Here is an interesting working paper that assesses the effect of new market-rate housing construction on the low-income housing market:
Increasing supply is frequently proposed as a solution to rising housing costs. However, there is little evidence on how new market-rate construction—which is typically expensive—affects the market for lower quality housing in the short run. I begin by using address history data to identify 52,000 residents of new multifamily buildings in large cities, their previous address, the current residents of those addresses, and so on. This sequence quickly adds lower-income neighborhoods, suggesting that strong migratory connections link the low-income market to new construction. Next, I combine the address histories with a simulation model to estimate that building 100 new market-rate units leads 45-70 and 17-39 people to move out of below-median and bottom-quintile income tracts, respectively, with almost all of the effect occurring within five years. This suggests that new construction reduces demand and loosens the housing market in low and middle-income areas, even in the short run.
This paper is not suggesting that everything will be fine so long as you build lots of new and expensive market-rate housing. But it is suggesting that a “filtering” of housing downward does oftentimes take place.