Struggling to apply the Implicit function theorem 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}
 A: 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?
