Because Detroit has fewer amenities than, say, San Francisco, its housing prices have to be lower than those in places with better amenities and stronger opportunities. If that weren't true, people would move out of Detroit until housing became sufficiently cheap that there were no longer reason to move from one to the other.
One reason for this is that houses are durable goods. Detroit built up a large stock of housing when the city's industry could support a much larger population. If people took their houses with them when they moved, housing costs wouldn't adjust downwards as much when there were substantial out-migration.
Another reason can be time-to-build in high-amenity locations. Incoming migrants push up the price of the existing stock of housing; that provides a signal to developers to convert more houses into higher density uses and to expand on the fringes of town. High housing costs relative to incomes are then a disequilibrium phenomenon - they're something that happens in the interim until developers are able to get new housing on-stream. And you get a nice price gradient in the standard simple model where housing close to downtown amenities is very expensive to housing farther away, with the slope of the gradient depending on things like the ease of commuting and the desirability of the amenities. At the edges of the city, the cost of a house should be the cost of building it, plus the cost of providing basic infrastructure, plus the underlying base value of land in its next best alternative use.
So median house prices relative to incomes can tell us a few different things. High prices would be associated with strong amenity values. But if there are also very high prices at the city fringes, or if the very high prices persist for a long time, there's also something else going on - something related to building new houses and apartments. You don't get sharp drop-offs in land values at metropolitan urban limits, or apartment buildings built to only 10 stories when another 5 stories would cost less to build than the apartments' sale prices, unless something else is up.
What else can be up? Well, you could be in a place like Hong Kong or Singapore where there is little land available, and where much of the land that can be turned into high-rise apartments already has been. In places with hard physical constraints against further building, benefits of productivity increases or stronger agglomeration effects get capitalised into the price of existing land. Why? Spatial equilibrium: if people are just more productive in those places, that draws in more workers, which bids up the price of housing, which can't really draw in more housing supply, and then confers rents on the holders of existing land.
In other places, it's zoning constraints. Both the metropolitan urban limit and the city height limits seem binding: Arthur Grimes showed strong discontinuities in prices at the MUL, and that apartment buildings are being constrained by height restrictions. Yes, Auckland has physical constraints on land supply too, but there are plenty of places that could be upzoned (and haven't) and land at the boundaries that could be brought into use as suburbs were it not forbidden.
Peter Nunns really doesn't like Hugh Paveltich's median multiplier measure of housing affordability. Nunns says that because it ignores differences in amenities, it's worthless as a measure.
Maybe I'm an easier grader than Peter, but I do find good value in Hugh's measure. Cities that persistently have high median multiplier measures and no particular physical constraints on further building likely have issues in regulations around land supply.
If Detroit implemented strong restrictions against upzoning or new suburbs, it wouldn't show up in a median multiplier measure because the regulations wouldn't be binding - there's a large stock of housing available relative to demand. A high median multiplier does not necessarily follow from bad regulations, nor does a low median multiplier necessarily follow from good regulations in places with binding physical constraints. But a persistently low median multiplier in a city with a growing population likely signals accommodative regulations and a persistently high one in a city without strong physical constraints likely signals regulatory issues.
The measure simultaneously tells us that places like Auckland have highly desirable amenities, and that they have pretty binding regulatory constraints against new building.
