (x) '-- and
(1
--II
log
( I -I-
/''
(.T) -.:o
-\ !
-rlog
alld
lim j (.v)
.;
...
=o
xnd lixn
r
0
-
/ (.Y]
I,
Tlicse are Professor Kurz's :i.;suniptio1rs.
Piit
'111c.n
I
11
G
- and
x
Q
_= I
-7
,
--I
x.
(I
+ xj .
provided o < (3 < I .
As for the case where (3 = I ,
lim
X+O
1%
(1
+4
-I.
X
Then
and
We see that f (x)/xP does not approach a positive constant for
any p such that u < p 5 I . In this case there exists no quasi Golden
Age Path, and hence a fortiori no Golden Age Path.
4.
Both Professors Swan and Kurz assume that the production
function is always increasing in one of two variables, i. e. in Swan’s
case K and in Kurz’s case N . This assumption is vital to their results.
The following fixed coefficient production function which appears in
the literature is of course not Cobb-Douglas. But Hicks neutrality
is compatible with a Golden Age Path.
F ( K , N ) = Mia
As easily seen,
I--,
n
b
8 F ( K , N)
SF ( K , N )
or
can be zero
8K
6M
if the value of K or N is large, respectively.
5.
Professor Kurz assumes the existence of a Golden -4ge Path fGr all
o
< s < I , n > o , A > o . This is too much. We can obtain the same
result as his under the following weaker assumption. If there exists
a Golden Age Path for every o < s < E and some fixed values of n > o
and A > u , the production function is Cobb-Douglas and the growth
rate does not depend on s . Here E can be taken as any small positive
number. That is, we need not assume that both n and A take any
positive values, and need not assume that s takes any value of full
range ( u I ) .
-
70
-
As shown above, the slope of the dividing line between the CobbDouglas domain and the possibly non-Cobb-Douglas domain is
Here both A and p do not depend on s . If s tends to zero, the
slope tends to infinity, and thus the Cobb-Douglas domain covers
the entire positive orthant. This proves the results by Professor Kurz.
6.
We have been concerned with the constant saving ratio model.
Another frequently imposed assumption on the saving consumption
pattern, which is also familiar to economists, is that capitalists don’t
consume and 1aboi.trers don’t save. Assuming the ccmpetitive market,
the growth model becomes
Again we assume the existence of a Golden Age Path. Then,
we can show by almost the same reasoning as above that F ( K , N )
is a Cobb-Douglas function in the eventually relevant domain and
that any dynamic path is a q z m i Golden Age Path. As is well known,
the Golden Age Path for this saving consumption pattern has a special
feature and is called the Golden Rule Path. That is, sustainable per
capita consumption level becomes higher on the Golden Rule Path
than on any other Golden Age Path. I t is, however, noted that +er
cafiita consumption grcws at the same rate g - n on all Golden Age
Paths, i. e., it is expressed by
Y eto-n)ts
and the value of y attains maximum on the Golden Rule Path.
-
71
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REFERENCES
[I] KURZXI.: Q Substitution Versus Fixed Production Coefficients: A Com-
Econometrica, Vol. 31, 1963, 209-17.
4 A Contribution t o the Theory of Economic Growth H,
Quarterly Journal of Economics, Vol. 70, 1956, 65-94.
[3] SWANT. W.: (( On Golden Ages and Production Functions o, in Kenneth
Berrill (ed.) Economic Develo9ment with S$ecial Referelzce to East A s i a ,
Chapter I , London, MacMillan and Co., 1964.
[4] UZAWA
H.: (( h-eutral Inventions and the Stability of Growth Equilibrium ,)b
Review of Economic Studies, Vol. 2 8 , 1961, 117-24.
ment
)),
[z] SOLOWR . M.:
Prof. MANLIO REST& Direttore responsabile
Registrato il 2 luglio a1 n. 67 del Registro del Tribunale di Trieste
Copyright Editore-Tipografo L. Cappolli. Trieste
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