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Complementarity demand functions and pricing models for multi-product markets

Published online by Cambridge University Press:  06 May 2009

WANMEI SOON ,
GONGYUN ZHAO  and
JIEPING ZHANG
WANMEI SOON
Affiliation:
Mathematics and Mathematics Education Academic Group, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore637616 email: wanmei.soon@nie.edu.sg
GONGYUN ZHAO
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore117543 email: matzgy@nus.edu.sg
JIEPING ZHANG
Affiliation:
Judge Business School, University of Cambridge, Trumpington Street, Cambridge CB2 1AG, UK email: jz251@cam.ac.uk
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Abstract

In contrast to single-product pricing models, multi-product pricing models have been much less studied because of the complexity of multi-product demand functions. It is highly non-trivial to construct a multi-product demand function on the entire set of non-negative prices, not to mention approximating the real market demands to a desirable accuracy. Thus, many decision makers use incomplete demand functions which are defined only on a restricted domain, e.g. the set where all components of demand functions are non-negative. In the first part of this paper, we demonstrate the necessity of defining demand functions on the entire set of non-negative prices through some examples. Indeed, these examples show that incomplete demand functions may lead to inferior pricing models. Then we formulate a type of demand functions using a Nonlinear Complementarity Problem (NCP). We call it a Complementarity-Constrained Demand Function (CCDF). We will show that such demand functions possess certain desirable properties, such as monotonicity. In the second part of the paper, we consider an oligopolistic market, where producers/sellers are playing a non-cooperative game to determine the prices of their products. When a CCDF is incorporated into the best response problem of each producer/seller involved, it leads to a complementarity constrained pricing problem facing each producer/seller. Some basic properties of the pricing models are presented. In particular, we show that, under certain conditions, the complementarity constraints in this pricing model can be eliminated, which tremendously simplifies the computation and theoretical analysis.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 5 , October 2009 , pp. 399 - 430
Copyright
Copyright © Cambridge University Press 2009

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