What's the cheapest way to reduce sound transference between floors?

Our neighbors below complain constantly about sound transference through our floors and their ceiling. We’re arguing about what to do. Is it cheaper to remove their ceilings or our floors to add sound blocking materials? What are our options?

Cut holes in the drywall between the ceiling joist, and have an insulation company spray foam insulation in to the cavities between the ceiling joists.
Save the pieces of drywall you cut out, you can use those later to patch the ceiling with.

It’s cheaper to remove the ceilings, insulate, and use quietrock (https://www.quietrock.com/) to minimize foot-fall and other noise coming from upstairs.

Removing floors is more costly than sheetrock but there are sound deadening floor underlayments that can also be used to deaden sound transfer between floors.

If the ceiling is gypsum board (sheetrock), than the least expensive way to solve the problem would be to remove the ceiling below and insulate with any number of materials made specifically for their sound deadening properties. I have personally used products from Roxul with good results (and quite inexpensive)

If the ceiling is plaster, this could get quite messy and expensive as lathe and plaster is quite labor intensive to remove.

Just to make you are aware of potential issues, opening the ceiling or floor and exposing plumbing and/or electrical can in some jurisdictions require you to correct any issues found. You may want to ask what rooms are the most egregious to the neighbor downstairs and fix that area first.

If you’re looking for a remedy that needs very little change and/or contracting, there are 3 options that stand out in my past experience with this problem.

  1. Stop wearing shoes in your place. No shoes, just barefoot, socks or soft slippers. This is an enormous improvement to your neighbors over shoes.

  2. Area rugs. Added to the fact that you stopped wearing shoes, this is a great help.

  3. Cork. We’ve used interlocking cork pieces and they have deadened the sound transfer of music, footsteps, and other noise nuisance problems.

None of these steps are going to guarantee the cessation of sound to your neighbors, but trying all three would help enormously, with little cost/s involved.