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> Here the idea is that Seattle would run its own voucher program to subsidize rents for low-income people.

The problem with subsidy programs like that is that the rents then simply go up by the amount of the subsidy.

You're right that subsidy tends to increase prices, but the rate by which that happens isn't fixed the way you describe. Vouchers result in an average increase in buying power. Over the whole population of renters (those eligible and those ineligible), this would be a marginal increase. And yes, it's likely rents would achieve a new equilibrium consistent with that aggregate increase in buying power.

But, on the other hand, the voucher programs would increase the buying power of the absolute neediest people at a greater marginal rate than for the population as a whole. The end result is that poor people's ability to participate in the housing market still improves, in that their ability to meet rent becomes more commensurate with people more affluent than them.

The solution is increasing the supply (which is how you actually drive down prices).

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