You can run a 1" belt on a grinder designed for 2" belts....but not vise-versa. I actually have a hook on my belt rack that has nothing but used belts of 220 and finer grits that have been torn down to various widths....from as narrow as 1/4" up to 1'+. The narrower belts are mostly for specific applications....I use them to get into tight areas that I otherwise could not with wider belts. Places like the inside radius of guards, etc.
90% of the time I'm using a 2" belt, but the narrow ones come in handy for those "special" needs.