General rule of thumb: rent if strongly believe you'll be in a place for less than three years, and consider buying if staying longer.
Say a potential buyer has a mountain of cash. They can scoop up a property with no issue. Financially, owning makes sense to them (no interest payments due to a mortgage), especially if they pick location correctly and see a property value increase.
Next, say a potential buyer has a small pile of cash. A mortgage will almost certainly be needed to finance property, though this buyer should be able to cover the down payment. If location is right, it should work out well for this person, but it will take decades to see a positive return.
Finally, say a potential buyer has little-to-no cash, living month-to-month on rent. They simply cannot afford to purchase a property and/or get approved for a mortgage.
Let's consider each of these three categories of buyer in your scenario, where a local government makes it more difficult to seek rent. The wealthy buyer doesn't care, unless they own many rental properties, in which case they'll possibly have a harder time filling their units. The second buyer will benefit the most, potentially allowing more of this class to purchase properties (theoretically, more properties will be on the market). However, the third buyer is out-of-luck because they simply don't have the financial resources to outright purchase a property. Decrease the number of rental properties, and rents will increase, unless the number of new buyers balances existing renters.
In summary, in your scenario, I would bet the wealthy will sell some properties and cash out, while the middle class will scoop up more properties and be able to pay for higher rents. The poorest, however, will likely need to settle for a lower overall standard of living.