I am studying "The Simple Mathematics of Income Determination", by Paul Samuelson (from book is called "Macroeconomia (artigos selecionados)", by APEC-CAEN. It is the 1st chapter of the book, the 13th page), and I am having some troubles in understanding a calculus passage.
First, Samuelson states the income equation:
$(1)$ $ Y = C (Y - \bar{W}) + \bar{I} + \bar{G} $
Where $Y$ corresponds to income, $C$ corresponds to consumption, $\bar{W}$ corresponds to taxes (actually, I don't exactly know the odd reason why the letter W was picked as taxes instead of a more common letter as T), $\bar{I}$ corresponds to investment and $\bar{G}$ corresponds to government spending). For purpose of simplification, these three last variables are treated as constants.
Then, $Y$ is differentiatied in $\bar{G}$ (which I suppose it is an implicit differentiation, in the form of $\frac{dy}{dx} = - \frac{F_x}{F_y}$):
$(2)$ $\frac{dY}{d\bar{G}} = \frac{1}{1 - C'(Y - \bar{W})}$
But my doubt emerges in the following passage:
$(3)$ $\frac{dY}{d\bar{(-W)}} = \frac{C'(Y - \bar{W})}{1 - C'(Y - \bar{W})} = \frac{dY}{d\bar{G}} - 1 $
I realize that Samuelson did the same implicit differentiation as in step 2 and I can visualize that
$\frac{C'(Y - \bar{W})}{1 - C'(Y - \bar{W})} = \frac{1}{1 - C'(Y - \bar{W})} * C'(Y - \bar{W}) = \frac{dY}{d\bar{G}} * C'(Y - \bar{W})$
but, how does $\frac{dY}{d\bar{G}} * C'(Y - \bar{W})$ turns to $\frac{dY}{d\bar{G}} - 1 $? I mean, how did this $-1$ replace the multiplying term $C'(Y - \bar{W})$
I tried to get it, but unsuccessfully. I hope you could shed some light on it.