In my younger years of working it was known as job and finish.I think you miss a point and another really good point.
The first one is you don't pay a person for a day they don't work, you pay them to get their 5 days of work through increased productivity over 4 days. Why would they step up and do that? Because they get a three-day weekend with family or to do other things without a drop in income.
The second one is, if you're smart, you'll have a second employee working 4 days (their days could overlap within a normal working week) under the same arrangement and productivity kpi's, and effectively you get an 8-day week from 2 people, higher production from happier staff and you potentially save 3 days pay because you didn't have to employ a 3rd person. As the number of staff goes up depending on the industry, productivity continues and staff numbers deliver.
Won't work with all industries perhaps but would with most.
Job done to the bosses expectations and you get the rest of the day off
