Local Housing Allowance meant that 'the taxpayer' gave those in need an amount which was calculated to allow them to rent enough rooms for their family. If they could find a 3 bed for a 2-bed rate of LHA, then well done them, or if they had a need (older child at Uni, etc) for a 3rd bedroom, and could scrape together the extra cash, they could top up the LHA to pay for a 3 bed.
When I was claiming LHA, I was entitled to the 2 bed rate. I looked around, and the options (that I would consider clean and liveable in) for that amount was a 'swanky' city centre bedsit, or an ordinary 3 bed house in the suburbs. There were NO NO! 2 bedroom places for rent for anywhere near the LHA sum. The only 1 or 2 beds were swanky flats, going for twice the sum of LHA. I spent the same amount of 'taxpayers' money', but by careful choosing of my priorities, I lived somewhere that better suited my family.
I needed support for a while. I worked hard. I've been a higher rate taxpayer for longer than I was ever on benefits.
Had the 'bedroom tax' been in place, I would have got less money for my sensible choice of the 3 bed house, but more money had I chose to get into debt, and rent a swanky 2 bed I couldn't afford. Totally barmy.