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

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.