The Chancellor’s budget decision to cap income tax relief has caused a hullabaloo. Wealthy donors and their beneficiaries are in open revolt, saying that charities will be hit hard by the measure. And a lobbying alliance of the wealthy and the charity sector is not something that politicians are likely to defy.

That there will be some cap seems certain. But in the face of this firestorm the government has been curiously reticent in defending its plan. Pointing to the need to curb the minor problem of fraudulent charity giving understandably angered many. There are good arguments for capping donations tax relief, which tend to get less of a hearing. A look at the evidence also points to some ways in which the Chancellor could appease the charity sector while keeping most of his savings.

So what possible justification could there be for cutting tax breaks on giving?

First it’s worth asking the question of how much charities actually benefit from tax relief on donations. This depends how donors respond. If they aim to give a fixed amount of their post-tax income, regardless of government policy, then the charity can expect to get the full value of any tax break that applies.

But what if donors want the charity to get a fixed amount – say a round million pounds? In this case, the availability of a tax top-up might cause them to cut their net donation from what it would otherwise have been. Here the donor benefits but the charity does not. Cutting relief in the first case would hit the charity, but in the second, the total received would be unchanged.

Which of these effects dominates is an empirical question. Several studies [2] suggest [3] that charities get significantly less than £1 for every £1 of tax relief paid out, because people reduce the amount they give in response to the top-up. The evidence isn’t conclusive but a reasonable approximation would be that perhaps two-thirds of tax relief gets to the charity. The residual ends up in the pockets of donors.

So since charities get less than the government spends on tax relief, the state has a dilemma. The cap is expected [4] to save the Treasury up to £100m per year from charity donors. So should it spend that extra £100m on schools or the NHS, services that the electorate as a whole (not just wealthy donors) want to see provided? Or should it reverse its policy and spend that money on tax relief for only £66m to go to privately favoured charities, ranging from famine relief to donkey sanctuaries? The case for doing the latter is perhaps weaker at a time when public services being cut to the bone and ministers lose sleep about the government’s creditworthiness.

Nevertheless, the growing clamour now looks very likely to force some kind of concession from the Treasury. And here the evidence has interesting things to say about how the Chancellor could recast his cap to make sure that government saves some cash and charities maximise giving.

Recent research [5] shows that how tax relief is offered really matters to maximising donations. Where the charity directly claims the tax rebate on behalf of the donor, as with Gift Aid, the scheme looks more like a matching proposition. You give £1 and the government will match it with a further 25p. Under Gift Aid for higher rate taxpayers, the basic rate half of their tax break goes straight to the charity in this way. But they reclaim their rebate on the other 20% - the gap between basic and higher rate income tax - through self assessment.

Field experiments indicate [6] (pdf) that the matching design can wring up to three times as much in donations for every pound spent on the match as the tax rebate version. And this is in spite of the fact that economic theory would suggest that how the tax relief is delivered should have no impact on donor behaviour.

Yet under the government’s current proposal both parts of the donors’ tax relief will be subject to the cap. This makes little sense. The smart move for Mr Osborne would be to un-cap the tax relief that boosts giving while screwing down the cap on the rebate. Both the Big Society and the broke state would be the winners.