But I don't think the competition of the official prize with the black market is relevant at all.
Right now a big proportion of exploits come from security researchers, partially because they're looking for exploits, but also because they do have a strong incentive to find and report vulnerabilities. I don't think a cash prize is going to change their calculation much.
The place a prize could make a difference is in ordinary developers. I suspect a lot of bugs are partially discovered multiple times before they're officially reported. Some developer is working with the software, notices some weird behaviour, but doesn't follow up because they lack the incentive. A cash prize increases the incentive and potentially turns some of those dev hunches into new bug reports.
The way the black market comes into play is the devs are competing against the black market. If the bug discovery rate goes up the price of zero-day exploits goes down (since they're shorter lasting) as does the incentive to discover them (since good devs are competing for the same bugs). So you can significantly impact the black hat market without approaching the black hat rate.