L.R.
I'm mostly asking questions here because your post says very little about this move and your and your son's relationships with boyfriend and his son. I'm not saying move or don't move, but I've got some questions I'm wondering if you have asked yourself and worked through with your boyfriend:
Have you really thought through this move from the perspective of what is best for him? Saving money is clearly important, no doubt, but is this move the best thing for him and have you prepared him for it adequately?
He will be gaining an instant "brother." How do these boys get along right now? Do they already know each other and get along? Do they have little interaction so they're mostly strangers to each other? How does the older boy treat your younger one now?
Even if it's positive between the boys now -- and I"m assuming here they do see each other now, before you move in -- that could change. The older boy will have a younger kid on his turf, in his space. Your son will lose his familiar home overnight and suddenly become a "sibling" in effect if not in reality. Your son may not be able to articulate this to you yet, but he is plenty old enough to realize on some level that he is being forced (even if you don't feel it's 'forcing,' he will) into a relationship with an adult and child who are not his family. Right now he has you to himself. Overnight he will have to share you, his space, his stuff. It is a HUGE and profound change for him. Is it the right change?
You do not mention some extremely important factors here other than finances. Where do these things stand:
What relationship do your son and boyfriend have right now? Your son may say he likes boyfriend, but that does not mean son is ready to live with him.
Boyfriend may think that "The boys will be playmates for each other, that's great!" and so may you, but that does not necessarily follow and isn't realistic. Living under the same roof is likelier to cause friction than bonding. Ask siblings who fight.
Are you and boyfriend ready to handle it together, on the same page and maturely if the boys do not get along? Every time? Is boyfriend mature and objective enough to say it when his own son is wrong and yours is right? Are you able to do the same with his son?
Does boyfriend have the same values, attitudes and techniques as you on discipline, for instance? If boyfriend is a confirmed spanker and you not -- problem. If you are easygoing and let things be and choose battles, and he is tougher -- problem. If you are the "bad guy" who enforces all discipline but he is the "good guy" -- problem. Will his son accept discipline from you? Will yours accept discipline from him? Will each boy respect the other boy's parent?
How involved will your boyfriend be in decisions about your son--schooling, activities, friendships? And vice versa for your involvement with his son? Or will it be like there are two separate households under one roof because each parent is parenting a child differently?
If boyfriend's son's mom is in the picture, are you ready and williing to be seeing her and interacting with her on a regular basis? Are you prepared for any comments she might make about your moving in? Are you prepared to work with her when it's pickup time or dropoff time and boyfriend isn't around to handle it?
Will your son have to change schools because of this move? You don't mention that. Again, a huge change for a child, especially one who is also undergoing a huge change of an instant family. Do you want to bring two such large changes into his life at once?
It sounds just based on the post like you are looking at the finances and logistics but not at the fact you would be blending families. Unless you and boyfriend are willing to work on being a family and raising both boys the same way with the same, consistent discipline and values, and unless you are willing to make your son feel VERY secure, and unless you are willing to work hard with boyfriend if the boys do not get along -- and at times they will not -- I would not do this but would find any other way to save money. It's NOT about how much you love your boyfriend; it's about what is best for your son.
If you have thought through all these things and just haven't mentioned them in the post, and you and boyfriend have actually talked extensively about your child-rearing attitudes and agree on things -- that's different. But your son still will need help because he still will know he's no longer the center of your attention.