A functional approach to prove complementarity

Some complementarity among a firm's activities is an important source of its profits. In this paper, we focus on the way to prove complementarity. Though there are many studies on complementarity as supermodularity or the increasing differences of a function, we introduce the notion of self increasing differences with respect to a single activity, which is an essence of convexity from the viewpoint of complementarity, and investigate some interrelations among these three notions of complementarity. Mathematically, we give a sufficient condition for a composite function to have self increasing differences. This proposition is deeply related to Topkis' (1998) Lemma 2.6.4 on sufficient conditions for a composite function to be supermodular. Both propositions are combined and applied to yield and/or strengthen complementarity in an organization, which will also disclose the functional structure of an organization's activities.