By the poor's definition of "cheaper" it is cheaper to rent. By my definition of cheaper it is cheaper to own, in most situations. (It is not cheaper to own if you live in an area/time will falling real-estate prices, or a highly volatile real-estate market. It is better to rent when you are short term in the area, or your needs will change in just a few years.)
The reason I think this is worth arguing is that I hear the comments "the poor can't afford houses" from my politically right leaning friends. The falls into the two narratives on the cause the financial/housing crisis. 1) The housing crisis was caused by policies to make it easier for the poor to buy houses. The poor can't afford to own and this caused the crisis. 2) The crisis was caused by deregulated banks taking on too much risk to make greater profits.
I would like to believe that making it easier for the poor to buy instead of rent can work, with in reasonable guidelines. The reason for that is that it is the best/easiest way to move people out of poverty, build wealth that can be passed down to future generations.
Whether it be it from putting 10 or 20% down payment, home appreciation, or simply paying off that much of the mortgage, once you have equity in the property you have a way to tap a moderately sized pool of credit at a reasonable interest rate. If the home needs repairs because the heat or A/C went out, then you can borrow at a good rate against that equity to pay for the repairs. Then repaying over the next 2 to 5 years at a rate much better than credit card, or predatory lenders. If you have enough equity, you can use it to get a loan to buy a car.
Even if one has no other assets, owning all or part of a home means having something of value to leave to children when you die. This gives the children a leg up on the ladder out of the lower classes. That is why many feel like this is the best way to fight poverty.
It is false to believe that over a reasonable time horizon that it is cheaper to rent than to own. If it were, no one would want to be a landlord. (However there are times and places where this might be true. In a housing price bubble the cost to own might be higher than renting. If apartments have been over built, but homes under built, the market might swing to renting.) The landlord will not rent for less than their cost, unless forced to by weird economic conditions and policies. To do so means they are losing money/wealth, and no one likes that. In normal conditions the landlord is trying to charge at least enough to pay his/her loan on the property, taxes and insurance, to fund savings for future repairs, and make a profit on top of all of that. The difference between the cost to rent and cost to own is the landlord's profit (or loss). A smart landlord is not raising the rent next year to pay for a repair this year; they have been setting aside money above and beyond their costs all along knowing future expenses will come.
All that said, I understand that if you can't save, inherit, or win the money for a down payment, that all of this is a moot point. If you don't have the down payment you "can't afford" to buy, but that doesn't make renting cheaper.