Inada conditions

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]

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

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

All these conditions are met by a Cobb–Douglas production function.

References

↑ 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.
↑ Uzawa, H. (1963). "On a Two-Sector Model of Economic Growth II". The Review of Economic Studies. 30 (2): 105–118. JSTOR 2295808.