A Study of Piecewise Smooth Demand Functions and Pricing Models

A good model of the relationship between the market demand and prices plays a fundamental role in much economic decision-making. However, many decision-making models use either incomplete demand functions which are defined only on a restricted domain, or functions that do not reflect market reality. In the first part of this paper, we explain our motivation behind a complete reasonable definition of demand functions using some examples. Indeed, these examples show that incomplete demand functions may lead to inferior pricing models. Then we formulate the demand functions using a Nonlinear Complementarity Problem (NCP). We will show that such demand functions possess certain desirable properties, such as the law of demand. In the second part of the paper, we consider an oligopolistic market, where producers/sellers are playing a noncooperative game to determine prices of their products. When the model of demand functions 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. MSC 2000 Subject Classifications: 90C33, 90C90, 91B24.