Today’s Euro summit document commits all members to a fiscal rule in which “the annual structural deficit does not exceed 0.5% of nominal GDP.” It also commits to “The specification of the debt criterion in terms of a numerical benchmark for debt reduction (1/20 rule) for Member States with a government debt in excess of 60%.”

I’m on the record as being in favour of numerical benchmarks for debt reduction. Indeed, I argued for a more stringent one than the one-twentieth rule that has been proposed by the Commission and has been adopted today.

However, I wonder whether those proposing the limit of 0.5% of nominal GDP on the structural deficit have thought about what this implies for debt ratios. I take this proposal to mean that the average deficit, going through the cycle, should not be more than 0.5% of GDP.

Now suppose a country consistently ran a deficit of exactly 0.5% of GDP. What would happen to its debt-GDP ratio?

Here’s a little note describing the dynamics of the debt-GDP ratio in a simple world with a constant deficit ratio, d, and a constant growth rate of nominal GDP, g. It shows that the debt to GDP ratio converges over time to (1+g)*d/g. (One could add random fluctuations in the growth rate or the deficit ratio and then the debt ratio would cycle around this long-run average value. Also, the timing assumption could be changed so that the current-period debt is determined by last period’s deficit, in which case the (1+g) would dissappear, but that wouldn’t make much difference to the calculation.)

Let’s assume a modest long-term growth outlook for the Euro area of 3 percent nominal GDP growth, i.e. 2 percent inflation and 1 percent growth in real GDP. In this case, the long-run implication of a 0.5 percent of GDP deficit ratio is a debt-GDP ratio of 1.03*0.005/0.03 = 0.172.

Since 0.5 percent of GDP is to be a maximum for the average deficit, this is a fiscal rule that would see long-run debt-GDP ratios below 17 percent of GDP in all Eurozone member states.

This is, of course, a long way from where we are now in most member states. Many countries currently have excessive debt ratios and there is a need to get debt and deficit ratios down over the medium term. It would take a very long time for countries like Ireland to end up with this very low debt ratio, so these limits may work fine as a medium term rule for high debt countries.

However, taken on its own merits, this rule doesn’t seem to make much sense as a long-run legally binding rule. As an alternative, an average deficit of 1.5 percent of GDP could combine with a nominal growth rate of 3 percent to produce a stable and manageable average debt-GDP ratio of 51.5 percent. This would seem like a more sensible benchmark.

Of course, if a government followed such a policy, normal cyclical fluctuations would likely take the economy above a 3 percent deficit fairly often without in any way jeopardising long-run fiscal stability. So the elevation of a three percent deficit limit to sacred cow status (“As soon as a Member State is recognised to be in breach of the 3% ceiling by the Commission, there will be automatic consequences unless a qualified majority of euro area Member States is opposed”) has little grounding in the actual economics of fiscal stability.

There is little doubt that Europe needs to act to reduce debt levels over the medium term and better institutional fiscal frameworks are required. However, these rules, however much they may appeal to the Swabian housewife instinct, are overly restrictive and have little connection to fiscal arithmetic. They are all the more likely to be flouted in future because of their poor design.