Of all the reasons given by optimistic Brexiteers as to why the British government will win concessions from its European counterparts, the one that's least convincing is that the U.K.’s departure will blow a hole in the EU’s budget.

The theory is that the bloc needs the U.K.’s money and, thus, will do everything necessary to prevent it making a sudden and disorderly exit. The EU, so the Brexiteers would have it, is anxious that the U.K. carry on paying fees for continued membership of the single market for two years — an idea that Theresa May floated in her Florence speech two months ago.

Earlier this month, Belgian, Dutch and Spanish newspapers sounded the alarm about the effect of Brexit on their receipt of EU funds that are supposed to help the poorest regions and sections of society. They reproduced numbers from a discussion paper from the European Commission’s department for regional policy, which had posited three post-Brexit scenarios for spending in the regions: the status quo; a reduction of 15 percent; or a reduction of 30 percent.

Champions of Brexit should, however, be wary of exaggerating either the size of the EU’s budget or the extent of the U.K.’s contribution

Now, if Brexit (and Brexit alone) were to trigger a 30 percent reduction in the EU’s regional spending, I could see why Brexiteers might hope to find some of the EU27 governments ready to sue for peace.

Gisela Stuart, a former Labour MP who chaired the Vote Leave campaign, warned on Tuesday that “if the EU loses its second largest net contributor … there may be some things that the EU can no longer do unless other people are prepared to pay more.” (Whatever Stuart thinks, the most recent figures from the Commission put the U.K. as the third largest net contributor to the EU budget, behind Germany and France.)

Champions of Brexit should, however, be wary of exaggerating either the size of the EU’s budget or the extent of the U.K.’s contribution. To do so might lead to delusions about the extent of the EU27’s need for the U.K.

Back in 2013, the member countries set the EU’s budget for 2014-2020 at very close to 1 percent of gross income. A back-of-the-envelope calculation shows that the U.K. contributes about 12.5 percent of the EU’s annual budget, which is about €160 billion — so roughly €20 billion a year.

According to the Commission’s estimates, about €7 billion a year of EU money is spent in the U.K. So that leaves a net flow of around €13 billion a year to the rest of the EU and beyond.

But there are reasons to doubt the U.K.’s financial leverage, buried in the details of the EU’s recent deal on an annual budget. In the small hours of Saturday morning, representatives of the European Parliament and the Council of the EU nailed down an agreement on the precise figure for the EU’s spending next year: €160.1 billion.

The main political battle was over the 2018 budget, which cast into the shade an accompanying agreement to amend the EU’s budget for 2017: a decision to pay back to the member countries some €10.9 billion.

EU rules don’t allow the Commission to carry over a surplus into the next financial year, and the repayment is partly explained by income from the EU’s role as an anti-trust regulator: €3.2 billion came from cartel and anti-trust fines, which was €2.2 billion more than forecast. But the bulk of the repayment (€7.7 billion) stems from a downgrading of the estimates of how much is needed to support regional aid programs and projects. Put simply, the structural funds aren’t being spent as fast as the EU intended or the member countries hoped.

Running out of excuses

This is not a new phenomenon, though it’s not supposed to be the norm. Last year there was an amending budget to pay back €7.3 billion of unspent structural funds. Back then, Commission officials tried to explain it away as a glitch in timing: The member country administrations that are responsible for the vast bulk of such spending were busy winding up projects and programs from the 2007-2013 spending period and were struggling to get up to speed with the new rules governing the 2014-2020 period. Those excuses are wearing thin.

Indeed, the extent to which the bills are still not coming in suggests that in some countries there are serious problems with managing the EU’s programs. Some national administrations are struggling to find projects that meet the EU’s objectives. It could be the conditions attached are off-putting. Perhaps the paperwork required is not worth the financial rewards. The origins and extent of these problems will vary from country to country, but the cumulative effect is that the EU is not spending to support poorer regions and sectors of society on the scale or schedule intended.

It is still likely that spending will speed up in the latter part of the 2014-2020 cycle, though it will be 2023 or later before the last claims are in and the bills paid. However, it is possible, albeit increasingly unlikely, that the EU will spend all the money initially envisaged for structural funds in the 2014-2020 cycle. So the failure to spend the full allocation of money in 2016 and 2017 does not necessarily mean a permanent saving.

But it does set the political context. For a start, it’s already made the negotiations for the 2018 budget easier than normal: It’s hard for the Parliament to make a convincing case for a bigger budget when the bills are not coming in.

As far as Brexit is concerned, the underspending means that the EU and the member countries are not as cash-strapped as some Brexiteers seem to imagine.

The threat to the budget from Brexit is not that a rushed British departure would trigger a cash crisis and bring ongoing EU business to a halt. If there is a danger, it is that the U.K. will refuse — during the negotiations over the divorce bill — to meet its commitments during the 2014-2020 cycle. (For all that May has promised the U.K. will meet its obligations, the nature of those commitments is still a matter of dispute between Michel Barnier and David Davis.) But even if the U.K. does back out, leaving some part of its 12.5 percent contribution to the 2014-2020 program uncovered, the gap will not be felt quickly.

Countries in Central and Eastern Europe that are supposed to receive large allocations from the EU’s budget — notably Poland and Hungary — fear a move to cut programs in the wake of Brexit, but given how much the overall spending is behind schedule, the pain can probably be put off until well after 2020.

The sluggish overall performance of the EU’s internal aid programs is not something that its supporters like to shout about — perhaps because it reflects badly both on the national administrations and the EU. But it may yet provide a cushion against the disruptive effects of Brexit.

Tim King writes POLITICO's Brussels Sketch.