Debt & deficits: the maths

Do we need a budget surplus to get the ratio of government debt to GDP down? There seems to be some confusion on this, so let me clarify.

George Osborne said today:

Government must ensure that debt continues to fall as a percentage of GDP, including using surpluses in good years for this purpose.

That word "including" is doing some work here, because surpluses are not always necessary to get the debt-GDP ratio down. A simple formula tells us this. It says the primary budget balance compatible with a stable debt-GDP ratio is equal to:

d * [^(r-g)/_(1+g)]

where d is the debt-GDP ratio, r is the long-term real interest rate and g the long-term real growth rate.

So, let's take d as 0.8 (the OBR's forecast for the peak ratio in 2015-16 and g as 0.02, a bit below the OBR's forecast. Long-term real interest rates are now negative, but let's assume they rise to 1%. Then our formula tells us that we can stabilize the debt-GDP ratio with a deficit of 0.78% of GDP.

To see this, imagine the primary balance were zero. Then with growth above the interest rate, GDP will rise by more than interest payments add to the debt, so the debt-GDP ratio will fall.

In this context, Richard Murphy is half-right to say:  

We do not need a surplus: when the government has control of its currency it can run a deficit equivalent to debt multiplied by the interest rate and stand still.

He's right to say we don't need a surplus. But he's wrong to think the government's control of the currency matters in this context; the above formula holds, regardless of whether the government controls the currency or not.

However, the government is going far beyond this. The OBR envisages the primary balance - the balance excluding interest payments - going into surplus after 2016, and hitting 2.9% of GDP in 2018-19. There are three possible justifications for this:

1. As Tim says, a surplus might be necessary for demand management purposes, to cool the economy down. This, though, raises all sorts of questions of whether such management is best done by fiscal or monetary policy.

2. The government fears that interest rates will rise faster than I've assumed. A real rate of 4% - a return to what we have in the 80s - would require a surplus of 1.6% of GDP to stabilize the debt-GDP ratio. This, though, raises questions of whether "secular stagnation" will keep rates low or not. It's in this context that Richard's point about the government being able to print money does matter; printing money to buy government debt might be a way to hold interest rates down, if this is necessary (though this raises other issues).

3.It believes the debt-GDP ratio is too high now, and must fall faster than a primary deficit would permit. Again, though, this poses some tricky questions about whether debt is a burden on future generations and if so whether this is a bad thing.

Now, I don't want or need to take a view on these matters. All I wanted to do is clarify the maths of the relationship between debt and deficits. And this tells us that, at current interest rates, we don't need a surplus to reduce the debt-GDP ratio.

Comments

[TickyW]

Elegant.

Far more elegant than my brutal and clumsy take here:

http://theuxbridgegraduate.wordpress.com/2013/06/02/public-debt-growth-and-fiscal-multipliers/

[Agog]

Justification 4. They think Labour (or Lab/Lib) will be stupid enough to match this aim, run a surplus early in the next parliament, and induce a recession.

[Roy Lonergan]

Chris

So we can run a sustainable primary budget deficit at any debt/GDP ratio so long as g>r. Which means I guess that the effect of running a smaller deficit or a surplus compared to the sustainable deficit is to reduce the nominal debt or to spend on HS2 or whatever (buy bitcoins or tulip bulbs perhaps).

If there is a swifter increase in r then it doesn't change the story too much unless r is close to g and I assume that increasing r would tend to go along with increasing g anyway? But even then I just need to balance the budget rather than run a surplus.

Is that right?

Roy

[Boffy]

"printing money to buy government debt might be a way to hold interest rates down, if this is necessary (though this raises other issues)."

Printing money to buy debt might at best hold the yield down on those bits of debt that are bought, but would probably cause the yield on other debt to rise instead. We see that with the steepening of the yield curve in the US currently.

But, money printing devalues the money that is printed, thereby causing inflation. There has been little inflation in commodity prices, because there has been a massive rise in global productivity over the last 30 years, that has reduced the value of commodities. Instead, there has been a massive inflation of asset prices.

If commodity price inflation rises as a result of the money printing - now likely as productivity gains weaken - then bondholders will fear that the real value of the debt they hold will fall, so they sell it, causing yields to rise.

My guess is that state bureaucrats know that whatever commitment Carney, Yellen or other central planners give about keeping interest rates low, its not in their gift, but a matter of the demand and supply of money-capital. In fact, whenever, Carney has committed to keeping interest rates frozen, market rates have risen as he was speaking!

Interest rates are heading much higher.

[Boffy]

@Roy,

"If there is a swifter increase in r then it doesn't change the story too much unless r is close to g and I assume that increasing r would tend to go along with increasing g anyway?"

r may rise because g increases causing the demand for money-capital to rise. But, r may rise because the supply of money-capital falls to its demand. That would be because the rate of profit, for example, falls.

If r rises, and a lot of the debt is owned by foreigners, then there will be a flow of money-capital out of the economy, which means aggregate demand may also fall. So, rising r may go along not with rising g, but with declining g.