Jonathan Catalan does an excellent job dispatching a simplistic take on the national income identity.
In response to his government spending post at Mises.org (which actually I disagreed with), he was asked the question: "As we have studied, Y=C+I+G+(X-M). The
author did not really address the fundamental question of how, at times
of low private consumption and investment, national income can be
boosted without government expenditure."
This is an identity, of course. Jonathan could have just answered "well you could make policy such that C and I increased and G doesn't change". I don't think that would have satisfied his questioner, but it would have made sense.
The components of the national income equation have their own behavioral laws and G doesn't just influence Y additively - it impacts everything else as well.
I think Jonathan is often wrong about how the components behave and interact. We both have a simplistic view of how they behave and interact (how could we not?) but we disagree on the primary determinants and interactions of interest. That's how we get different answers.
But one thing no one should do is just assume that the summation of the aggregates is the end of the story.