# Why is my elasticity of substitution wrong?

I am calculating elasticity of substitution for the following production function:

$$F(K,L) = A(aK^{-\gamma}+bL^{-\gamma})^{-\mu/\gamma}$$

where $$A, a, b, \mu, \gamma$$ are constants. $$A, a, b, > 0$$, $$\mu \neq 0$$ and $$\gamma \neq 0; \gamma>-1$$.

We learned in our course that to calculate elasticity of substitution we should first calculate marginal rate of substitution and then calculate elasticity with respect to $$K/L$$. I calculated MRTS to be:

$$MRTS = \frac{b}{a} \left(\frac{K}{L}\right)^{\gamma+1}$$

Then I try to calculate the elasticity of the above with respect to $$K/L$$ which gives me:

$$EL_{MRTS_{K,L}} = \frac{d MRTS}{d (K/L)} \frac{K/L}{MRTS} \\ = \frac{b}{a} (\gamma+1)\left(\frac{K}{L}\right)^{\gamma} \left( \frac{K/L}{\frac{b}{a} \left(\frac{K}{L}\right)^{\gamma+1}}\right) \\= \gamma +1$$

However, the textbook says the correct answer is $$\frac{1}{\gamma+1}$$. I think I just made some small mistake somewhere because the answer provided by book is just an inverse of my answer. But I don't know where I am making the mistake.

Please could someone help me with this? Thanks.

• Your calculation is correct just a small definitional issue. The definition of Elasticity of substitution is the inverse of what you have taken. Nov 13, 2020 at 8:49
• @Dayne consider expanding it into full answer
– 1muflon1
Nov 13, 2020 at 10:06
• Okay. Sure will do and try to enrich it further Nov 13, 2020 at 13:57