Multivariate implicit differentiation (Samuelson, 1948)
We start from the equation:
$$
Y = C(Y - \overline{W}) + \overline{I} + \overline{G}
$$
First, we totally differentiate with respect to $(-\overline{W})$ (taking into account that $Y$ is a function of $\overline{W}$, $\overline{I}$ and $\overline{G}$)
$$
dY = C'(Y - \overline{W})dY + C'(Y - \overline{W}) d(-\overline{W}).
$$
Bringing the $dY$ terms on one side gives:
$$
(1 - C'(Y - \overline{W}))dY = C'(Y - \overline{W}) d(-\overline{W}),\\
\to \frac{dY}{d(-\overline{W})} = \frac{C'(Y - \overline{W})}{1 - C'(Y - \overline{W})}
$$
Next, we totally differentiate with respect to $\overline{G}$:
$$
dY = C'(Y - \overline{W})dY + d \overline{G}
$$
Again gathering terms gives:
$$
(1 - C'(Y - \overline{W})) dY = d \overline{G}\\
\to \frac{dY}{d \overline{G}} = \frac{1}{1 - C'(Y - \overline{W})}
$$
Therefore:
$$
\begin{align*}
\frac{dY}{d \overline{G}} - 1 &amp;= \frac{1}{1 - C'(Y - \overline{W})} - 1,\\
&amp;=\frac{1 - 1 +C'(Y - \overline{W})}{1 - C'(Y - \overline{W})},\\
&amp; = \frac{C'(Y - \overline{W})}{1 - C'(Y - \overline{W})},\\
&amp;= \frac{dY}{d(- \overline{W})}
\end{align*}
$$