Structural analysis of optimal investment for firms with non-concave revenue

Qualitative properties of optimal investment strategies for a firm with quadratic costs that faces network externalities in its revenues are analysed. Organising the qualitative information in a bifurcation diagram, it is found that the structure of the bifurcation manifolds is determined by a so-called swallow-tail singularity. This implies the existence of up to three saddle equilibria of the state-costate system, whose local optimal stability for positive discount rates is determined by a three-dimensional bifurcation diagram. Taking a particular two-dimensional section of this diagram, it is shown that the bifurcation curves divide this into 19 different parameter regions; within each region, the corresponding phase systems, and also the optimal solutions derived from them, are structurally stable. In particular, stable configurations with threshold (Skiba) points are found in this way.