Today's news that CPI inflation has fallen to its lowest rate since October 2009 poses the question: what is the relationship between unemployment and inflation?

The fact that the fall comes after months of falling joblessness is awkward for conventional Phillips curve-type thinking, which says that lower unemployment should lead to higher inflation.

In truth, the latest episode is not unusual in this regard. My chart plots unemployment against CPI inflation in the following 12 months since the Bank of England was given independence in May 1997. It's clear that there's no neat single Phillips curve. In fact, taking all the data together gives us a positive correlation between unemployment and subsequent inflation; it is higher unemployment that leads to higher inflation, not lower.

I suppose you could espy a Nairu at around 5.3% and another at around 7.8%. But this doesn't help us with the current inflation outlook, as it merely poses the question: does falling unemployment mean there's the risk of inflation picking up, or does it mean the Nairu is falling?

There are several possible reasons for this apparently perverse relationship, most of them mutually consistent. Maybe this shows that inflation expectations matter; at a given unemployment rate, higher expected inflation will lead to higher actual inflation. Maybe it shows that shifts in the Nairu are of big practical significance. Maybe (relatedly) it tells us that supply shocks matter a lot; it could be that inflation and unemployment have both fallen because of a mix of lower commodity prices and a pick-up in productivity. Or maybe it means that the idea of an inflation-unemployment trade-off is dubious in an open economy:

In the closed economy there is a unique unemployment rate consistent with constant inflation. By contrast, in an open economy, there is a range of unemployment rates consistent with the absence of inflationary pressure. (Carlin and Soskice, p343)

Whatever the reason, the conclusion is the same. Statements such as "there is every chance inflation will pick up as economic slack is eroded" are more questionable than Phillips curve intuition suggests. Yes, inflation might rise. But you need to do much more than point to falling unemployment to believe it will.

This poses the question: why, then, are such statements so common?

One possibility is that we are unduly influenced by closed economy thinking in which there should be an unemployment-inflation trade-off. Because there's such a trade-off in the US - and because it's plausible that mass unemployment is causing low inflation in the euro area - we tend to think the same must be true in the UK. But it ain't necessarily so.

A more benign possibility is that such thinking is a way of smuggling demand management into monetary policy. The belief (or statement!) that a big output gap would reduce inflation allowed the Bank to slash rates in 2008-09. In truth, the claim "bugger inflation: let's try and save the economy" would have done just as well, but waffle about spare capacity allowed the Bank to appear to reconcile demand management with inflation targeting.

There is, though, a nastier possibility which Michal Kalecki famously pointed out - that the possibility of fuller employment makes many capitalists rather jumpy. But then, Kalecki can't possibly have been right, can he?