# How to get from a CES production function with inverse elasticity on weights to the special cases Cobb-Douglas & Leontief

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.

• Commented May 25, 2022 at 16:01
• thank you @Bertrand, I tried to apply that but did not manage. Really what confuses me is the inverse elasticity on the $\beta$ Commented May 25, 2022 at 16:04
• @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. Commented May 25, 2022 at 22:01
• See here for an alternate track: economics.stackexchange.com/questions/51415/… Commented May 28, 2022 at 9:43