Limiting Functional Forms for Market Demand Curves

Abstract

On the basic assumption that individual consumption of a good is a stochastic phenomenon, the first part of this article shows that under general conditions market quantity demanded is asymptotically (as n, the number of individuals in the market, increases) distributed as normal with and variance a function of own price given all other prices and individual incomes. Next, by the use of integral transforms, it is shown that the unknown market demand function can be approximated by a specific functional form. The estimation problems involved with such a model are discussed in the last part of the paper. Two BASIC, but fundamental, problems facing any econometrician attempting to estimate market demand curves are the choice of functional form and the justification of the normal form for the distribution of the disturbance terms. This paper goes some little way toward meeting both problems. The first step is to regard quantity demanded as a random variable. It is assumed that the axioms of choice of modern demand theory refer to the mean quantities demand curves one can derive the normal distribution as a limiting form for it is assumed that the consumer in determining his preferences determines the parameters of the distribution function of quantity demanded. Market stochastic demand curves are obtained from individual stochastic demand curves by taking the sum of the quantities demanded over all individuals in the market. It is shown that under certain weak assumptions about the characteristics of individual demand curves one can derive the normal distribution as a limiting form for stochastic market demand curves. The limits are taken as n, the number of individuals in the market, approaches infinity. It is shown that under the assumptions of the problem both the and variance of the market stochastic demand function are decreasing functions of own price. The second step in the argument is to obtain approximations for the functional relationships between the and own price and between the variance and own price for the market curve. This is achieved by stating those conditions under which upper and lower bound functions can be defined. The approximations to the actual, but unknown, functions are obtained by considering the convex combination of both bound functions. The last section of the paper discusses the problems involved in estimating the parameters of the limiting form of the market stochastic demand function.