### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 211-227 |

Number of pages | 17 |

Journal | Taiwanese Journal of Mathematics |

Volume | 15 |

Issue number | 1 |

Publication status | Published - 2011 Feb |

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### Keywords

- Bandwagon effect
- Complementarity
- Economies of scale
- Increasing differences
- Supermodular

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Taiwanese Journal of Mathematics*,

*15*(1), 211-227.

**A functional approach to prove complementarity.** / Kido, Kazuo.

Research output: Contribution to journal › Article

*Taiwanese Journal of Mathematics*, vol. 15, no. 1, pp. 211-227.

}

TY - JOUR

T1 - A functional approach to prove complementarity

AU - Kido, Kazuo

PY - 2011/2

Y1 - 2011/2

N2 - 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.

AB - 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.

KW - Bandwagon effect

KW - Complementarity

KW - Economies of scale

KW - Increasing differences

KW - Supermodular

UR - http://www.scopus.com/inward/record.url?scp=79251637008&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251637008&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79251637008

VL - 15

SP - 211

EP - 227

JO - Taiwanese Journal of Mathematics

JF - Taiwanese Journal of Mathematics

SN - 1027-5487

IS - 1

ER -