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

It can be shown that the Inada conditions imply that the elasticity of substitution is asymptotically equal to one (although the production function is not necessarily asymptotically Cobb–Douglas).[3][4]

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.[5]

## 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. ^ Barelli, Paulo; Pessoa, Samuel de Abreu (2003). "Inada Conditions Imply That Production Function Must Be Asymptotically Cobb–Douglas". Economics Letters 81 (3): 361–363. doi:10.1016/S0165-1765(03)00218-0.
4. ^ Litina, Anastasia; Palivos, Theodore (2008). "Do Inada conditions imply that production function must be asymptotically Cobb–Douglas? A comment". Economics Letters 99 (3): 498–499. doi:10.1016/j.econlet.2007.09.035.
5. ^ Kamihigashi, Takashi (2006). "Almost sure convergence to zero in stochastic growth models". Economic Theory 29 (1): 231–237. doi:10.1007/s00199-005-0006-1.