Barelli, Paulo

Pessoa, Samuel de Abreu

English

Research Papers in Economics

Ensaios Econ￿micos
Escola de
P￿s-Gradua￿ªo
em Economia
da Funda￿ªo
Getulio Vargas
N 477 ISSN 0104-8910
Inada conditions imply that production func-
tion must be asymptotically Cobb-Douglas
Paulo Barelli, Samuel de Abreu Pessoa
Mar￿o de 2003
URL: http://hdl.handle.net/10438/1012Os artigos publicados sªo de inteira responsabilidade de seus autores. As
opiniıes neles emitidas nªo exprimem, necessariamente, o ponto de vista da
Funda￿ªo Getulio Vargas.
ESCOLA DE P￿S-GRADUA˙ˆO EM ECONOMIA
Diretor Geral: Renato Fragelli Cardoso
Diretor de Ensino: Luis Henrique Bertolino Braido
Diretor de Pesquisa: Joªo Victor Issler
Diretor de Publica￿ıes Cient￿￿cas: Ricardo de Oliveira Cavalcanti
Barelli, Paulo
Inada conditions imply that production function
must be asymptotically Cobb-Douglas/ Paulo Barelli,
Samuel de Abreu Pessoa ￿ Rio de Janeiro : FGV,EPGE, 2010
(Ensaios Econ￿micos; 477)
Inclui bibliografia.
CDD-330Inada Conditions Imply that Production Function must be
Asymptotically Cobb-Douglas
Paulo Barelli∗ Samuel de Abreu Pessôa†
February, 2003
Abstract
We show that every twice-continuously diﬀerentiable and strictly concave function f : R+ → R+
can be bracketed between two C.E.S. functions at each open interval. In particular, for the Inada
conditions to hold, a production function must be asymptotically Cobb-Douglas.
JEL Classiﬁcation: E13, E23
Key Words: Inada Condition, Production Function, Elasticity of Substitution
1I n t r o d u c t i o n
The celebrated Inada conditions, that a (per-capita) production function f : R+ → R+ should
satisfy
f(0) = 0, f0(0) = ∞, f0(∞)=0 , and f(∞)=∞ (1)
on top of being strictly increasing (f0(k) > 0) and strictly concave (f00(k) < 0) for all k ∈ R+,
are widely used in the applied literature. In 1963 Inada noticed that those conditions had been
implicitly used by Usawa in his series of two-sector growth models, and that those conditions were
suﬃcient to ensure existence of equilibria. In addition, those conditions are intuitively very plausible
and easily justiﬁed. It is then not surprising that the assumptions (1) have not yet been subject
to a more thorough investigation. In this note we show that they impose strong restrictions on the
∗Department of Economics, Columbia University, 420 W 118th Street, New York, NY 10027, USA. Email Address:
pb230@columbia.edu
†Graduate School of Economics (EPGE), Fundação Getulio Vargas, Praia de Botafogo 190, 1125, Rio de Janeiro,
RJ, 22253-900, Brazil. Fax number: (+) 55-21-2553-8821. Email address: pessoa@fgv.br
1asymptotic behavior of the elasticity of substitution between capital and labor. In particular, for
(1) to hold we must have that the production function is asymptotically Cobb-Douglas (that is, its
elasticity of substitution is asymptotically equal to one), as k approaches either zero or inﬁnity.
2T h e R e s u l t
We assume that f ∈ C2(R+) is increasing and strictly concave. The elasticity of substitution
between capital and labor is given by
σ (k) ≡−
f0 (k)[f (k) − kf0 (k)]
kf (k)f00 (k)
≥ 0, (2)
and is assumed bounded and continuous.
In order to prove Proposition 1 below, we will make use of two Lemmas, which show that
any production function can be approximated both from above and from below by suitable C.E.S.
functions. Let σ² ≡ σ (0) − ², σ² ≡ σ(0) + ² and deﬁne
h(k) ≡
f (k) − kf0 (k)
f0 (k)
, Cσ
k ≡
Ã
h(k)+k
h(k)+k
1
σ
! σ
σ−1
and ασ
k ≡
h(k)
h(k)+k
1
σ
.
Lemma 1 For every ²>0 there exists k>0 such that
f (k)
Cσe
k
³
1 − ασe
k + ασe
k x
σe−1
σe
´ σe
σe−1 ≤ f (x) ≤
f (k)
Cσ²
k
³
1 − ασ²
k + ασ²
k x
σ²−1
σ²
´ σ²
σ²−1
for all x ∈ [0,k].
Proof. Continuity and boundedness of σ(k) assure that for any ²>0 there exists k>0 such
that
σ² ≤ σ (x) ≤ σ² for all x ∈ [0,k].
From the deﬁnition of h above we have
dh(x)
h(x) = 1
σ(x)
dx
x .H e n c e
1
σ²
dx
x
≤
dh(x)
h(x)
≤
1
σ²
dx
x
for all x ∈ [0,k].
2Integrating and substituting for h we get:
h(k)
³x
k
´ 1
σ² + x ≤
f (x)
f0 (x)
≤ h(k)
³x
k
´ 1
σ²
+ x for all x ∈ [0,k], (3)
where the inequality follows from h(k) > 0,w h i c hc o m e sf r o mf being concave. Integrating again:
f (k)
µ
h(k)
k
1
σ²
+ x
σ²−1
σ²
¶ σ²
σ²−1
µ
h(k)
k
1
σ²
+ k
σ²−1
σ²
¶ σ²
σ²−1
≤ f (x) ≤ f (k)
µ
h(k)
k
1
σ² + x
σ²−1
σ²
¶ σ²
σ²−1
µ
h(k)
k
1
σ² + k
σ²−1
σ²
¶ σ²
σ²−1
for all x ∈ [0,k].
Now use the deﬁnitions of Cσ
k and ασ
k and we are done.
Lemma 2 If σ(∞) is well deﬁned, then for every ²>0 there exists k>0 such that
f (k)
Cσe
k
³
1 − ασe
k + ασe
k x
σe−1
σe
´ σe
σe−1 ≤ f (x) ≤
f (k)
Cσ²
k
³
1 − ασ²
k + ασ²
k x
σ²−1
σ²
´ σ²
σ²−1
for all x ∈ [k,∞),w h e r en o wσ² ≡ σ (∞) − ², σ² ≡ σ(∞)+².
Proof. Just repeat the proof of Lemma 1 with proper adjustments.
We are now in position to state
Proposition 1 1) If σ(0) < 1 then f (0) = 0, f0 (0) < ∞ and limk&0
kf0(k)
f(k) =1 ;
2) If σ (0) > 1,t h e nf (0) > 0, f0 (0) = ∞ and limk&0
kf0(k)
f(k) =0 ;
3) If σ (∞) < 1,t h e nf (∞) < ∞, f0 (∞)=0 ,a n dlimk%∞
kf0(k)
f(k) =0 ;
4) If σ (∞) > 1,t h e nf (∞)=∞, f0 (∞) > 0,a n dlimk%∞
kf0(k)
f(k) =1 .
Proof. We can set ² i nL e m m a s1a n d2s u c ht h a t :
1) σ(0) < 1 ⇒ σ² < 1;
2) σ(0) > 1 ⇒ σ² > 1;
3) σ(∞) < 1 ⇒ σ² < 1;
4) σ(∞) > 1 ⇒ σ² > 1.
Taking respectively the limit for x & 0 in Lemma 1 and for x % ∞ in Lemma 2 the results for
f (0) and f (∞) follow from the asymptotic properties of the C.E.S. function.
To derive the results for
kf0(k)
f(k) we divide (3) (and the equivalent expression from the proof of
Lemma 2) by x and take the limit. Likewise, to get the results for f0 (0) and f0 (∞) we divide the
i n e q u a l i t i e si nL e m m a s1a n d2b yx, and then take the limit.
3In particular, if we force f (0) = 0 and f0 (0) = ∞,t h e nσ (0) must be equal to 1. Since the
Cobb-Douglas functional form is the one that has σ(k) constant and equal to 1, we conclude that
the Inada conditions force the production function to be asymptotically Cobb-Douglas. Whenever
evidence points out that the Cobb-Douglas functional form is not appropriate for some application,
we are forced to give up the full force of the Inada conditions, and perhaps some of its implications.
References
[1] Inada, Ken-Ichi 1963. “On a Two-Sector Model of Economic Growth: Comments and a
Generalization,” T h eR e v i e wo fE c o n o m i cS t u d i e s ,30(2): 119-127.
4

On a Two-Sector Model of Economic Growth: Comments and a Generalization,” T h eR e v i e wo fE c o n o m i cS t u d i e s

Inada conditions imply that production function must be asymptotically Cobb-Douglas

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Inada conditions imply that production function must be asymptotically Cobb-Douglas

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