Games tend to have one of two philosophies on running out of pieces: the Risk philosophy, where pieces are theoretically infinite and you can use any suitable substitute; and the Monopoly philosophy, where components are limited and you cannot take an action requiring a component if the game is out of that component.
In Monopoly itself, the rules say you cannot build more houses and/or hotels than the pieces supplied with the game. Unlimited houses and hotels is a common enough house rule that it is included as an option in many computer game adaptation of Monopoly. Implementing this house rule is fairly easy to do if you have multiple monopoly sets, and even if you don't, the even build rule means that you can indicate the build state of an evenly developed monopoly with 2+ houses by simply putting the houses/hotel on one property in the group.
So why does Monopoly have this rule in the first place? Is this just a historic artifact where early 1900's board game designers thought people would get confused about using substitute pieces or alternative representation of the game state? Is there some compelling strategic depth that this component limit adds to the game?