I'm trying to compute the derivative $\frac{d c_{t+1}/c_t}{d (1+r_{t+1})}$ of this function: $$\frac{c_{t+1}}{c_{t}} = \left( \beta (1+r_{t+1}) +\gamma \frac{(s-c_{t}+z) ^{-\Sigma }}{c_{t+1}^{-\sigma}} \right)^{\frac{1}{\sigma}}.$$
The difficulty is that the ratio $c_{t+1}/c_t$ depends itself on the level $c_t$, and this level will itself depend on $c_{t+1}$ and $r_{t+1}$. I think I need to apply the implicit function theorem, but I'm not sure how to do so. Any help would be greatly appreciated!
To simplify the function, I could assume that $\sigma=\Sigma=\gamma=1$, yielding $$\frac{c_{t+1}}{c_{t}} = \beta (1+r_{t+1}) + \frac{c_{t+1}}{s-c_{t}+z} .$$
Ultimately, I'm interested in the elasticity $$\varepsilon = \frac{d c_{t+1}/c_t}{d(1+r_{t+1})} \frac{1+r_{t+1}}{c_{t+1}/c_t}.$$ A common results in economics is that if $\gamma=0$, this elasticity simplifies to $1/\sigma$.