Incentive comes from profit, not just there being a 'problem'. If rent was allowed to naturally increase due to an increase in demand, investors would quickly

Incentive comes from profit, not just there being a 'problem'. If rent was allowed to naturally increase due to an increase in demand, investors would quickly be there to build more appartments. But with rent control, you have little opportunity to offer something unique. It's also harder to compete with appartments that are already close to the center, as they are artifically cheapened. Of course it's made more complex by additional regulation enforcing you can't build up, more than n stories in most european cities.