No, it makes much more financial sense to not work and earn 0 while getting income with begging or criminal actions than work 10h/day for 1/3 of minimum wage that is for sure not enough to survive.
In my country, according to income data provided by the national data collection agency, ~14% of the adult population make less than $10,000 per year. Minimum wage where I am is $14/hr (with some variation across the entire country). Someone working full-time at minimum wage will earn just shy of $30,000 here. Very conveniently, the yearly earnings of that 14% of the population works out to your chosen 1/3 of minimum wage figure almost exactly.
I haven't heard any reports of 14% of adults dying off each year due to not making enough money, and that would be a pretty big news story, so it seems that they have somehow managed to survive. And while the data indicates that some have chosen $0 (~4%), and maybe a life of begging and crime (the data is not sufficient to see that correlation), it seems some dollar value greater than zero is still the preferred option for most even when their earning potential is limited.
I haven't heard any reports of 14% of adults dying off each year due to not making enough money, and that would be a pretty big news story, so it seems that they have somehow managed to survive. And while the data indicates that some have chosen $0 (~4%), and maybe a life of begging and crime (the data is not sufficient to see that correlation), it seems some dollar value greater than zero is still the preferred option for most even when their earning potential is limited.