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I am dealing with a CES production function, and I have attempted some of the "traditional" ways to derive the Cobb-Douglas (logs & l'Hôpital) but I am not sure how to deal with the elasticities on the weight parameters.

$$Q = \big(\beta^{\frac{1}{\sigma}}L^{\frac{1-\sigma}{\sigma}} + (1-\beta)^{\frac{1}{\sigma}}K^{\frac{1-\sigma}{\sigma}}\big)^{\frac{\sigma}{1-\sigma}} $$

I am trying to show that when $\sigma \rightarrow 1$ then $Q = L^\beta K^{1-\beta}$ and when when $\sigma \rightarrow 0$ then $Q = \min\big(\frac{L}{\beta}, \frac{K}{1-\beta}\big)$. The only CES that I found that looks somewhat similar to the one above is utility function of the Armington Model, but I don't think that fits the bill here.

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  • $\begingroup$ See here: economics.stackexchange.com/questions/361/… $\endgroup$
    – Bertrand
    Commented May 25, 2022 at 16:01
  • $\begingroup$ thank you @Bertrand, I tried to apply that but did not manage. Really what confuses me is the inverse elasticity on the $\beta$ $\endgroup$
    – user862800
    Commented May 25, 2022 at 16:04
  • $\begingroup$ @Bertrand, I managed to solve for the Leontief using the sandwich proof from the post, I had to tweak aspects of the simplification. Still, it is not trivial. Any ideas for the Cobb-Douglas? I am stuck with factoring out the weights for the first order Taylor approximation. $\endgroup$
    – user862800
    Commented May 25, 2022 at 22:01
  • $\begingroup$ See here for an alternate track: economics.stackexchange.com/questions/51415/… $\endgroup$
    – Bertrand
    Commented May 28, 2022 at 9:43

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