Well, why not? Scanner data shows that goods prices are actually quite flexible even in the short-run, contrary to what a "menu cost" argument would have you believe. Financial prices of course adjust at the level of microseconds. So it's not that prices in general are inflexible downwards -- just wage prices.
Bryan Caplan raises this issue, arguing that workers who offer to work for less "sound weird." That sounds about right. But why? Here's what's going on on either side of the labor market relationship:
1. From the employees' point of view: they're aware that a situation with a lot of unemployment generates a market for lemons. The long-run unemployed tend to be different from those who are employed -- they may not even compete for the same jobs. Yet workers have much more private information on their own about their own capacities than employers.
So you have asymmetrical information. How do workers signal to employers that they really are more qualified than all those people in the unemployment rolls that really do lack skills? These sort of labor search questions are a bigger deal during a severe recession, in which you have a greater mix of people with and without usable skills, when you worry about the depreciation of human capital among the long-run unemployed, etc.
If you walk up to an employer and say, "I'll be willing to work for 10% off;" you basically signal to that employer that you are worth less than the other job-seekers queuing up at the door. If employers were able to perfectly measure skills, they might be willing to take a chance on you anyway knowing that they're either getting a good worker at a lower price, or at worst a worse worker at a dearer price. All they can see in reality is that you have some level of ability that's hard to measure -- but you apparently lack enough confidence in your own ability to get a job that you're willing to lower your wage. That's not a good signal. If you have access to many other job-seekers (say, it's a recession and many people are unemployed) it's easier to go for those workers.
2. From the employer's side, you worry about the morale and productivity among all of your current staff. It's very damaging to morale, it seems, to simply cut the wages of current workers -- perhaps they consume many commitment goods, and even small wage cuts result in costly cutbacks among the few categories of discretionary spending that workers have. Maybe they're just psychologically drawn to "make more" over time, and don't like going backwards.
Either way, you worry about the productivity of workers that you get pay on the cheap. Instead, you want to pay wages that are a little above the market wage to grab workers of slightly better quality. Perhaps you manufacture an O-Ring style product in which outstanding effort from all workers is essential to the final product. In that case, you really worry about the maximum productivity of your worst employee. Grabbing minor wage savings from a marginal employee is a second-order consideration relative to lowering company-wide productivity that can happen from offering even a few workers lower than market wages.
Again -- you're not going to be hiring the best workers at a lower wage. The best workers are confident that they can actually get a good wage. The information asymmetry problem here is tough, and prices here are used as signals more than incentives.
There's a huge literature on this subject I'm not familiar with, so I'm sure others have had similar thoughts. This is what I would look to though.