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]
A Cobb-Douglas -type function satisfies the Inada conditions when used as a utility or production function.
Given a continuously differentiable function
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
, where
X
=
{
x
:
x
∈
R
+
n
}
{\displaystyle X=\left\{x\colon \,x\in \mathbb {R} _{+}^{n}\right\}}
and
Y
=
{
y
:
y
∈
R
+
}
{\displaystyle Y=\left\{y\colon \,y\in \mathbb {R} _{+}\right\}}
, the conditions are:
the value of the function
f
(
x
)
{\displaystyle f(\mathbf {x} )}
at
x
=
0
{\displaystyle \mathbf {x} =\mathbf {0} }
is 0:
f
(
0
)
=
0
{\displaystyle f(\mathbf {0} )=0}
the function is concave on
X
{\displaystyle X}
, i.e. the Hessian matrix
H
i
,
j
=
(
∂
2
f
∂
x
i
∂
x
j
)
{\displaystyle \mathbf {H} _{i,j}=\left({\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}\right)}
needs to be negative-semidefinite .[3] Economically this implies that the marginal returns for input
x
i
{\displaystyle x_{i}}
are positive, i.e.
∂
f
(
x
)
/
∂
x
i
>
0
{\displaystyle \partial f(\mathbf {x} )/\partial x_{i}>0}
, but decreasing, i.e.
∂
2
f
(
x
)
/
∂
x
i
2
<
0
{\displaystyle \partial ^{2}f(\mathbf {x} )/\partial x_{i}^{2}<0}
the limit of the first derivative is positive infinity as
x
i
{\displaystyle x_{i}}
approaches 0:
lim
x
i
→
0
∂
f
(
x
)
/
∂
x
i
=
+
∞
{\displaystyle \lim _{x_{i}\to 0}\partial f(\mathbf {x} )/\partial x_{i}=+\infty }
, meaning that the effect of the first unit of input
x
i
{\displaystyle x_{i}}
has the largest effect
the limit of the first derivative is zero as
x
i
{\displaystyle x_{i}}
approaches positive infinity:
lim
x
i
→
+
∞
∂
f
(
x
)
/
∂
x
i
=
0
{\displaystyle \lim _{x_{i}\to +\infty }\partial f(\mathbf {x} )/\partial x_{i}=0}
, meaning that the effect of one additional unit of input
x
i
{\displaystyle x_{i}}
is 0 when approaching the use of infinite units of
x
i
{\displaystyle x_{i}}
The elasticity of substitution between goods is defined for the production function
f
(
x
)
,
x
∈
R
n
{\displaystyle f(\mathbf {x} ),\mathbf {x} \in \mathbb {R} ^{n}}
as
σ
i
j
=
∂
log
(
x
i
/
x
j
)
∂
log
M
R
T
S
j
i
{\displaystyle \sigma _{ij}={\frac {\partial \log(x_{i}/x_{j})}{\partial \log MRTS_{ji}}}}
, where
M
R
T
S
j
i
(
z
¯
)
=
∂
f
(
z
¯
)
/
∂
z
j
∂
f
(
z
¯
)
/
∂
z
i
{\displaystyle MRTS_{ji}({\bar {z}})={\frac {\partial f({\bar {z}})/\partial z_{j}}{\partial f({\bar {z}})/\partial z_{i}}}}
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]