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Interesting, although none of this has to do with overhead, and by extension, my question...
"overhead" is that $1000/week I pay Joe. I'd love to understand this plan a little more, but so far no one is explaining the economics of it to me.
You are right in saying you have two option, pay Joe based on work output (i.e. decrease his pay for the 4 widgets he now produces or take less profit.).
But there's more to you then just Joe and the widgets. There's the market buying the widgets, now everyone is working four days, has the demand for widgets changed? If this is a market where the demand is increased, you need more widgets, you can sell them at a higher price and hire more workers to increase output. If the market has now shrunk, Joe's reduced output is fine.
It took a lot of effort to reduce the work week down from around 80 where it was at the beginning of the century and from the purely economic perspective of "we will have less output and the country will fail" it didn't. Businesses did and guess what, it's likely because they weren't viable to begin with. Most workers on minimum wage currently can't survive an unexpected expense, the current system isn't paying enough to begin with. I'd argue lots of business right now should fail, because they aren't being run for those actually making the widgets. Joe is burning his labour for cash and the output is widgets. In this scenario, what are you doing to earn the $20? Supplying Joe the chance to make widgets? Is that worth $20 from Joe's labour?
Joe now has 72 hours off to rest and use his time more wisely. Joe uses this time to further his education follow a career in a different field other than making widgets. This may not make him more productive but he is happier. Eventually he will leave with his further education and move onto something more fulfilling, or just using the extra day to spend more time with his kids or doing whatever recreation he really loves. He will work on having a fulfilling life. You will move production to Bangladesh and bunch of people you are paying $5 a hour will die in a widget factory fire.
I own the equipment that produces the widgets. I pay the insurance policy that covers Joe getting injured at work. I pay for distribution and marketing of the widgets. If Joe feels like taking on all these responsibilities, he is free to quit his job and start a competing widget manufacturer.
What happens if Joe doesn't show up for work?
No... $1000/week is Joe's salary. That's payroll. Overhead is things like utilities at your workshop/factory.