As mysteries go, it doesn’t rank with Murder on the Orient Express or the identity of Gerald the mole in Tinker Tailor Soldier Spy but the economics profession is currently gripped by its own drama.

This story won’t make it to the silver screen with starring roles for Kenneth Branagh or Judi Dench, but a lot depends on how it ends. For central banks, in particular, the question of Who Killed the Phillips Curve? is a nail-biter. They badly want to know how it ends.

Here’s the plot. Back in the 1950s a brilliant London School of Economics professor called Bill Phillips looked at UK wage and unemployment data stretching back to the 1860s and found a clear relationship: as unemployment rates fell wage growth picked up; as the dole queues lengthened pay growth eased back. Phillips plotted a century’s worth of data and came up with a nice smooth curve.

Q&A What is inflation and why does it matter? Show Hide Inflation is when prices rise. Deflation is the opposite – price decreases over time – but inflation is far more common. If inflation is 10%, then a £50 pair of shoes will cost £55 in a year's time and £60.50 a year after that. Inflation eats away at the value of wages and savings – if you earn 10% on your savings but inflation is 10%, the real rate of interest on your pot is actually 0%. A relatively new phenomenon, inflation has become a real worry for governments since the 1960s. As a rule of thumb, times of high inflation are good for borrowers and bad for investors. Mortgages are a good example of how borrowing can be advantageous – annual inflation of 10% over seven years halves the real value of a mortgage. On the other hand, pensioners, who depend on a fixed income, watch the value of their assets erode. The government's preferred measure of inflation, and the one the Bank of England takes into account when setting interest rates, is the consumer price index (CPI). The retail prices index (RPI) is often used in wage negotiations.

Policymakers loved it because it meant a choice from a menu of options: they could have a bit less unemployment if they were prepared to tolerate a tad more inflation. Conversely, they could ease cost of living pressures by nudging the jobless total up a bit.

The first attempt to kill off the Phillips curve came in the 1970s, and then there was no mystery about who fired the shot. Milton Friedman said that accepting higher inflation as a way of reducing unemployment was a pointless strategy because wage bargainers would build the assumption of a higher cost of living into their pay demands. Inflation would ratchet upwards without any fall in unemployment. The stagflation of the 70s – high unemployment and high inflation at the same time – seemed to prove Friedman right.

But the Phillips curve was not killed off that easily. New models of the economy were developed incorporating the Friedman critique, which drew a distinction between the long-term and the short-term. In the long-term, it was accepted that Friedman was right: any attempt to reduce unemployment below its “natural” rate would lead to ever-higher inflation. But in the short-term the Phllips curve relationship held, provided consumers and businesses were confident that inflation would be kept low.

Central banks loved the idea of the short-term Phillips curve and it became central to their models. If unemployment was higher than its natural rate, central banks could stimulate demand without any risk of breaching their inflation targets. When all the slack in the labour market was used up, they would remove the stimulus. The 1990s and the early 2000s were the heyday for fine-tuning the economy using quarter-point increases or cuts in official interest rates.

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The financial crisis and global recession of 2007-09 brought this golden age to an abrupt end. Factories closed, trade dried up and unemployment rose. Central banks cut interest rates aggressively, aware that if they didn’t do so they would risk under-shooting their inflation targets. Deflation was seen as the real threat.

A second Great Depression was averted and unemployment started to fall. But here’s where things started to get tricky because the central banks had to decide when and how quickly to remove the stimulus they had provided in late 2008 and early 2009 and their Phillips-curve models were not much help. As unemployment rates have come down, central banks have waited for wage growth to pick up. A decade on they are still waiting.

The past week has provided two examples of the dilemma policymakers are facing. In the UK, the Bank of England raised interest rates because it thinks earnings growth is bound to pick up with unemployment at a 42-year low of 4.3%. In the US, the unemployment rate is even lower, at 4.1%, and the markets believe the Federal Reserve will need to raise interest rates four times next year to prevent inflation rising above its 2% target.

But in neither the UK nor the US is there any hard evidence of rising wage pressure. Average earnings growth in Britain is stuck at just over 2% and the Bank’s own agents – the eyes and ears of the monetary policy committee – said in their latest report that labour costs per employee were subdued. In the US, more than 250,000 new jobs were created last month but average hourly earnings were unchanged.

Put simply, central banks are relying on their Phillips-curve models to set interest rates at a time when the Phillips curve relationship appears to have broken down. When Mark Carney arrived at the Bank of England in 2013 he thought wage pressure would only start to build when unemployment fell below 7%. In the same year, the American economist Robert Gordon said the Fed would not be able to reduce unemployment below 6.5% without inflation picking up.

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Gordon is one of the economists who finds it hard to contemplate a world without the Phillips curve. Without a correlation between unemployment and inflation, he said in his 2013 paper, the Fed would not be able to calculate the natural rate of unemployment or the amount of slack in the economy. “In such a world, the Fed would be operating like a captain of a giant ocean liner operating in a fog, with no instruments to warn of icebergs to the left or to the right.”

This looks like a pretty accurate summation of how central banks have been operating these last few years. It is clear that their estimates of the natural rate of unemployment are stabs in the dark. Five years ago when the actual level of unemployment was 8%, the Bank of England thought the natural rate was around 7%. Now that the actual rate is 4.3%, it thinks the natural rate is 4%. But who’s to say unemployment can’t be pushed to 3% or 2% before wage pressure builds? It seems entirely plausible that a combination of globalisation, weaker trade unions and a deep recession mean that economies can operate at much lower levels of unemployment without triggering inflation.

Trust us, the Bank says. There is now very little slack left in the economy and once that is used up inflation will start to rise. But estimates of spare capacity are no more reliable than estimates of the natural rate of unemployment.

It’s not hard to see why central banks are so wedded to the Phillips curve. As the economist Gavyn Davies said recently: “Without the [Phillips curve], the whole complicated paraphernalia that underpins central bank policy suddenly looks very shaky. For this reason, the PC will not be abandoned lightly by policymakers.”

Rather like the supposed murder victim in an Agatha Christie book, the Phillips curve might simply be feigning death. At some point the old unemployment-wage inflation relationship may again show up. But for the time being central banks are winging it. They need to tread carefully.

