Hell yeah. If you're fighting a losing battle to maintain your favorite pre-hair loss hairstyle and having good and bad days (good merely pacifying you and bad seriously bumming you out) trying to conceal it do yourself a favor and shave it and tan yourself. If we're vain enough to care this much about hair loss, then pouring time and money into stuff like getting a tan, going to the gym a lot, eating a seriously healthy diet, taking vitamins, taking extra good care of teeth and being mindful of smelling good all the time should be the way to go. If hair loss is affecting your mood negatively and consistantly channel your negative energy into positive self improvement in every other way. Learn a new language. Learn a to play a musical instrument. Make yourself more interesting. You'll be a better person only with a shaved head.
Note to self: this post will probably be a self fulfilling prophesy