Basic economics dictates that if you introduce a cap that makes letting out a property less lucrative, fewer landlords will choose to let out their second homes.
Rent caps limit how much a landlord can charge tenants in rent. This limits their return on investment on their rental property. With such limits in place, it is likely fewer landlords will buy to let or be willing to invest the time and effort required to lease out a second home. This would mean fewer properties in the rental market, potentially causing a housing shortage. After the 1994 rent cap expansions in San Francisco, researchers found the city’s stock of rental properties depleted by 15% as landlords sold off rental units. 
While this may have played out in San Francisco, this is the exception not the rule. Rent caps introduced in New Jersey and Massachusetts did not lead to a housing shortage. Properties were constructed in the same volume as in cities without rent controls. In New Jersey, the volume of properties on the rental market actually increased, as landlords saw financial benefits in splitting larger properties into multiple units. Studies indicate that the housing and labour markets do not follow the established supply and demand economic models of other industries. Rent controls do not lead to fewer houses on the rental market, nor do they disincentivize the construction of new buildings.
Prices dictate supply. When prices rise, suppliers are incentivized to increase stock. When prices fall, suppliers are de-incentivized to increase stock.
[P1] Rent caps de-incentivize letting out properties. [P2] This causes landlords to sell rather than let. [P3] This reduces the volume of properties available to let. [P4] This is harmful to those who can't afford to buy and drives prices up in the long term. [P5] Therefore, rent caps are not a good idea.
Rejecting the premises
[Rejecting P4] Rent caps do not de-incentivize new builds. Therefore, the same amount of properties are on the market. In some places, rental property stock increased as a result of rent caps.