So the interesting thing about our money system is that it is Debt (or Credit) based. Money is created at the bank, when you borrow it. People usually think that they are borrowing someone else's savings, but that hasn't been true for over 100 years.
GDP is increased by government spending in the standard calculation, however that's a horribly bad way to look at government spending. Governments create nothing, so saying that they increase GDP by spending is wrong. In fact, Governments take what could be used by the people and re-direct it into places that the people might not choose. This is actually destructive to overall GDP. Getting a handle on how much is the hard thing to do.
Now thinking about inflation, inflation occurs when either more money is created than goods or more goods destroyed (or used up) than money is destroyed. Deflation is the opposite. And yes, money gets destroyed. That's what the FED wants to prevent, believing that deflation is bad.
So consider two graphs below:
In a credit based system, total money is basically cash + credit. The funny thing is that one person's credit is someone else's debt. Now the next number is subject to speculation, but I believe I'm giving the right range. Credit in the US is something like 4 - 7 times larger than cash.
So when household debt declines as it has been, that is deflationary to the total money supply. The FED saw this beginning to happen in the end of 2008 and wanted to inflate. Always the best and easiest way to inflate is to give money to the big banks which in turn buy government debt or pay off bad debts. Deflation in a credit based system with high credit levels will cause exactly what we saw happen: high price items, like homes, lose their value rapidly due to price contraction effects of deflation in the money supply.
The really interesting thing about it is that people are required to look at the value of their "owning" in terms of price instead of in terms of the value of the item itself to them, when they have borrowed money on it. If the sale price goes down, the people and the bank both want to either re-negotiate or get out of the obligation. So the credit based system becomes a feedback loop that brings the prices down rapidly, in an attempt to remove the unneeded products from the market place, or at least reprice them at their needed level.
The government sees this as destructive to the market (and especially to the banking system) and attempts to prevent the declines. That's why there has been so much additional government spending from 2008 to now.
So the short answer is actually two-fold: government spending as a percent of total GDP is low and household debt has been declining, which is deflationary.
Enjoy,
-MikeMark