Elasticity of intertemporal sustitution with composite CRRA function In the usual CRRA $\frac{c^{1-\sigma}-1}{1-\sigma}$ function we have that the intertemporal elasticity of sustitution  $\partial\frac{c_{t+1}}{{c_{t}}{\partial r}}$ is $\frac{1}{\sigma}$.
But how can i calculate the IE of S when the CRRA function is the composite:
$\frac{(c-G())^{1-\sigma}-1}{1-\sigma}$
Thanks in advance
Suppose $c_t$ is some composite function of interest rate $r$, e.g. $c_t(G(r_t))$.
First the intertemporal elasticity of substitution is actually (IES) given by $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial r}$ (or also $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial \ln( u'(c_{t+1})/u'(c_t))}$).
You can take the derivative above using chain rule for composite functions which says that $dF(G(x))/dx = \frac{dF}{dG}\frac{dG}{dx}$.
So in the above case we would get:
$$\frac{\partial \ln(c_{t+1}(G(r))/c_{t}(G(r)))}{\partial r} = \frac{\partial \ln(c_{t+1}(G(r))/c_{t}(G(r)))}{\partial G(r)} \cdot \frac{\partial G(r)}{\partial r} $$
You can directly apply this to the problem you mention in your question.