You quote $180. They come back with "someone else said $140." This happens constantly, and most owners handle it in one of two losing ways: they discount, or they get defensive about the other company.
There is a third way, and it works more often than discounting does.
Do not attack the competitor
The moment you criticise the cheaper business, you sound worried. Worse, the customer has to defend a decision they have not made yet, and people defend decisions by committing to them.
Say nothing about them at all. Talk about what you do, specifically enough that the difference becomes obvious without you naming it.
Make the comparison concrete
"You get what you pay for" is not an argument, it is a shrug. Give them something to compare:
"That may well be the right choice for you. Before you decide, it's worth asking them three things: are they insured, are their cleaners background-checked, and who is liable if something gets broken. Some are and some aren't, and the honest answer is that the price often reflects which."
You have not said a word against anyone. You have handed the customer three questions, and you already know how you answer all three.
Then make yours checkable
This is where accreditation does its work. Anyone can claim to be insured. Almost nobody can point at a dated, independent record:
"Here's our verification page — our insurance, licensing and background checks were reviewed by an accreditation council, with the dates on each. Takes about ten seconds to look at."
Text it while you are still talking. Most competitors at the lower price point have nothing equivalent to send, and the customer discovers that on their own — which is far more persuasive than being told.
Price the risk, not the hours
A cheaper quote is usually cheaper for a reason: uninsured, cash-paid help, no background checks, no cover if a cleaner is hurt in the house. Those are real exposures that land on the homeowner, and most homeowners have never once thought about them.
Say it plainly and without drama:
"If a cleaner is injured in your home and isn't covered, that can become your problem. Our coverage is part of what you're paying for."
Let some of them go
Some buyers want the cheapest price and nothing else. Chasing them costs you margin and they leave the moment somebody undercuts you. Be pleasant, leave the door open, and move on:
"Completely understand — if it doesn't work out, we're here."
A surprising number come back, usually after a no-show or a bad clean, and they come back at your price.
Fix the quote itself
Most of this conversation can be avoided upstream. If your quote is a number in an email, you are inviting a comparison on the only variable you supplied.
Put the inclusions in it. What is covered, what is not, how long it takes, who is coming, that they are insured and checked, and a link to your verification page. Now the customer is comparing a specification against a number — and the $140 quote has to explain itself.
If you do discount, get something
Never drop the price for nothing; it tells the customer your first number was invented. Trade it: a lower rate for a recurring booking rather than a one-off, for a mid-week slot that fills a gap in the round, or for three months committed up front.
That is not a discount. That is a different deal, and it protects the price you quote everybody else.