Jason Schrock, the Chief Economist of Colorado Governor's Office of State Planning and Budgeting sent along this interesting graph.
At first, the relationship seems obvious, but I think it is a bit more subtle than it first appears.
I have previously tried to look at housing expense as a function of things like population change, housing permits, and net domestic migration as a proportion of population. The problem with some of those measures is that some cities have low growth because they have dysfunctional housing policies and some cities have low growth because of some other source of stagnation.
This graph solves that problem by measuring housing affordability against (housing starts / population growth). This is interesting, because after you think about it for a little bit, there isn't necessarily a reason why this would have to be the case. The relationship must be somewhat subtle. In cities that aren't building enough new housing units, there will be behavioral and structural changes about housing usage to utilize the available units more intensively. So, I don't think this is a direct relationship as much as an indirect measure of how those secondary effects of constricted housing supply would show up in the same cities where households are bidding up the existing housing stock. But, even though it is slightly subtle, I think this tells a strong story.
Also, notice that the relationship doesn't exactly look linear. As with the other measures I have been looking at, I think there are three subgroups of cities here. There are the Open Access cities, which do not have a dysfunctionally constrained housing market, and therefore do not tend to have desperate marginal behavior that leads to more intensive usage of housing units. Among those cities, there is no relationship. The regression line is horizontal, or very slightly downward sloping. Then there are the few Closed Access cities that are simply in a class of their own dysfunctionality. Then there are a few cities that are sort of at a crossroads, that might end up as a full-fledged Closed Access city within a couple of decades if they continue to attract workers without expanding their housing availability.
I suspect that if this graph ended at 2005, there would be a few more of those in-between cities, since at the time, some cities, like Phoenix and Las Vegas, appear to have temporarily had more housing demand, either from migration from the high cost cities, or from investor activity as homeowners captured capital gains from the high cost cities and reinvested, than their local housing permitting processes could accommodate.