Cambridge. Why? Both are good enough that you'll NEVER get turned away for interview on the basis of your degree, so forget employment figures - a degree from either of these institutions is fine. Now, we start to look at other factors: academic prestige; international prestige; quality of life/experience.
I think that, although Economics is LSE's flagship course, no one would argue that it has greater (although perhaps it does have equal, in this course) academic prestige than Cambs.
In terms of international prestige, I think Cambridge edges it (think world rankings, the first two UK universites average Joe American can name etc.). All of this so far is pretty even, though.
Quality of experience: this is where I think Cambridge wins hands down. Yes, LSE has that 'London factor' (whatever that is), but Cambridge is a magical place. The university is ancient, full of amazing architecture and fascinating traditions - I'm told you truly feel as though you're becoming part of history, walking in the same grounds that some of the most famous scholars of all time used to walk at your age. Compare this to LSE's notoriously stagnant, overseas student-dominated student body. I think you'd meet more amazing people at Cambridge, too - children of former presidents, that sort of thing. Also, the collegiate system is amazing.
As Knogle and others have said: choose on the basis of where you'd rather spend 3 years of your life as opposed to future prospects; I think you'd graduate as a slightly different person from both of these places.