Where do public savings go? In Mankinw's Macroeconomics, he mentions that public and private savings form national saving, which ends up in the financial sector, which then people and firms use to purchase investment goods. But what happens in real life? If the US federal government runs a budget surplus, where does it go?
Most governments (outside of typically oil-producing nations that have sovereign wealth funds) are net debtors, so public deficits show up as increases in government debt outstanding, and public surpluses (public saving) show up as decreases in debt outstanding.

In the model used in Mankiw’s book, the assumption is that government spending never constitutes investment (i.e, it doesn’t create productive capital). In that model, it’s also assumed that households hold government debt: government debt just soaks up some of the funds that would otherwise be available as “loanable funds” to private firms to produce “productive capital,” shifting the supply curve to the left. 

So, in that model, the quantity axis represents something like “private investment,” and government saving shows up as an outward shift in the supply curve, which results in lower interest rates and more private investment. 

There’s another way of thinking about it, which doesn’t require that one assume that government spending never forms capital like roads, bridges, airports, etc., nor that government deficits are paid for directly out of private saving. In that model, government deficits just show up as increases in demand for loanable funds, which increases interest rates. Implicitly, this does still mean that less private sector investment will occur. The opposite is of course true for government surpluses. The interpretation of the quantity axis then changes to something like “amount of financial intermediation.”

Either model of loanable funds is, of course, a radical simplification of how financial intermediation works, and like anything in an intro course, it’s worth remembering that tons of caveats apply.