Urban Institute and National Housing Conference recently looked at the financials from a number of housing projects in Denver that sought affordable housing fund loans and tax credits last year. Denver is a relatively good national case study: It doesn't have a white-hot housing market like Washington, but new housing construction there is healthier than in many places, and it's not overly regulated. Denver is like Minneapolis or Charlotte.

There, the researchers posed this common question about housing for the relative poor: "Why is it that the private market just can’t build housing at a cost people can afford?" asked Erika Poethig of the Urban Institute.

For renters who make 30 percent of the local median income, or even 60 percent, new multifamily buildings were simply impossible to build without public help once you factor in the costs of acquiring land, paying designers, constructing buildings, maintaining them and servicing loans. The Urban Institute has built a revealing interactive — based on composites of all of those real Denver deals — that lets you adjust the details to try to make the math work.

Nudge down the design and development fees, pay the construction workers less, drop the interest rate as low as it will go, spend nothing on maintenance, even assume that someone gave the land for free — and the buildings still aren't feasible. A 50-unit apartment is still millions short.

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In the Urban Institute's model, there are really only three ways to solve the equation. You can demand renters pay more than what's considered affordable to them, or 30 percent of their income in rent. Push that assumption closer to 100 percent — assuming the poor don't need much for groceries, or bus tickets, or school clothes or prescriptions — and you may get there. Or you can push the target income for the renters higher, designing a building for families at 80 or 120 percent of the median income -- and that may do the trick, too. But then you're not really building affordable housing for the poor.

Or you can bridge the gap with public subsidies. That's the bottom line.

"Building affordable housing is truly a public-private partnership," Poethig says, "and the private only takes you so far."