0

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.

Giskard
  • 29,387
  • 11
  • 45
  • 76
user862800
  • 1
  • 1
  • See here: https://economics.stackexchange.com/questions/361/how-can-i-obtain-leontief-and-cobb-douglas-production-function-from-ces-function?rq=1 – Bertrand May 25 '22 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 $\beta$ – user862800 May 25 '22 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. – user862800 May 25 '22 at 22:01
  • See here for an alternate track: https://economics.stackexchange.com/questions/51415/confusing-on-the-crs-property-of-ces-function/51434#51434 – Bertrand May 28 '22 at 09:43

0 Answers0