I would tend to mostly agree with most who have already commented on your question in that breadcrumbs serve no purpose on the homepage.
Besides the resource already mentioned by @Brian above, baekdal.com includes a relevant point:
Front page breadcrumbs are often a mistake because it only contains "Home" - it is not even linked. Your visitors will most likely be confused about this single word, and not get the idea before they visit a sub page.
Breadcrumbs are a trail back to the homepage, and since you're already there (and you've mentioned it's obvious where you are), it would just be another redundant link to the same page -- taking up unnecessary space.
Note: The one case you could possibly argue for (although not a very good point) is that all sub-pages may have breadcrumbs and thus the layout could be considered inconsistent by not having the same bar on the homepage.