## Trying to see how the deficit is a different thing to the debt

I, like lots of people, have a morbid fixation with the state of the nation’s finances at the moment. You often hear arguments about whether it is the debt or the deficit that is the problem. The fact that one is a running total of the other doesn’t mean they are interchangeable. It’s basically the same thing as the difference between your speed and acceleration, over time, one is obviously related to the other, but you can always have one low while the other is high. For many I’m sure it’s not that complicated but some people do seem determine to mix them up.

The deficit is literally how much more the government spends than it takes in taxes. The debt is more or less just how much more the government has ever spent than it has ever taxed. That is, if the government had one big bank account, the debt would be the balance and the deficit would be the sum of the payments less the withdrawals in a given year.

This is a nice simple formula:

$D_y = f(D_{y-1})$

So y is just a number: the year. $D$ is the “debt”, and $D_y$ is the debt in year $y$ and $D_{y-1}$ is the debt in the year before $y$. The function $f$ tells you the deficit – it gives the debt for the next year given the debt for the current year.

Actually, that’s not quite right, what you usually see referred to as the deficit is another function, call it $g$, which is given by:

$g(y) = f(D_y) - D_y$

$g$ is a function of the year $y$, not the current debt like $f$ is. But given the current debt and $f$ you can find $g$, because all it is is next years debt minus this years debt.

You see $g$ plotted all the time, it looks like this:

This is data for the UK government deficit which I got from this handy Guardian blog entry. They even have a chart that looks like a pretty version of the mine! I also got some inflation data here. To be clear what I’ve done is plot the solid red “Deficit/surplus” line by simply adjusting the yearly figure given by the Guardian so that it’s roughly in “todays money”.

The dashed blue “National debt” line is a bit of a weird measure I made up, but it’s just the cumulative sum of those deficit/surplus figures over time, assuming that we started at a debt of £0 just after the war when the data begins. So you can read it as sort of “how much more money the government had this year than they had in 1945 (in todays money)”. Remember that the UK government had a massive debt in 1945, so where the “National debt” that I have made up appears to go positive (i.e. for the whole time until very recently), the UK was still really in some considerable debt, it was just doing better than in 1945. Anyway, the blue line shows a reasonable value of  $D_y$ that corresponds with the plot of $g$.

Anyway, so everyone says, the debt is just the running total of the deficit, so they are related right? And you can see it in the above graph – the debt looks mostly like a delayed version of the deficit, so when there’s a peak in the deficit, slightly later there is a peak in the debt.

But this apparent relation is inevitable if you define the deficit this way, as a function of the year. I decided it would be interesting to view the deficit as the function $f$ rather than $g$. In other words, the deficit really looks like this:

Remember that $f$ just tells you the current debt $D_y$ given last years debt $D_{y-1}$. That relation is shown by the blue dots. Along the x-axis is $D_{y-1}$, and along the y-axis is $D_{y}$. The red line is a sort of debt “equilibrium”, i.e. that is where $D_y = D_{y-1}$, the deficit is zero and the government is spending just as much as it taxes. Being below the line means you are spending more, probably not a great thing, and being above the line means you are taxing more, so at least your finances are ok.

The year is not shown on this chart, because the deficit it not based on the year, it is based on the change in debt (sort of, see notes below).

What can you tell from this chart? Well not a lot actually, the deficit seems quite boring, in that the yearly change is pretty much nothing compared to the range of debts we’ve seen. Right now however, we do seem to be quite a long way below the red line, and that does look bad to me.

I’ve chosen to construct my function f on a year to year basis, but you could use a month long delay, two year delay, five year delay or anything. This makes this technique a little tricky – too short a delay, and you are bound to see only points right next to the red line. Too long a delay, and the relationships might be very complex and not meaningful. At least with the year to year form, we can see that the debt is normally (in the recent past) very close to the line, but isn’t right now.

So anyway, I’m not trying to make points about economic policy, just that the deficit and the debt are really not the same thing. At all. I would like to know if there is any better way to look at this, as I’m not sure that my way has really worked.

Notes:

• For mathematical purists the way I’ve defined my functions above is probably a bit horrifying. $f$ is not really a function because for a given input there is potentially more than one output. And clearly, it does change over time, because the deficit changes over time. I’m really just trying to find a lazy way of expressing what the chart is and the relation to the deficit as we normally see it. Kantz and Schreiber call this a delay representation.
• Don’t forget the position of the origin on the second chart is arbitrary and not important – it’s just the point where by my calculation we have the same debt (not deficit) as we did in 1945, but that isn’t particularly special apart from being the earliest time that it’s easy to compare with.
• Often you see debt and deficit as a percentage of GDP, whereas I’ve done it in “real terms” money. Doing it with GDP tends to reduce the apparent volatility in recent years, because GDP has gone up so much that the amount of money seems less significant. Maybe using GDP is better, maybe it’s not, I won’t comment.﻿