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I know that the Constant elasticity of substitution production function is given by:

$$Q=\gamma\left[\delta L^{-\alpha}+\left(1-\delta\right)K^{-\alpha}\right]^{\frac{-1}{\alpha}}$$

where $\gamma$ is the efficiency parameter, $\delta$ is the distribution parameter and $\alpha$ is the substitution parameter.

I can also show that the $$CES = \frac{1}{1+\alpha}$$.

But does all this mean that the "production function represents the average of two inputs $L$ and $K$ for different values of $\alpha$, given that $\gamma>0$ and $0<\delta <1$"?

I could not imagine this to be true graphically since the shape of the Isoquant depends on the parameter $\alpha$. For instance, when $\alpha=1$ it takes convex shape, $\alpha=-1$ it takes downward sloping straight line shape, and when $\alpha=\infty$ it takes L shape.

Edit 1: The source of the question is the Indian Economic Services General Economics Paper 1, 2019. This is an exam conducted by the Government of India to hire economists.

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    $\begingroup$ This production function is decreasing in $L$ and $K$ for all $\alpha\neq 0$. It does not look like a sensible production function. $\endgroup$
    – Amit
    Commented May 27 at 11:01
  • $\begingroup$ @Amit Sir Thank you for the response. There was a typo. I have corrected it. Can you provide your insights about it now? $\endgroup$ Commented May 27 at 11:30
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    $\begingroup$ What do you mean by average of two inputs $L$ and $K$ for different values of $\alpha$? $\endgroup$
    – Amit
    Commented May 27 at 14:25
  • $\begingroup$ I think here it means the generalized weighted mean taking specific forms as I also mentioned in the question. On considering your comment I found this. $\endgroup$ Commented May 27 at 15:11

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Yes, all this does mean that the production function represents the average of two inputs $L$ and $K$ for different values of $\alpha$, given that $0<\gamma<1$. The key thing to consider here is the broader definition of a mean. The initial equation itself is a weighted generalized mean. And later depending on the value of $\alpha$ it gets transformed to other forms of mean like weighted mean, and geometric mean.

It is has been shown in detail in a previous answer on Stack Exchange here.

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