Only if people actually live in those houses, and they're bought by people living in the cheaper houses right now who then move up in to the new housing stock.

If people buy the new houses as investments and leave them empty most of the year then it has no impact on the rest of the housing market, or if people from outside buy them and move to the city then it just increases the population of the city with no effect on the lower social strata. This is precisely what happens in places like London. There's a boom in house building but it's often apartments bought by foreign investors (allegedly for money laundering... ).

Directly building more social housing would have a far greater impact than trying to affect cheap housing stock by 'trickle down' methods.