How would you respond to someone who said the following?
GDP by definition is consumption plus investment plus government spending - C+I+G. Therefore increasing government spending increases GDP.
The right answer, of course, is that higher G only increases GDP holding all else equal. It is quite possible that raising G by $100 B reduces C+I by $100 B. Net effect: zero, but GDP still equals C+I+G. The deeper lesson is that accounting identities can't resolve empirical questions.
Now note that this example exactly parallels NRO's argument that government borrowing can't crowd out private borrowing:
Will private borrowers be crowded out [by increased government deficits]? Impossible. The causation is “loans create deposits,” as taught on day one of every traditional money and banking class. The act of borrowing itself creates exactly that same amount of new liabilities (deposits). The process is “self funding” and circular, as a matter of accounting.
As a matter of accounting identities, loans create deposits. But this sheds no light on the substantive question of crowding out. Even if a $100 B deficit leads to $100 B fewer loans to private borrowers - full crowding out - the books still balance.