It's not a good idea to force someone else to do something you think is good for them, because:
(a) they have a right to do what they want
(b) you could well be wrong
(c) and you would not like them forcing you to do things for your own good
In this case, the belief that "vacations increase productivity" is one of those comfortingly counterintuitive Gladwellian nostrums that collapses under close examination.
Counterintuitive: "But vacation means working less, so how could it mean more productivity? Ah, the vacation increases efficiency during working hours to such an extent that the integral of efficiency over time is greater!"
Collapsing under close examination: "Clearly increasing the vacation proportion p to 1.0 would push the working hours (1-p) to zero, reducing integrated efficiency to zero. Thus if an effect exists[1], it would have (at least one) local maximum between 0 < p < 1. And it is not obvious what side of this local maximum we are on, nor whether the location of that local maximum is constant from industry to industry & person to person.
Hence it is not at all obvious that forcibly increasing vacation time would raise the overall efficiency. Indeed, it is highly unlikely that the claim holds in anything like the asserted generality."
QED
[1] To be precise, by postulating that an effect exists, we are stipulating that there exists a p' such that E(p') > E(0), as the efficiency is (supposed to be) greater at that p=p' than at p=0. We also note that no matter how large E(p), the integral of it over time at p = 1 is zero as there are no hours worked. These two observations mean that the integrated productivity is increasing at the beginning & decreasing at the end. The classic such smooth function is a parabola; there may be multiple wiggles depending on exactly how efficiency evolves with productivity.
The first thing the auditors do is go through the HR records, find out who hasn't taken off any vacation time or sick leave for three years. That person is raising purchase orders and signing and counter-signing the approval forms or otherwise doing something so Byzantine that if they leave it for a single day, it will come crashing down on them...
I find I'm most productive if I work for 6-8 weeks, then have a week off, then another 6-8 weeks, etc. It helps clears the mental grogginess that builds up and kills my productivity. So that's about 5-8 weeks a year (but they have to be evenly distributed).
No one (so far as I know) is claiming that it's obvious that taking more vacations (than is normal in the US) increases productivity; only that it's true. Your argument is entirely devoted to demonstrating that it's not obvious, and has nothing to say about whether it's true.
(Is it true? I don't know, though it seems very plausible to me. It's certainly commonly said, and really not all that counterintuitive.)
It's not a good idea to force someone else to do something you think is good for them, because:
(a) they have a right to do what they want
(b) you could well be wrong
(c) and you would not like them forcing you to do things for your own good
In this case, the belief that "vacations increase productivity" is one of those comfortingly counterintuitive Gladwellian nostrums that collapses under close examination.
Counterintuitive: "But vacation means working less, so how could it mean more productivity? Ah, the vacation increases efficiency during working hours to such an extent that the integral of efficiency over time is greater!"
Collapsing under close examination: "Clearly increasing the vacation proportion p to 1.0 would push the working hours (1-p) to zero, reducing integrated efficiency to zero. Thus if an effect exists[1], it would have (at least one) local maximum between 0 < p < 1. And it is not obvious what side of this local maximum we are on, nor whether the location of that local maximum is constant from industry to industry & person to person.
Hence it is not at all obvious that forcibly increasing vacation time would raise the overall efficiency. Indeed, it is highly unlikely that the claim holds in anything like the asserted generality."
QED
[1] To be precise, by postulating that an effect exists, we are stipulating that there exists a p' such that E(p') > E(0), as the efficiency is (supposed to be) greater at that p=p' than at p=0. We also note that no matter how large E(p), the integral of it over time at p = 1 is zero as there are no hours worked. These two observations mean that the integrated productivity is increasing at the beginning & decreasing at the end. The classic such smooth function is a parabola; there may be multiple wiggles depending on exactly how efficiency evolves with productivity.