Hogan is right that there are always exceptions, but basically it isn't called "rent". The local promoter (who may be the theater owner or a group renting the space) splits the gross proceeds with the touring company. In the Dark Ages when I ran a local house, the usual split was 75% to the tour, 25% to the local promoter (my boss).
The local promoter provides the space (including "rent", if any) and any local personnel, including box office, maintenance, and the local crew and musicians.
The tour producer pays royalties, provides the sets and costumes, and pays any traveling personnel, including cast, lead crew and musicians.
At our theater, we also split advertising 75/25.
Bottom line: somebody is paying rent, certainly. In the classic model, it is not the touring company, but the local entity controlling the theater.
Hope this helps. I should add the disclaimer that things like non-Equity tours were virtually unknown when I was working in theater management; I'm sure a lot has changed.
(There used to be (and probably still is) another model in which a group of theaters (e.g., the "Starlight" theaters in the Midwest, the "Kenley Players" in Ohio, plus individual theaters such as Papermill and the Miami Beach Theatre for the Performing Arts) where the theaters themselves mount a tour. In this model, we used to pay the usual "local" expenses PLUS a flat fee calculated by dividing touring production costs by the number of participating theaters. For this type of tour, we kept 100% of our gross.)
Updated On: 7/16/16 at 05:44 PM