eternumviti
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The short answer is: no. But I'll do a longer one. Let's start backwards. QE is the government 'buying up' bonds, so the very reverse of bond sales. (Edit - important addition: to create money to buy bonds then say the government sells bonds to borrow money clearly does not make sense). Bond sales are used as a monetary policy tool as a reserve drain, to remove excess reserves so they don't need to pay interest on the reserves (and so that it meets the overnight bank interest rate). So yes bonds are sold, but not for 'borrowing'. It will make the post overlong if I go through the main three reasons why they are issued, so I'll find where I've said it elsewhere and put a link. What I will say is that they end up being a mere asset swap since the money people use to buy these gilts comes from money loaned from the bank in the first pace.
Back to QE though, the purchase of bonds by the BoE ends up as the swelling of banks reserves. Since monetarist economists actually think banks lend out reserves; that they need them to make loans. Be assured that banks do not like being forced to hold excess reserves, they prefer interest-bearing bonds. But that's what QE is: reserve increases.
But the question was: "what does the government do to make up the budget deficit". The first question is 'what is the deficit?' It's a problem question, not because it can't be answered, but because the answer often leads people astray. A deficit is a government spending more than it taxes. That doesn't mean they failed to get enough tax, therefore they had to somehow to fill a hole. It means they spent more than they destroyed with tax. Only that. Knowing they issue the money they use, this a logical conclusion. To assume they borrow is illogical, since if you had sole power of currency issue, would you borrow that currency? If so, why and from whom?
Deficit spending is not 'debt'. Debt (bonds) is something government issues by choice. Again they don't borrow for deficit expansion, so there can't possibly be an actual debt. And since the so-called debt is in the money of account anyway the interest payments on it can always be serviced. Always. The usual example is Japan, whose level has been enormous for decades. They never default, because they won't. Compare also pre-Euro Italy whose 'debt' was 20 times higher than Louisiana, but Louisiana 'defaulted'. How? Because it is in a currency union with imposed rules, it can't issue currency or service debt.
It certainly is counter-intuitive, because the common intuition (which is actually not intuition but the spread of erroneous theory) is wrong. There is an initial question to consider: why does the the private sector use/accept the government's currency? Because they need it to pay the tax. The idea that the money government issues somehow already exists first in the economy and then the government has to get it back in order to spend it, doesn't even make sense. Here is the very basic order of events:
Government sets legal tax liabilities in the money of account (this initially drives currency acceptance) >
People now owe a tax and are effectively unemployed until they enter into productive labour (in any sense) so they can get the currency to pay the tax (this is resource mobilisation) >
Spending into the economy (this includes money issued as normal loans) >
Productive activity creates goods/services. There is a profit motive to make the effort worthwhile, though a good deal of the production ends up as social product >
The tax is paid. This destroys a portion of spent money.
That is a skeleton outline.
None of that is 'state intervention' as popularly conceived by the right-wing. Their preferred governments in our lifetimes are the ones who have employed use of an unemployment buffer stock as inflation control. And so seem to prefer to have people idle (after creating national unemployment by default).
Not 'high levels', but levels that reflect the private domestic sector's desire for income/saving. Obviously things like surpluses, austerity don't achieve that. Can't possibly achieve it.
OK, grateful for this, a long post in which you've gone to considerable trouble to explain. It's going to take me a while to process it all, and in places it seems to raise even more questions than it answers.
I do take one point as read, and that is that a country (such as the UK/Japan) which has the ability to issue currency and service debt is far more resilient than one such as Italy (today) which doesn't. This has always been the logical argument against EMU, which places widely disparate economies into a monetary straightjacket. However, I find your analysis behind that conclusion very difficult to grasp.
One small question, what do you mean by 'money of account'?