List of production functions that satisfy the Inada conditions

Question

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?

Answer 1

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.

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

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

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.

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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.

Answer 2

The power family of production functions satisfies Inada's conditions over a set of admissible parameter values. In the case with two inputs, the functional form is given by:

$$ y = \alpha_{11} x_1^{\gamma_1} + \alpha_{22} x_2^{\gamma_2} + \alpha_{12} x_1^{\gamma_1/2}x_2^{\gamma_2/2}, $$

The power production function $f$ satisfies $f(0)=0$, $f(x) \geq 0$ and is increasing in each $x_i$ if all $\alpha_{ij}>0$ and $ \gamma_{i}>0 $. It is concave in $x$ if in addition all $\gamma_{i}<1$, as a sum of concave functions.

The partial derivatives are: $$ \frac{\partial f}{\partial x_i}( x ) = \frac{ \alpha_{ii} \gamma_i }{ x_i^{1-\gamma_i} } + \frac{ \alpha_{ij} \gamma_i/2 }{ x_i^{1-\gamma_i/2} }{x_j^{\gamma_j/2}} $$ The partial derivatives' limiting behavior are compatible with Inada's conditions: $$ \lim_{x_{i}\rightarrow 0}\frac{\partial f}{\partial x_{i}}\left( x\right) =+\infty ,\qquad \lim_{x_{i}\rightarrow +\infty }\frac{\partial f}{\partial x_{i}}\left( x\right) =0. $$

This power production function can be generalized to the case of $J$ inputs as follows: $$ y = h( \sum_{i=1}^J \sum_{j=1}^J \alpha_{ij} x_i^{\gamma_i/2}x_j^{\gamma_j/2} ), $$ where function $h$ satisfies $h(0)=0$, and is increasing and concave. See for instance Diewert (1971) for a contribution with $\gamma_i=\gamma_j=1$. A similar argumentation should apply to the Box-Cox family of production functions.

Diewert, W. E., 1971, "An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function," Journal of Political Economy, 79, 481-507.

Answer 3

I believe this question is too broad. As discussion in the comments indicates, even functional forms that are commonly used for their ability to satisfy the Inada conditions can fail to satisfy them for allowed values of their parameters. On the other hand, as also noted, one can always define a piecewise-continuous function that satisfies them.

With that said, it's been shown that every Inada-satisfying function is asymptotically Cobb-Douglas. https://www.sciencedirect.com/science/article/abs/pii/S0165176503002180