600 a month to babysit?

My bf is pist because he thinks we don't need to pay my mom to babysit our baby when I go back to work, I feel like he is being cheap. She is 6 weeks old and my mom doesn't work so she will be watching her. I'll be working full time making min and he works full time so I think we're actually being selfish. That's 150/a week! He thinks the whole point of family helping out it's because they are helping out without pay and more because my mom is her grandma and shouldnt have to want to charge us. But like I said my mom doesn't work and it would be nice if we gave her money for herself while she is actually doing US a favor! Daycare will cost more, now he's insisting that I shouldnt work which would be nice but we can't really afford that. He then insists his 17 year old sister to babysit but she is in hs and she shouldn't have to worry about having this responsibility plus I do trust my mom more. Am I wrong? Is 150 actually selfish? My mom has done a lot for us.

*she didn't ask, I just thought it was the right thing to do..my dad got injured at work a few months ago so I know they will be tight on money since my mom doesn't make a income and he isn't getting paid. My bf says this isn't our problem which I understand I just thought it was kind of selfish to not pay at all and I didn't know 600 was a lot I usually hear daycare is expensive so I never looked it up lol 😁

*thank you guys, I appreciate the honest comment and help 300 sounds good.