As nitram says the LA do have to find a placement even if it's over their stated rate, however, the LA are bulk purchases so get a better deal than a one off purchaser.
I went into my local small supermarket and Mrs Next Door paid with her loyalty card, 4 pounds off my bill straight away, loyality schemes and bulk purchase get you a better deal, LA's money is council tax money, so ours and the arguement about whether what is essentially a cross subsidy should exist is a different matter.
As for the figure you quote, location, location, location as the TV programme used to be called, plus are we talking care home or nursing home, there is a difference, often quite a considerable difference.
I'll shut up for now. K