# Deriving average productivity from a CES production function

Following the work of Raurich et al. (2012) I got stuck trying to derive the average productivity starting from the following CES production function:

$$Y=A\left [ \alpha K^{\frac{\sigma -1}{\sigma }}+(1-\alpha)(AL)^{\frac{\sigma -1}{\sigma }} \right ]^{\frac{\sigma}{\sigma-1 }}$$

The result they get is: $$\left (\frac{Y}{AL} \right )^{\frac{1-\sigma }{\sigma }}=\frac{1}{\alpha \left ( \frac{K}{AL} \right )^{\frac{\sigma-1 }{\sigma }}+(1-\alpha )}$$

Is there anyone who can help me with this?

• I checked just to be sure. But actually the exponent of the LHS is right. – Alessandro Sep 29 '18 at 23:52

This is the solution to your difficulty. By the way, which text are you following?

• I think OP's confusion is over why average productivity is defined in this manner. To me, average labor productivity (in efficiency units) would just be $\frac{Y}{AL}$. The paper in question is here (see page 6 of the paper). Maybe it's just a language issue -- it is a working paper. – Kenneth Rios Sep 30 '18 at 16:35
• Yes, I'm also trying to understand this. Anyway, it is not a working paper, it has been published on Journal of Macroeconomics (Volume 34, Issue 1, March 2012, Pages 181-198). – Alessandro Oct 1 '18 at 11:42
• I think this is a language issue. I see no relevance in raising average productivity to a function of elasticity of substitution and then calling it average productivity. If you find an answer, please do let me know. – DrStrangeLove Oct 1 '18 at 17:53
• Agreed that this is likely just semantics. The language preceding the expression given for average productivity is indicative to me. – Kenneth Rios Oct 1 '18 at 21:02