Just as in the realm of politics, in the arts world — and here, particularly regarding the performing arts — different countries adopt different policies depending on their historical and economic circumstances.
The United States, for example, takes a free-market stance on the arts in a similar way to its "liberal" approach to business in general, while in several European countries, including France and Germany, a government ministry or department is closely involved in the operation and financing of the arts. In Britain, it's halfway, with a so-called arm's-length system channeling government subsidies through the Arts Council, while keeping a certain distance from both arts policy and funding from the private business sector.
So what is the system in Japan, I wondered? Well, on its home page the Agency of Cultural Affairs explains its role as: