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:

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: $\lim_{x_{i} \to +\infty} \partial f(x)/\partial x_i =0$

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

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.

Further reading

• Barro, Robert J.; Sala-i-Martin, Xavier (2004). Economic Growth (Second ed.). London: MIT Press. pp. 26–30. ISBN 0-262-02553-1.
• Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 176–178. ISBN 3-540-60988-1.
• Romer, David (2011). "The Solow Growth Model". Advanced Macroeconomics (Fourth ed.). New York: McGraw-Hill. pp. 6–48. ISBN 978-0-07-351137-5.