Inada conditions

In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada, 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.

The six conditions for a given function $f(x)$ are:

1. the value of the function $f(x)$ at 0 is 0: $f(0)=0$
2. the function is continuously differentiable,
3. the function is strictly increasing in $x_{i}$: $\partial f(x)/\partial x_{i}>0$,
4. the second derivative of the function is negative in $x_{i}$ (thus the function is concave): $\partial^{2} f(x)/\partial x_{i}^{2}<0$,
5. the limit of the first derivative is positive infinity as $x_{i}$ approaches 0: $\lim_{x_{i} \to 0} \partial f(x)/\partial x_i =+\infty$,
6. the limit of the first derivative is zero as $x_{i}$ approaches positive infinity: $\lim_{x_{i} \to +\infty} \partial f(x)/\partial x_i =0$