Let me propose a little thought experiment about the minimum wages in cities like we are seeing more of today.
Assume that minimum wage proponents are correct, and that minimum wages are raised and through whatever mechanism there is no impact on employment. And also assume that, like seems to be the case in many of today's big cities, the supply of housing is pretty fixed due to regulatory supply constraints.
What happens in this scenario? One common way to model local economic shocks, for example Rosen-Roback, is to assume peoples' utility functions in a particular city depend on wages plus amenities less the cost of rent available in that city. One general conclusion from this is that local shocks to supply and demand are capitalized in land prices in equilibrium.
In this case, if minimum wages go up and employment doesn't, then workers will bid up the price of housing, and be willing to accept higher cost of rent. If a landlord knows that all workers are making $15 an hour at the minimum, it's easy to charge more without seeing demand for housing decline. In this case, the gains to workers from a higher minimum wage all get eaten up by housing costs, and landlords are the real winners.
On the other hand, if minimum wages go up and employment does decline, then housing demand will decline as workers move out of the city and price of housing will fall.
However, this doesn't imply that rents will fall enough to make it worth staying for low-skilled workers and they will be no worse off. Instead, a decline of rents and low-skilled employment falling imply that higher skilled workers who aren't affected by the minimum wage increase will move in to the lower rents units.
In fact there is empirical evidence that supports this. In a recent working paper, John Monras uses changes finds that changes in the minimum wage lead to a decline in the low-skilled share of the population. This is different from the typical results that focus only on employment, and instead focuses on the resident population.
Monras argues for the use of spatial equilibrium model to consider the effects as a minimum wage, and I think his work sets the stage nicely for more research on this issue. For now it remains an understudied issue. For example, Monras doesn't look at the effect on minimum wage hikes on house prices or rents. In addition, Glaeser and Gyourko argue that populations decline very slowly in response to negative shocks.If the minimum wage causes low-skilled people to leave a city or state, the full effects could take much longer than the few years considered in most minimum wage studies.
Yes, believe it or not, the minimum wage remains an understudied topic that we continue to learn quite a bit about. It is a far more dynamic area of study than portrayed by those who argue there is no evidence on the other side.
