Non-concavity of the Dynamic Optimization of a Proprietary Two-sided Platform

We formulate the dynamic of the platform size from aggregation of the best response dynamic of agents who have different exogenous cost to participate the platform, e.g. transportation costs. Then, concavity in the dynamic is no more thought as standard, because it means just concavity of the cumulative distribution function of the heterogeneous costs. We argue that non-concave (convex) one fits the reality when we think of a shopping center in a rural area as a two-sided platform. Although the non-concavity of the dynamic and the c.d.f does not imply that of the static optimization problem of the platform size and the static optimal platform size can be inferior even in the convex case, the dynamic optimization shows divergence of the platform size and the path can be non-monotonic. We relate two distinctive results from the concave case and the convex case with structural change in logistics of sellers in a shopping center. Further, we interpret the difference between the dynamic optimization and the static one as the the endogenous adjustment cost during disequilibrium process, proposing an “inertial entry game” which is a static approximation of the original dynamic entry game. Nonconcavity of the dynamic optimization means that of the static optimization in the inertia entry game.