In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada,[1] are assumptions about the shape of a production function that guarantee the stability of an economic growth path in a neoclassical growth model. The conditions as such had been introduced by Hirofumi Uzawa.[2]

Given a continuously differentiable function ${\displaystyle f\colon X\to Y}$, where ${\displaystyle X=\left\{x\colon \,x\in \mathbb {R} _{+}^{n}\right\}}$ and ${\displaystyle Y=\left\{y\colon \,y\in \mathbb {R} _{+}\right\}}$, the conditions are:

1. the value of the function ${\displaystyle f(\mathbf {x} )}$ at ${\displaystyle \mathbf {x} =\mathbf {0} }$ is 0: ${\displaystyle f(\mathbf {0} )=0}$
2. the function is concave on ${\displaystyle X}$, i.e. the Hessian matrix ${\displaystyle \mathbf {H} _{i,j}=\left({\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}\right)}$ needs to be negative-semidefinite.[3] Economically this implies that the marginal returns for input ${\displaystyle x_{i}}$ are positive, i.e. ${\displaystyle \partial f(\mathbf {x} )/\partial x_{i}>0}$, but decreasing, i.e. ${\displaystyle \partial ^{2}f(\mathbf {x} )/\partial x_{i}^{2}<0}$
3. the limit of the first derivative is positive infinity as ${\displaystyle x_{i}}$ approaches 0: ${\displaystyle \lim _{x_{i}\to 0}\partial f(\mathbf {x} )/\partial x_{i}=+\infty }$,
4. the limit of the first derivative is zero as ${\displaystyle x_{i}}$ approaches positive infinity: ${\displaystyle \lim _{x_{i}\to +\infty }\partial f(\mathbf {x} )/\partial x_{i}=0}$

In the class of CES production functions only the Cobb–Douglas production function meets all of these conditions.

## References

1. ^ Inada, Ken-Ichi (1963). "On a Two-Sector Model of Economic Growth: Comments and a Generalization". The Review of Economic Studies. 30 (2): 119–127. JSTOR 2295809.
2. ^ Uzawa, H. (1963). "On a Two-Sector Model of Economic Growth II". The Review of Economic Studies. 30 (2): 105–118. JSTOR 2295808.
3. ^ Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. pp. 125–126. ISBN 0-521-31498-4.