A common mistake amongst Ph.D. students is to place too much weight on the ability of mathematics to solve an economic problem. They take a model off the shelf and add a new twist. A model that began as an elegant piece of machinery designed to illustrate a particular economic issue, goes through five or six amendments from one paper to the next. By the time it reaches the n'th iteration it looks like a dog designed by committee.
Mathematics doesn't solve economic problems. Economists solve economic problems. My advice: never formalize a problem with mathematics until you have already figured out the probable answer. Then write a model that formalizes your intuition and beat the mathematics into submission. That last part is where the fun begins because the language of mathematics forces you to make your intuition clear. Sometimes it turns out to be right. Sometimes you will realize your initial guess was mistaken. Always, it is a learning process.