A note on demand functions with uncertainty
Introduction
As Kreps [15] notes, “the central figure in microeconomic theory is the consumer”, defined as “an entity who chooses from some given set of feasible options”. Economic analysis, therefore, begins with a model of consumer behavior. The theory of demand is developed from these models, and demand functions used in applied economic analysis are based on an underlying model of consumer behavior.
In many applications, however, functional forms for demand are specified without making the underlying consumer behavior model explicit. A particular demand function is guaranteed to be based on a utility-maximizing representative consumer if and only if certain conditions on the partial derivatives of demand with respect to prices are satisfied. These conditions, known as Slutsky symmetry, are described in Section 2.
A key aspect of demand in operations management models, as opposed to traditional models in economics, is the presence of uncertainty. Random-demand functions are sometimes generated from an underlying random-utility consumer model, for example from an “address” (Hotelling-type) model. Most papers in the operations literature, however, simply start with a particular functional form for random demand.
Consider the demand for a product . In its most general form, the demand function with uncertainty can be represented as , where is the price vector and is the uncertainty parameter. In most cases, however, uncertainty is introduced into deterministic demand functions through an “additive” or “multiplicative” uncertainty parameter (see [24]), or a combination of both (e.g. [12], [7], [1]). In the additive uncertainty model, ; in the multiplicative model, . The underlying, deterministic, demand functions, , are usually well-established demand functions from the economics literature.
Demand functions with additive or multiplicative uncertainty do not, in general, satisfy the conditions necessary to guarantee utility maximization by a representative consumer. This issue has largely been overlooked in the literature. Two specific technical problems arise.
First, these demand functions may not satisfy the “Slutsky symmetry” conditions. While it is not necessary for aggregate or market demands to satisfy Slutsky symmetry, these conditions must be satisfied if we assume that demand functions are generated by a “representative consumer” (see [21]). These conditions are routinely imposed on demand functions in applied economics models because, without a clear understanding of the underlying utility model, which the representative consumer model provides, it is impossible to do a meaningful analysis of consumer surplus and total welfare. The inability to perform welfare analysis is a significant handicap for operations researchers working in areas where welfare implications are important, such as in contract theory and antitrust regulation and in other public sector applications.
Second, models with additive uncertainty allow negative demand to occur in some states of the world. Unfortunately, the imposition of non-negativity constraints is not trivial. Non-negativity constraints can create problems with the sufficiency of first-order conditions in monopoly models, and the existence of equilibria in an oligopoly. These challenges have been ignored in some operations models. The aim of this methodological note is to set out more carefully the constraints that must be met in operations models that adopt demand functions as their starting point.
The paper proceeds as follows. After presenting a brief overview of demand theory, I discuss (1) the potential failure of Slutsky symmetry in commonly used demand functions with uncertainty and (2) the problems associated with the imposition of non-negativity constraints in additive demand models.
Section snippets
Overview of classical demand theory
This section presents a brief overview of classical demand theory. For more details, the reader is referred to the microeconomic texts by [15], [21], [28], from which this section draws.
Failure of symmetry in demand functions with uncertainty
In a model with demand uncertainty, note that symmetry has to be satisfied for all realizations of demand. For example, suppose that the demand for product 1 and product 2 is and , respectively. The symmetry condition requires that for every pair of realized random variables ().
In the multiplicative case, even if unless . In other words, even if the
Imposition of a non-negativity constraint on demand
Consider a single-product monopolist (the price-setting newsvendor) who has to choose price and inventory before observing demand. Assume that the firm faces the demand function , which is non-increasing in . Define the “choke” price, , as the price at which demand falls to zero. (Karlin and Carr [14] refer to this as the “null” price.) In traditional demand models, the choke price can be finite (e.g. linear demand; ) or infinite (e.g. iso-elastic demand; ). Note
Discussion and conclusions
For demand functions with additive and multiplicative uncertainty, the failure of Slutsky symmetry makes welfare analysis infeasible. Many models in the supply chain contracting literature, however, rely on these functions. Research based on these models, unfortunately, will be unable to influence public policy discussions about the welfare implications of contracts and will therefore be unable to impact contracting practice. In addition, models with additive uncertainty often ignore the
Acknowledgement
I thank the Associate Editor, Maurice Queyranne and Ralph Winter for several helpful comments and suggestions.
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