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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!
Personally, during harsh winters we have a hard time motivating our construction crew to clean up as much as possible. It's really hard when it's negative degrees outside to maintain the perfect construction site.