And all you need to do to make it cheaper is use the money you're saving to employ a bunch of civil servants to determine what the cut off point is (especially as it'll change depending where the person is across the country). Then a campaign to ensure people who should be getting them actually apply for this benefit, in a way that'll make them forget they live in a country that has spent the last two decades labelling anyone who gets benefits lazy scroungers. Then have those civil servants determine who is and isn't eligible to receive the benefits amongst those who apply. Then a bunch more costs in people who deal with appeals. Then a bunch more people in other departments catching the people who inevitably fall through the cracks.
But it might stop one family receiving these benefits when they don't need to, so who cares about cost? Oh you, about ten minutes ago when you were using it to excuse Labour not binning the policy in the first place.