If your country's debt-to-GDP gets too out of control, markets might doubt that you'll be able to pay on your obligations, and then you've got problems. The debt is made up of many tiny obligations (treasury bills, bonds, and notes) that get paid on a regular basis every day, and the US government has always paid every one of them. When these individual obligations are due, they need to be paid, and investors need to be very confident they will be. The point at which this becomes a problem is not a hard and fast rule, but it depends on a lot of situational factors that go even beyond the realm of math and into geopolitics.
I don't think you really answered my questions, or at least I'm not smart enough to understand.
What would out of control mean, and why? What are these problems? I struggle to understand what exactly it is that is entirely not a problem until this difficult to define point where it can become a problem, and what that problem is.
The US is only able to borrow tens of trillions of dollars for very low interest rates is because investors, big and small, both in the US and around the world, see the US as essentially a risk-free way to store value. People (or investment organizations, or businesses, etc) with many millions or billions of dollars have to put it somewhere. Using vaults to store that much money is expensive to maintain and causes them to lose value to inflation and the cost it entails to run that operation. Basically, the default way to store a lot of money in an extremely safe way (in terms of it being there when you want it, and not losing tons of value) is to buy US treasuries. Financial markets consider them to be zero-risk -- the benchmark by which all other investments are compared.
Now, if this were to change, and financial markets thought that buying US treasuries might not be a guarantee that they can get that money back later, a lot of things would be disrupted. In the immediate term, interest rates would likely increase significantly for everything, as many interest models base their rates on treasury interest rates, and treasury interest rates would skyrocket due to the now non-zero risk. That would in turn mean that the US's interest on their outstanding debt would skyrocket, further compounding the issue. Also, because so much of the world economy is based on the assumption that US treasuries are zero risk, this would upend investor confidence in many other financial markets globally that rely on those instruments for their function. Those who were forking over trillions to the US for safekeeping may look for other places to keep their money instead. This would be bad for the US governments ability to raise money to do the things it needs to do, in likely a significant way. And if a default does happen, creditors could seize assets, and depending on who is stuck holding the bag, war has happened in these situations before, although the UN currently disallows it.
But at a minimum, the global economy would be hit very hard.
Then paying down debt reduces the risk of exposure to interest rates and this lack of confidence situation, and also reduces interest repayments. I would think these are reasons to reduce debt you would weigh along the reasons to keep or increase debt, but apparently not?
