How much charity giving is just a subsidy to advertising?

One of us has been rather surprised, after perhaps a decade of not paying attention to the medium, to find that the Americans can actually make reasonably good TV shows. The resultant binge watching has led to exposure to the standard British advertising associated with such habits. A bit of which raises an interesting question.

How much of the government subsidy to charitable giving actually just ends up as a subsidy to such advertising? 

As we understand it this is not about gift aid, the return of tax that has been paid upon money that is then donated. Rather, upon the advertisements (from the likes of Save the Children, Action Aid and so on) there is a claim that, up to a certain amount, the government will 100% match donations made. So, when a viewer texts in with a £2 a month donation there is a match to that from public funds.

Well, OK. More taxpayer money goes to those organisations which, by their actions, people think should have more money. That's better than some committee somewhere deciding upon the allocation. And yet, and yet.....

Standard theory tells us that some amount of this matching money will end up not as an increase in the amount spent upon alleviating famine, plague and destitution, but as an increase in the advertising being done and thus a subsidy to the TV stations.

Imagine, sans subsidy, a charity pays 50 pence (purely made up numbers) in order to gain a £2 donation. That's £1.50 more to be spent upon alleviation. Now add in that subsidy. It makes sense for the charity to be paying up to £3.50 on advertising in order to gain that £2 donation plus the matching £2. For the net effect upon their funds is still that £1.50 increase in resources.

Theory tells us that some of this will almost certainly happen. Over time that it certainly will. What theory doesn't tell us is how much of this will happen. It is possible that it will be a little tiny bit at the margin and that the net effect is a substantial increase in the funds allocated to famine, plague and destitution relief. It is also possible that the arms race to advertise to gain donations will swallow up all of that matching funding, in fact theory tells us that net funding allocated to famine etc could actually fall.

Note the similarity here to the tax incidence argument - who pays the cheque isn't necessarily the same as who bears the economic burden. We are discussing much the same thing here. That more cash is nominally allocated to plague relief doesn't mean that more gets there - it's entirely possible that it ends up instead in larger advertising budgets and thus in the coffers of the TV stations. 

We don't know the answer here but we insist that it's an interesting question. How much of that matching charitable donations by government actually ends up as a subsidy to advertising to gain donations? Theoretical answers range from fractionally above 0% to over 100%. But what does reality say? And shouldn't we find out given that it is we taxpayers coughing up for this?