To not pay the cleaner more??

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We currently have a cleaner every week. I want to cut this down to every other week since a) it's expensive and b) we don't need someone every week.

I asked her today if that was ok, and said I'd understand if she wanted to leave us to find someone who could hire her every week. She said she'd stay and do every other week but only if we paid her more.

IMO this is undermining what we set out to do (i.e. save money), plus we'd basically be giving her a salary rise for no reason (since it wouldn't be twice as dirty - I'd do the cleaning in her week off).

DH thinks we should go with it, but I think we're being ripped off.

I'm happy to find someone else so don't mind if she leaves us entirely.

AIBU in refusing to pay her more?

she can find other work in the week you don't need her.
why should you pay more?

Of course YANBU. Like some others have said (although I'm surprised by the proprotion who say YABU), you can only be expected to be paid for the hours you work. It's unfortunate if your hours are cut, but it happens. You (as an employer) can't justify paying someone for something that doesn't need doing.
That said of course, you will also run the risk of losing her if she can find another job, but that's a chance you take.
I like the suggestion ^^ of asking around neighbours / friends to see if anyone is looking for a cleaner to help her out though - a lot of people like the idea of a cleaner but don't get round to the hassle of finding someone reliable. You could be helping out 2 people at once