<p>I think so. I didn't work out the proof myself but found in Wikipedia another version of the CES production function that has the same shape as your cost production function by replacing <span class="math-container">$1-\sigma$</span> with <span class="math-container">$r$</span>. Here is the <a href="https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution" rel="nofollow noreferrer">link</a>. The exponents in the parameters <span class="math-container">$\alpha_i$</span> should cause no further complication.</p>

CC BY-SA 4.0
2019-05-27T16:54:31.980Z
2019-05-27T16:54:31.980Z
Regio
<p>Imagine I have a CES production function</p>

<p><span class="math-container">$$ Y_{\sigma}   = Z [ \sum_{i=1}^N  \alpha_{i} X_i^{\frac{\sigma}{1-\sigma}}]^{\frac{1-\sigma}{\sigma}}$$</span></p>

<p>I know that as <span class="math-container">$\sigma \to 1$</span>, the corresponding production function becomes the Cobb-Douglas production function</p>

<p><span class="math-container">$$Y_1  = Z  \prod_{i=1}^N X_{i}^{\alpha_i}$$</span></p>

<p>My question is, is the same true for the corresponding cost functions? That is, if I have the CES cost function</p>

<p><span class="math-container">$$C_{\sigma} = \frac{1}{Z} [\sum_{i=1}^N \alpha_i^{\sigma} P_i^{1-\sigma}]^{\frac{1}{1-\sigma}}$$</span></p>

<p>Is it true that as <span class="math-container">$\sigma \to 1$</span>, <span class="math-container">$C_{\sigma}$</span> converges to the following Cobb-Douglas cost function?</p>

<p><span class="math-container">$$C_1 = \frac{1}{Z} \prod_{i=1}^N P_i^{\alpha_i}$$</span></p>

<production-function>
Continuity of Prices for Constant Elasticity of Substitution Production Functions
Asterix
# Continuity of Prices for Constant Elasticity of Substitution Production Functions

(Asked by: username_0 on 2019-05-27)

Imagine I have a CES production function

$$ Y\_{\sigma} = Z [ \sum\_{i=1}^N \alpha\_{i} X\_i^{\frac{\sigma}{1-\sigma}}]^{\frac{1-\sigma}{\sigma}}$$

I know that as $\sigma \to 1$, the corresponding production function becomes the Cobb-Douglas production function

$$Y\_1 = Z \prod\_{i=1}^N X\_{i}^{\alpha\_i}$$

My question is, is the same true for the corresponding cost functions? That is, if I have the CES cost function

$$C\_{\sigma} = \frac{1}{Z} [\sum\_{i=1}^N \alpha\_i^{\sigma} P\_i^{1-\sigma}]^{\frac{1}{1-\sigma}}$$

Is it true that as $\sigma \to 1$, $C\_{\sigma}$ converges to the following Cobb-Douglas cost function?

$$C\_1 = \frac{1}{Z} \prod\_{i=1}^N P\_i^{\alpha\_i}$$

**Tags:** production-function

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## 1 answers

### Answer by username_1 on 2019-05-27. Score: 1 | Accepted answer

I think so. I didn't work out the proof myself but found in Wikipedia another version of the CES production function that has the same shape as your cost production function by replacing $1-\sigma$ with $r$. Here is the [link](https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution). The exponents in the parameters $\alpha\_i$ should cause no further complication.


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