Long run multiplier equality

Question

I don't understand the following equality:

$$ \beta_0 + \sum_{j=1}^\infty \phi^{j-1} (\phi\beta_0 + \beta_1) = \beta_0 + \frac{\phi\beta_0 + \beta_1}{1 - \phi} $$

It calculates the long run multiplier from an ADL(1,1) model. Could someone explain it or maybe show some steps in between?

Thanks

Answer 1

Hint. If $|x|<1$ then the geometric sum evaluation leads to $$ \sum_{j=1}^\infty x^{j-1}=\frac1{1-x} $$ giving $$ \sum_{j=1}^\infty \phi^{j-1}=\frac1{1-\phi},\qquad |\phi|<1. $$