Re: A bluewater live-aboard for under $20k
Another economist here. I'm going to suggest a modification to your plan. Why not start more slowly?
Your boat is going to eat your entire budget. Even if you can find a boat in your price range that will get you to the Caribbean, how are you going to eat on your trip? Pay for repairs? Budget for emergencies?
You should also consider your job search. I assume you are looking for degree-related employment. The market for MAs in economics is very thin. Particularly with the states in a budget crisis for the foreseeable future, it may take some time to find a job. I hire new MAs, and most of my candidates haven't completed their degrees yet. You should be budgeting some time and money to find a job.
Why not offer to crew on a longer trip while working on your thesis? That will give you a taste of cruising while minimizing cost and time away from starting your search. You'll also not be single-handing all the way to the Caribbean (if I understand your plans correctly).
The big markets for your degree are in DC and NYC. Both excellent sailing areas. Lots of people work a few years, save up a reasonable budget for a long trip, and take off.
Good luck, whatever you decide to do,