I'm not sure if I got the exact word for it (english isn't my first language) so I'll call it the "anchor price" effect.

Basically, if you are the one who mention the first number, you can set the "anchor" of the conversation. This mean that if you ask for 55$ first and they try to offer you 40$ You can focus on the fact that it's around 30% lower than your asking price.

On the other hand, if they mention 40$ first, you are the one who has to convince them to give you 30% more.

PS : I'm not saying you are wrong, just bringing another point of view on that.

That's probably true in some cases, like trying to land a contract with new client.

I am giving this advice based on the context that this person has a fairly good idea of what this company pays their employees & contractors, since he already has a relationship with them.