Shepherd, we tried something similar over a year ago but only for selected products and there was a noticeable improvement in rankings for those product pages which has remained to this day.
We undercut everyone by a mile and no competitors responded (although many have price match guarantees) so we still remain the most competitive on these products.
Things to note:-
1) Rich snippets show our product price which, being much cheaper, may have improved our CTR from Google for these products (which I believe is a powerful ranking factor).
2) Bounce rate improved for these products. I think this combined with a great CTR from Google is a potent combination.
3) Being ultra competitive for these products did generate more links, buzz, etc. than for other products.
Our conclusion was there were several factors potentially influencing the outcome which were all a consequence of the price reductions. My gut feeling is the improved CTR and bounce rate were the main factors responsible for the initial ranking improvements (within a week or two) and then links provided a longer term boost.
Unfortunately we couldn't afford to do it across the board so the effect was limited to these product pages. If we could be this competitive on everything I think the whole site would have seen a huge improvement in rankings.
Ultimately though what we did was make ourselves a little bit more unique.