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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?

$$C_1 = \frac{1}{Z} \prod_{i=1}^N P_i^{\alpha_i}$$

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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. The exponents in the parameters $\alpha_i$ should cause no further complication.

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