Inada conditions explained
In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada,[1] are assumptions about the shape of a function, usually applied to a production function or a utility function. When the production function of a neoclassical growth model satisfies the Inada conditions, then it guarantees the stability of an economic growth path. The conditions as such had been introduced by Hirofumi Uzawa.[2]
Statement
Given a continuously differentiable function
, where
X=\left\{x\colonx\in
\right\}
and
Y=\left\{y\colony\inR+\right\}
, the conditions are:
- the value of the function
at
is 0:
- the function is concave on
, i.e. the
Hessian matrix Hi,j=\left(
| \partial2f |
| \partialxi\partialxj |
\right)
needs to be
negative-semidefinite.
[3] Economically this implies that the
marginal returns for input
are positive, i.e.
\partialf(x)/\partialxi>0
, but decreasing, i.e.
\partial2f(x)/\partial
<0
- the limit of the first derivative is positive infinity as
approaches 0:
\partialf(x)/\partialxi=+infty
, meaning that the effect of the first unit of input
has the largest effect
- the limit of the first derivative is zero as
approaches positive infinity:
\partialf(x)/\partialxi=0
, meaning that the effect of one additional unit of input
is 0 when approaching the use of infinite units of
Consequences
The elasticity of substitution between goods is defined for the production function
as
\sigmaij=
| \partiallog(xi/xj) |
| \partiallogMRTSji |
, where
MRTSji(\bar{z})=
| \partialf(\bar{z |
| )/\partial |
zj}{\partialf(\bar{z})/\partialzi}
is the
marginal rate of technical substitution.It can be shown that the Inada conditions imply that the
elasticity of substitution between components is asymptotically equal to one (although the production function is
not necessarily asymptotically
Cobb–Douglas, a commonplace production function for which this condition holds).
[4] [5] In stochastic
neoclassical growth model, if the production function does not satisfy the Inada condition at zero, any feasible path converges to zero with probability one provided that the shocks are sufficiently volatile.
[6] Further reading
- Book: Robert J. . Barro . Robert J. Barro . Xavier . Xavier Sala-i-Martin . Sala-i-Martin . [{{Google books |plainurl=yes |id=jD3ASoSQJ-AC |page=26 }} Economic Growth ]. London . MIT Press . 2004 . Second . 0-262-02553-1 . 26–30 .
- Book: Gandolfo, Giancarlo . 1996 . [{{Google books |plainurl=yes |id=ouC6AAAAIAAJ |page=176 }} Economic Dynamics ]. Third . Berlin . Springer . 176–178 . 3-540-60988-1 .
- Book: Romer, David . David Romer
. David Romer . The Solow Growth Model . Advanced Macroeconomics . Fourth . New York . McGraw-Hill . 2011 . 6–48 . 978-0-07-351137-5 .
Notes and References
- Inada . Ken-Ichi . 1963 . On a Two-Sector Model of Economic Growth: Comments and a Generalization . . 30 . 2 . 119–127 . 10.2307/2295809 . 2295809 .
- Uzawa . H. . On a Two-Sector Model of Economic Growth II . The Review of Economic Studies . 30 . 2 . 1963 . 105–118 . 2295808 . 10.2307/2295808 .
- Book: Takayama, Akira . Mathematical Economics . New York . Cambridge University Press . 2nd . 1985 . 0-521-31498-4 . 125–126 . registration .
- Barelli . Paulo . Samuel de Abreu . Pessoa . 2003 . Inada Conditions Imply That Production Function Must Be Asymptotically Cobb–Douglas . . 81 . 3 . 361–363 . 10.1016/S0165-1765(03)00218-0 . 10438/1012 . free .
- Litina . Anastasia . Theodore . Palivos . 2008 . Do Inada conditions imply that production function must be asymptotically Cobb–Douglas? A comment . . 99 . 3 . 498–499 . 10.1016/j.econlet.2007.09.035 .
- Takashi . Kamihigashi . 2006 . Almost sure convergence to zero in stochastic growth models . . 29 . 1 . 231–237 . 10.1007/s00199-005-0006-1 . 30466341 .
