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Both can be costly. If you tear down, you may build it exactly how you want it - and potentially have less problems then trying to create something new out of something old. You never know what could potentially go wrong with an old building. It's kind of like buying an older car - it may be cheaper in the moment, but something may go wrong and have to be replaced after a year or so. If it's a good conditioned older building, then why not renovate it!! However, if it needs a lot of work, and the foundation is pretty shady, then tearing it down and starting from the bottom would logically be the best choice!
I guess it would really depend on what type of home, the age of it and where it's located. Most of your older homes (prior to 1940) have a lot custom details you want to keep intact. If it's really old, and hasn't been maintained, then restoring may not be an economical solution. A lot of the homes in our area (DC Metro) are being torn down because the lots are so valuable.
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