Do budget deficits cause inflation? Let me be a little more specific: does raising the level of debt and keeping it there when the economy is at full employment raise the price level? The conventional answer is: not if the central bank controls inflation. Sometimes economists say the same thing in a different way: not if the debt is not monetised. High debt may be problematic for other reasons (e.g. crowding out of private capital, default risk, increasing distortionary taxation), but not because it must cause inflation.
This post is about explaining this conventional view. The two ways of giving the answer reflect two different ways to describe the conventional view, and I think that tells us something interesting - although perhaps controversial - about the role of money.
In the textbooks, the conventional view starts by talking about the demand for the medium of exchange, money. The amount of money in the economy, it is assumed, is related to the amount of money created by the central bank, but not the amount of debt issued by the government. The demand for nominal money is proportional to the price level, which is what economists mean when they say people demand ‘real balances’. So, if the stock of nominal money does not change, neither can the price level. When economists talk about not monetising the debt, they mean that the central bank keeps nominal money fixed.
There are two elements in this argument. The first has to do with the relationship between the amount of money created by the central bank (‘base’ or ‘high powered’ money), and what the financial institutions that create money (private banks) can do. The second has to do with money demand, which is what I want to focus on first. To do so, imagine an economy where the only money is cash printed by the government.
It is trivial to show why there can be this tight and simple link between cash and the price level. The economic system is all about real variables: not just consumption and output, but also relative prices like real wages. Furthermore people want money to buy some real quantity of goods, so the demand function can be written as M/p = f(...) where (...) includes real output. So, if the price level only appears on the left hand side of this equation and nowhere else in the system of equations describing the economy, and the central bank controls the supply of cash M, then this will lock down the price level. This is the famous neutrality of money. Furthermore, the mechanism by which this lock down works is intuitive: if the central bank creates more cash, we have ‘too much money chasing too few goods’, so prices rise.
Once we allow private banks to create money, the story can get much more complicated. The textbooks try and short circuit this by teaching the money multiplier, which I think does a lot more harm than good. But we could just assume there is some mechanism by which the central bank can control the amount of money created by banks, and continue to tell our neutrality story.
So according to this conventional view there is no worry about government debt, as long as the central bank ignores debt when ‘fixing the money supply’. Whether it always will ignore debt, or whether it always can, is a separate issue for another post. The critical assumption I make here that allows me to avoid this issue is that the fiscal authority does adjust its taxes or spending to make the higher level of debt sustainable.
This story is missing a key ingredient, and to see why consider the following. Let all government debt be nominal (not indexed). Suppose that, just as there is a demand for real money, there is also a demand for a real quantity of government debt: B/p = g(....). The government, by cutting taxes for a period, raises the supply of nominal debt by a fixed amount. Suppose it keeps nominal debt at this level. In that case, using an argument analogous to the earlier one involving money, will the price level not increase, until the supply of real debt matches the demand for real debt? If so, higher government debt has raised the price level.
One argument here is to say that, as the government increases the nominal quantity of debt, the demand for debt also rises in step, so there is no need for prices to rise. This will happen if consumers are completely Ricardian, because they believe tax cuts today mean tax increases tomorrow, and they save to pay for those future tax increases by buying government debt. In this sense, the supply of government debt creates its own demand, so we do not need anything else, including the price level, to change.
Suppose, however, that this process is incomplete, perhaps because some consumers are credit constrained, and so spend rather than save their tax cut. Does that not mean prices will still have to rise a bit to match the increase in the supply of nominal bonds? However, if we still have a fixed nominal amount of money, then higher prices will raise the demand for money, giving us a contradiction. What squares this circle is that interest rates rise, which makes people economise on money, and also raises the demand for government debt without the need for prices to increase. So higher debt might raise interest rates, but it will not raise the price level if it is not monetised.
Now an interesting feature of this story is that we could cut out the stuff about money altogether. We could just talk about the supply and demand for nominal government debt, and how the demand for debt is positively related to interest rates and prices. If the government wants to borrow more, the demand for nominal bonds needs to rise, and this can happen either because interest rates rise or because the price level rises. If interest rates rise sufficiently there is no need for higher prices. Loanable funds vs liquidity preference and all that. 
It is a small additional step to just talk about the central bank controlling interest rates to fix the price level. Nowadays this is how many macroeconomists would explain why higher government debt does not raise prices: the central bank changes interest rates to make sure it does not. This explanation not only has the advantage of simplicity (we do not need to talk about the demand for money, or how the central bank controls its supply), but it also seems to match how central banks think.
Of course something about money is there in the background. When we talk about interest rates being varied to control inflation, and why therefore we can ignore the size of the stock of government debt as an influence on inflation, we are assuming that the central bank has the ability to control interest rates. This depends on the fact that the government can issue money, or more specifically that the central bank’s “liabilities happen to be used to define the unit of account”, to quote from
Michael Woodford’s Interest and Prices (p 37). So money is there, but like the impresario of a play, it does
not need to appear on stage.
In terms of the question posed by the title, both ways of describing the conventional view (with or without money) end up with the same answer to the question about government debt and inflation, which is good. However I remain puzzled about one thing. Do those who still tell the story using money think that telling the story just with interest rates is equally valid, or in some way misleading? When, with Campbell Leith, I first started using ‘cashless’ models of the Woodford type, a frequent complaint was ‘where was money?’. To appease potential referees we occasionally put money in, even though this added nothing to the main points of the paper. Yet I think those asking the question thought we might be missing something more fundamental, but I never discovered what it was. I remain genuinely curious.
 Recall that I am assuming full employment in all this. In a recession caused by people saving more, higher saving will raise the demand for bonds, so even if the supply of bonds also rises following budget deficits, interest rates or the price level could fall rather than rise.