I don’t think it’s that big of a difference between different cities. I personally don’t think it’s expensive to raise a child. My kids wear hand me downs. The new clothes they get are ones when I can find great deal for (i.e. ~$5/t-shirt and <$10 for jeans/shirts/jackets). We reused the same car seat/stroller for both kids, then we'll be giving it to my SIL. We got hand me down crib too before it fall apart and we have to buy a new one. Most of their toys are gifts. So, the materialistic things are pretty cheap.
However, private schools, piano lesson, swim lesson, various other activities are expensive. These items though are all luxury. So, if you don't have money, you don't have to do these luxury things. So, you can raise your kids on the cheap if you wanted to. But it can be quite expensive if you want to give them all the advantages you can afford.