List of production functions that satisfy the Inada conditions It is known that in the class of CES production functions, only the Cobb-Douglas production function satisfies the Inada conditions. Which other functions satisfy the Inada conditions?
The Wikipedia statement should come with a citation or proof, as it is not entirely accurate: Cobb-Douglas functions with increasing returns to scale do not necessarily satisfy the Inada conditions, e.g., $f(x_1,x_2) = x_1^2x_2^2$ does not.



There are many non Cobb-Douglas functions that satisfy the Inada conditions, albeit they may not have nice compact formulas. E.g., 
$$
f(x) = \left\{\begin{array}{ll}
x^{1/2} &amp; \mbox{ if } x \leq 1 \\
2x^{1/4} - 1 &amp; \mbox{ if } 1 &lt; x 
\end{array}\right.
$$
This satisfies the conditions:


$f(0) = 0$
The Hessian is negative semidefinite as $f$ is strictly concave.
$$\lim_{x \to 0}\frac{\text{d} f(x)}{\text{d} x} = \infty$$
$$\lim_{x \to \infty} \frac{\text{d} f(x)}{\text{d} x} = 0$$


You can easily manufacture similar functions. I just combined two strictly concave functions at a point of tangency and made sure that the piece going to zero would cross the origin.



I am not sure, but it seems to me that if you have an Inada satisfying function $h(\mathbf{x})$ then the function $f(h(\mathbf{x}))$ will also satisfy all the conditions.