The first is that in many older, established democracies, most prime ministers and finance ministers don't have much training in economics. Central bankers, for their part, typically tend to have more expertise, but even here, there aren't nearly as many PhDs as you'd expect. Here's the key chart (click to enlarge if it's too small):
One notable finding here is that countries in the euro zone are least likely to have prime ministers with economics degrees. "We had presumed that membership in an economic union, in particular the Eurozone, would increase the demand for more competent economic policymakers," Hallerberg and Wehner write in an accompanying VoxEu essay. Not so.
On the other hand, the authors discovered that newer, younger democracies like Slovenia are much more likely to slot economists into positions of power — presumably because this is their way of reassuring markets and investors after a period of instability.
Now, there's a key exception here: Countries of all types tend to turn to economists during financial crises. The authors found that "the appointment of an economics PhD as a central bank president is 22 percentage points more likely during a banking crisis." That's not too surprising, though it's possible that some of this is about signaling. The paper finds that left-wing governments are much more likely to turn to formula-wielding technocrats in crises, perhaps as a way of appeasing nervous investors.
In any case, the paper doesn't say much about whether having economists run a country actually leads to better outcomes. It's possible that they don't! In their VoxEU essay, the Hallerberg and Wehner note that "it is not a priori clear that technical competence in itself is a desirable trait." After all, most governments are already staffed with thousands of trained economists. It might be far more important for a leader to be a competent manager with political savvy than to have a lot of technical expertise.
And it seems most older, established democracies do tend to operate this way. It's useful to trot out the economists during crises — as Italy and Greece did in 2011 with Mario Monti and Lucas Papademos. But most of the time, for better or worse, technocrats aren't nearly as highly prized.