No, this is a stupid strawman. Productivity is per-capita. Generally estimated via GDP per-capita. At year 0 we have 500 people who want $2000 a month homes.

No, this is a stupid strawman. Productivity is per-capita. Generally estimated via GDP per-capita. At year 0 we have 500 people who want $2000 a month homes. In year 10 we have 750 people who want $2000 a month homes. Without changing the income distribution or economic output per person at all (and thus not changing productivity at all by definition) simple population increase will increase demand at every level, and now we have 250 additional people willing to pay $2000 but no increase in apartments formerly high enough quality to demand $2000, pushing $2000 rents down market. In the meantime there were 1000 people able to pay $1000 and now there are 1500 people able to pay $1000. They are likely to be forced out of their current neighborhood, so while they could only afford $1000 before, they will change their behavior and pack more people in by having multiple roommates or converting common spaces or dens into bedrooms. They still pay $1000 but the space formerly rented for $1000 may cost significantly more. Population increase often happens as a result of more jobs being available. More jobs does not imply an increase in productivity. Honestly you keep saying productivity, I think you mean total economic output normalized to housing available within a reasonable commute radius or something similar to that. In fact for the same economic output, lower productivity can cause higher rents because you can have more people employed who are forced to pay a higher fraction of their income for housing. Higher productivity at the same total economic output would cause houses to be abandoned as people have to leave to find work. This is less stark than it sounds because of service jobs, but it would still happen.