# Is CES production representing the average of inputs?

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.

• This production function is decreasing in $L$ and $K$ for all $\alpha\neq 0$. It does not look like a sensible production function.
– Amit
Commented May 27 at 11:01
• @Amit Sir Thank you for the response. There was a typo. I have corrected it. Can you provide your insights about it now? Commented May 27 at 11:30
• What do you mean by average of two inputs $L$ and $K$ for different values of $\alpha$?
– Amit
Commented May 27 at 14:25
• 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. Commented May 27 at 15:11

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.