One of the first concepts taught in any macroeconomics course is the identity that links the financial balance of the private sector, the financial balance of the government and the current account (which is the financial balance of the country as whole). In a closed economy the financial balance of the private sector has to be equal in size to the financial balance of the government but with the opposite sign. I find this identity very powerful when I teach my students as it links concepts that are normally seen as independent by those who have never studied macroeconomics before.
The identity is also a great source of confusion when it is used in a dynamic setting, to understand growth. For example, there are those who wrongly claim that this identity shows that government deficits cannot create growth because by definition the private sector would be saving every dollar that the government spends. This is wrong.
Via Mark Thoma I land on this interview published in Business Insider of Jan Hatzius (Goldman Sach's chief economist). Most of what I read in the interview is fine but I find, once again, that the identity is being misused to understand growth. Here are a couple of paragraphs from the interview:
"...every dollar of government deficits has to be offset with private sector surpluses purely from an accounting standpoint, because one sector’s income is another sector’s spending, so it all has to add up to zero."
This is only correct if we assume that the economy is closed, but that's not the point I want to make.
"That’s the starting point. It’s a truism, basically. Where it goes from being a truism and an accounting identity to an economic relationship is once you recognize that cyclical impulses to the economy depend on desired changes in these sector's financial balances. If the business sector is basically trying to reduce its financial surplus at a more rapid pace than the government is trying to reduce its deficit then you’re getting a net positive impulse to spending which then translates into stronger, higher, more income, and ultimately feeds back into spending."
This paragraph is misleading (I will ignore again the fact that in an open economy things are more complex). It states (at least this is the way I read it) that growth depends on the "desired changes in these sector's financial balance". This is not correct. I can imagine an economy where those financial balances are not changing at all where output is growing very fast (and I can also imagine another one where output is collapsing). There is no connection between growth and these financial imbalances. As long as demand (private or public) is feeding into production and income, the private or public sector might be spending more than last year but their income is also increasing which can make the financial balance remain at the same level as before.
If we believe that we are in a situation where the output gap is large, there are unused resources and, as a result, output is determined by demand, what matters for growth is whether demand increases relative to last year and not so much the change in the desired changes in the financial balances of either the private or public sector.