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.
