The search for a new England manager, new Tory leader and perhaps new Labour leader too brings an under-appreciated question into focus: what exactly can leadership achieve?

In the Times today, Matthew Syed decries the "ludicrous idea that everything would be well if only we could find a new messiah." He's talking about the England manager's job, but the words fit a lot of people's attitudes to the Labour leadership: Tories, I sense, are slightly more sensible.

This messiah complex is what I've called cargo cult thinking, the sort of thing that goes like this:

Leadership

?????

Success.

People don't fill in the ?????. They assume that the new messiah will perform some ju-ju and success will follow. They don't ask the question which the late great Andrew Glyn drummed into us: what's the mechanism?

I'd suggest three broad correctives to this messiah syndrome.

First, remember that what matters is the match, not just the man. Compare Louis van Gaal and Claudio Ranieri. Two years ago, van Gaal had by far the more impressive CV, with league titles in three countries and a Champions League medal. But Ranieri was a glorious success at Leicester whilst van Gaal suffered at Manyoo. The reason? Ranieri turned out to be a great match with City's squad, whilst van Gaal never fitted in.

The point generalizes. Boris Groysberg studied the fortunes of managers who moved from GM - generally regarded as a great training ground for bosses - to other firms. He found (pdf) that where the manager was a good fit for the new company he did well but where he wasn't, the firm suffered. This is despite the managers appearing equally competent beforehand. For example, if a firm needs to cut costs it shouldn't hire a marketing man, but if it needs to manage expansion it shouldn't hire an axeman. As the cliché goes, you need round pegs in round holes.

When we're looking for a leader, we must ask: what exactly is the defect we are trying to address? For England manager, it seems to be a need to overcome the mental block that so often strikes players in big tournaments - what Vincent Kompany calls the "psychological event" that afflicted the team against Iceland.

For Labour, I'd contend, it is a need to unite the PLP and grassroots. This requires emollience, charisma and person-management skills rather than a talent for policy development, because in the economic sphere at least this has been going well.

Whether such good matches are discovered by skilful hiring or by dumb luck is another question.

Secondly, we must remember that, as Nick Bloom and colleagues say, management is a technology (pdf). Those ????? are processes whereby performance is monitored and feedback gathered to enable the aggregation of marginal gains. We should ask of leaders: what processes have you put in place to facilitate improvement? It could be that the best such processes require less "strong leadership" and more decentralized or collegiate decision-making - although such a view is outside the Overton window.

The third principle is to be aware of the force for good and ill of organizational capital. Some organizations are so structurally weak that pretty much no manager can turn them around. As Warren Buffett said:

When a management with a reputation for brilliance tackles a business with a reputation for bad economics, it is the reputation of the business that remains intact.

This could well be England's problem. Fifty years of hurt under all different types of boss hint at deep structural problems; Matthew is right to say that it is the system that has failed. The same might be true of any Labour leader. Given the hostile ideological climate, (alleged?) splits between social conservatives and metropolitan liberals and a declining class base, it might be that nobody can be a truly successful leader now.

You might think this is a post about leadership. It's not. It's about inequality. Matthew says that it's no coincidence that Roy Hodgson was paid more than any other manager at the European finals. He's right. High pay for bosses arises from the messiah complex we have about them - from the superstitious nonsense that the best man will put everything right. Perhaps demystifying management and leadership would be one small step towards more sensible levels of pay and hence towards less inequality.