I'm struggling with applying the implicit function theorem for the following expression. As an economist I'm not sure if I'm missing something extremely easy, if so, I apologise. For other derivatives in the paper I'm successful using the IFT. Any help at pointing me in the right direction would be greatly appreciated.
Let $c^*$ be defined implicitly as \begin{equation} c^*=\frac{b(e)y+a\int_0^{c^*}c\;G(c)\;dc}{r+b(e)+aG(c^*)} \end{equation} where $G$ is a distribution. I would like to find the following derivative $\frac{dc^*}{de}$. The papaer I'm reading states that \begin{equation} \frac{dc^*}{de} = \frac{(y-c^*)b'(e)}{r+b(e)+aG(c^*)} \end{equation}
The idea is to use the implicit function theorem on $H(c^*,e)=c^*-\frac{b(e)y+a\int_0^{c^*}c\;G(c)\;dc}{r+b(e)+aG(c^*)}$ which should give \begin{equation} \frac{dc^*}{de} = - \frac{\frac{\partial H}{\partial e}}{\frac{\partial H}{\partial c^*}} \end{equation}
I find \begin{equation} \frac{\partial H}{\partial e} = -\frac{b'(e)(r+b(e)+aG(c^*))-b'(e)(b(e)y+a\int_{0}^{c^*}cG(c)dc)}{(r+b(e)+aG(c^*))^2} \end{equation}
\begin{equation} \frac{\partial H}{\partial c^*} = 1-\frac{ac^*G(c^*)(r+b(e)+aG(c^*))-aG'(c^*)(b(e)y+a\int_{0}^{c^*}cG(c)dc)}{(r+b(e)+aG(c^*))^2} \end{equation}
I found it easier to type the variable $c^* $ as $C$ in what follows.
P.S. This revised answer corrects a mistake I made earlier when I tried to type (1). However, even with this corrected equation for (1) as written below, I still am unable to reconcile the results that you reported occurred in the paper. Perhaps that paper has a typo??
The easiest way to apply the implicit function theorem is to first completely clear of denominators: re-write your initial identity as
(1) $$0= C (r + b(e) + a G(C)) -b(e)y - a \int _0^C cG(c)dc= H(C, e)$$
With this choice of the implicit defining function $H(e,C)$ you see readily that $$\partial H/\partial C= (r + b(e) + a G(C)) + a C G'(C) - a CG(C) = r+ b(e) + a CG'(C) + a G(C) - a CG(C)$$
and $\partial H/\partial e= (C-y) b'(e) $
hence
$dC/de=- \frac{\partial H/\partial e}{\partial H /\partial C}=- \frac{(C-y) b'(e)}{ r+ b(e) + a CG'(C) + a G(C) - a CG(C)}$
This answer disagrees with what you claim the paper asserts. Perhaps there was a typo in the paper you read, or perhaps you mistyped the equations that were in that paper?