Each employee has a timesheet (on Google Docs) where they report their hours along with a description of what they spent those hours on. This doesn't allow for fine grained analysis of how effective each worker is (or if they are embellishing slightly), but it's enough to ensure that any substantial miss reporting does not occur.

I'm not sure how things have been done in the past, but as of last week I'm in charge of the document production team and have access to editors timesheets.

I'd also suggest that you are overestimating the variance of speed (holding quality of the product constant) of workers who stick with the team.

Finally, tasks aren't always as clearcut as I think you imagine (not every task is, go convert this document and get it back to me) so that would complicate paying in a non-hourly fashion. Additionally clear cut tasks might vary in difficulty—converting one document might be easier then another—which means assigning a dollar figure isn't a trivial task and comes with it's own costs.

There may be other (notably legal) reasons to prefer to offer hourly wages rather than piece work. And if you're expecting your workers will misreport their hours, you have other problems!