In this paper, we consider a single monopoly platform provider which operates both platforms: an old and a new platform. These two platforms connect the user group with the suppliers, thereby leveraging the indirect network externalities in a two-sided market. We also incorporate a cross-platform externality which represents a potential backward compatibility of the new platform: i.e., users joining the new platform can also enjoy the products and services provided by suppliers using the old platform. Users and suppliers are uniformly populated over [0, 1] interval as in the Hotelling model, and play a subscription game to choose (exactly) one platform. The platform determines the pricing profile for the supplier market, and users and suppliers respond to the pricing profile. Our basic analysis for static equilibrium indicates that it is very unlikely that an interior equilibrium is stable. Furthermore, some specific types of boundary equilibriums, where at least one market side tips to a single platform, are stable under certain conditions. We also present a dynamic decision model of the platform provider, which tries to maneuver the markets toward a target state by controlling price profiles. Our analytical results from the optimal control theory assert that a bang-bang control with subsidization for a specific platform eventually leads the market to the corresponding boundary equilibrium. The cross-platform externality plays an important role for a co-existence of competing platforms under a certain condition.