What would happen if the entire world switched to the 4-day work week?
Specifically would someone's quality of life drop significantly in a developed or a non-developed country?
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Based on the OP's comment directed at @StevenLandsburg, it appears that what we are discussing here is "same $L$abor amount, more $H$eads". So if currently we have $H = L/40$ we will roughly have $H' = (1/0.8)\cdot(L/40) \Rightarrow H' = 1.25H$.
Assuming that there are $25$% more Heads to go to work, this is a redistribution of the same labor income from current earners to non-earners, possibly partly offset by the reduction of unemployment benefits and other transfers that are somehow related to non-earners, which could materialize as lower taxes.
Another interesting aspect here would be the effect on savings: if savings are non-linearly related to income, and then, reasonably expected to be a convex function of it (the higher the income, the higher proportion of it is saved), then such a redistribution would lower savings in favor of consumption (but only partially, since part of this +$25$% would come from households that had already some earners, thus tending to re-instate family income). In any case, lower savings would mean lower available funds for loans and investments. What would that do to capital accumulation, capital level and hence profit maximizing demand for labor?
Also, part of the non-earners that they will now go to paid work, may have been engaged in "non-market", "home production". What will happen with this aspect?
In other words, as comments suggested this needs a fully specified model, even if it is only a highly abstract one, and depending on the assumptions of the model, even within the "usual" variety of assumptions, conclusions will most probably differ.