<p>Yes, all this does mean that the production function represents the average of two inputs <span class="math-container">$L$</span> and <span class="math-container">$K$</span> for different values of <span class="math-container">$\alpha$</span>, given that <span class="math-container">$0&lt;\gamma&lt;1$</span>. 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 <span class="math-container">$\alpha$</span> it gets transformed to other forms of mean like weighted mean, and geometric mean.</p>
<p>It is has been shown in detail in a previous answer on Stack Exchange <a href="https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function?noredirect=1&amp;lq=1">here</a>.</p>

CC BY-SA 4.0
2024-05-27T15:40:13.857Z
2024-05-27T15:40:13.857Z
Pallak Goyal
<p>I know that the Constant elasticity of substitution production function is given by:</p>
<p><span class="math-container">$$Q=\gamma\left[\delta L^{-\alpha}+\left(1-\delta\right)K^{-\alpha}\right]^{\frac{-1}{\alpha}}$$</span></p>
<p>where <span class="math-container">$\gamma$</span> is the efficiency parameter, <span class="math-container">$\delta$</span> is the distribution parameter and <span class="math-container">$\alpha$</span> is the substitution parameter.</p>
<p>I can also show that the <span class="math-container">$$CES = \frac{1}{1+\alpha}$$</span>.</p>
<p>But does all this mean that the <em>&quot;production function represents the average of two inputs <span class="math-container">$L$</span> and <span class="math-container">$K$</span> for different values of <span class="math-container">$\alpha$</span>, given that <span class="math-container">$\gamma&gt;0$</span> and <span class="math-container">$0&lt;\delta &lt;1$</span>&quot;</em>?</p>
<p>I could not imagine this to be true graphically since the shape of the Isoquant depends on the parameter <span class="math-container">$\alpha$</span>. For instance, when <span class="math-container">$\alpha=1$</span> it takes convex shape, <span class="math-container">$\alpha=-1$</span> it takes downward sloping straight line shape, and when <span class="math-container">$\alpha=\infty$</span> it takes L shape.</p>
<p><strong>Edit 1</strong>: 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.</p>

CC BY-SA 4.0
2024-05-27T11:01:45.720Z
This production function is decreasing in $L$ and $K$ for all $\alpha\neq 0$. It does not look like a sensible production function.
Amit
CC BY-SA 4.0
2024-05-27T11:30:05.457Z
@Amit Sir Thank you for the response. There was a typo. I have corrected it. Can you provide your insights about it now?
Pallak Goyal
CC BY-SA 4.0
2024-05-27T14:25:17.473Z
What do you mean by average of two inputs $L$ and $K$ for different values of $\alpha$?
Amit
CC BY-SA 4.0
2024-05-27T15:11:10.343Z
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](https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function).
Pallak Goyal
<microeconomics><production-function><ces>
Is CES production representing the average of inputs?
Pallak Goyal
# Is CES production representing the average of inputs?

(Asked by: username_0 on 2024-05-27)

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.

**Tags:** microeconomics, production-function, ces

> **Comments:**
> **username_1**: This production function is decreasing in $L$ and $K$ for all $\alpha\neq 0$. It does not look like a sensible production function.  
> **username_0**: @username_1 Sir Thank you for the response. There was a typo. I have corrected it. Can you provide your insights about it now?  
> **username_1**: What do you mean by average of two inputs $L$ and $K$ for different values of $\alpha$?  
> **username_0**: 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](https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function).  

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## 1 answers

### Answer by username_0 on 2024-05-27. Score: 2

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](https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function?noredirect=1&lq=1).


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