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housing permits per capita is a convenient source for housing permits. Total annual units permitted each year, for Ann Arbor and for the county:

One "unit" of housing is a house, or an apartment, or a condo, etc., and the census gets these numbers by asking local governments each year about permits issued to build new housing units. Occasionally a permit is issued without the housing actually getting built, so this is a bit of an overestimate, but if it gives some idea of housing growth.

You'll note that there's more building each year in the county than in the city itself--for the obvious reason that it's bigger. If you want to make more of an apples-to-apples comparison, you can look at per capita numbers:

So, even on a per-capita basis, the county has generally gotten more housing development than Ann Arbor. A unit or two per 1000 residents per year just isn't very much.

(Note those zeroes for years before 1992. For some reason the census data set used by doesn't include population for the city until 1992. It might be interesting to get estimates from some other source, though the shape of the 1980-1991 graph wouldn't fundamentally be much different from that of the previous graph, since the population didn't change much through the eighties. Ann Arbor's biggest population increases happened before 1970.)

Anyway, the more interesting is how housing has (more likely, hasn't) kept up with demand. For that it'd be more helpful to compare with new jobs and students.