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Doing an econ PhD at one of the top programs is kind of like joining the Jedi order...they want you to start young. Your age may be a disadvantage, at least at the top twenty schools. I echo chateuheart's opinion that operations research may be a nice compromise between economics and your background. I know top programs include Georgia Tech, Harvard, Northwestern.
But, if you really want to give econ phd a go:
1. It is too late for the 2014 cycle. Applications are generally due in December and January.
2. GRE scores are important in that you essentially need to score above ~164 on the quantitative section to be competitive at the top programs and above ~158 to be competitive anywhere. If you are a native English speaker, you should have decent verbal scores (above, say, 150) but generally the verbal section does not matter too much. The analytical section is pretty new and I think some adcoms are using it as a criterion, but I'd say it's not all that important. GMATs are not useful; in fact, most programs do not accept GMAT scores.
3. You have a family, which is both a plus and a minus. I'm not enrolled yet, but based on what I hear, your first year will involve a lot of coursework and you will have little time to spend with your family (let alone sleep) unless you are incredibly talented and can solve your problem sets with ease. This may be tough. At the same time, your family can support you through this very stressful year. Many first-year grad students do not have a spouse or significant other and have little time to make friends, which can make the year pretty depressing. It gets easier/less busy after the first year, though!
4. I will add that it may be a strange financial situation. You're probably well-paid as an engineer; you're going to be getting a relatively measly stipend as a student, so it helps to have savings. This is another advantage you have over the fresh-out-of-college grad students who do not have much money.
5. You need to clear up what your research interests are. "Welfare economics" is very vague and generally has little to do with behavioral economics (though one point of overlap I can think of between public/welfare econ and behavioral off the top of my head is Raj Chetty's work on tax salience). I'd recommend browsing the top journals (American Economic Review, Journal of Political Economy, Econometrica) to get a feel for what's on the cutting edge of economic research. This may help focus (and at the same time flesh out) your interests. Having a good idea of what you want to research can hopefully come through on your Statement of Purpose essay and give you a boost in the admissions process for programs where you're on the fence.
6. You're going to need 2 (or likely 3) letters of recommendation. Generally, these letters come from Economics PhDs who work in academia and actively do research. This is going to be tricky for you, since you've been out of college for a while and it sounds like you didn't take many econ classes to begin with. I'm sort of at a loss for how you're going to overcome these hurdles, but maybe you or someone else could think of something.
In: Princeton, Northwestern, Harvard KSG, Berkeley ARE, Wisconsin, Michigan, Duke, UBC, UC Davis, Arizona Out: Harvard, MIT, Yale, Stanford, UCSD