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., $$ f(x) = \left\{\begin{array}{ll} x^{1/2} & \mbox{ if } x \leq 1 \\ 2x^{1/4} - 1 & \mbox{ if } 1 < x \end{array}\right. $$ 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.

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.

  • $\begingroup$ (+1) Thanks. Yes, I forgot to add that the Cobb-Douglas production function must be CRS to satisfy the Inada conditions. Do you know of any nice functions that are used in the literature and that satisfy the Inada conditions? It should be possible to construct functions satisfying Inada conditions by taking linear combinations of CRS Cobb-Douglas production funtions too. $\endgroup$ – Elias Feb 4 '20 at 9:40
  • $\begingroup$ Sorry, I am not a macroeconomist :) $\endgroup$ – Giskard Feb 4 '20 at 9:42
  • $\begingroup$ P.s. It occurs the me that the Hessian of $f$ is not defined at $x=1$. One could also "smooth" this over, but I do not want to tinker with the formula further. The idea is just to draw a strictly concave function that satisfies 1,3, an 4. $\endgroup$ – Giskard Feb 4 '20 at 9:42
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    $\begingroup$ Exactly. One can simply fill in the discontinuities. $\endgroup$ – Elias Feb 4 '20 at 9:43
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    $\begingroup$ @Brennan It is my understanding that negative definite is a subset of negative semidefinite? My point was that Inada needs the Hessian to be negative semidefinite, which is fulfilled due to $f$ being strictly concave and hence having a negative definite Hessian. $\endgroup$ – Giskard Feb 4 '20 at 20:24

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.


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


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