Some new approximation results for utilities in revealed preference theory

When dealing with consumer demand in economic modeling, researchers often solve the optimization problem which maximises the utility for a given budget constraint. The real data on consumption are used to fit parameters of various a-priori prescribed mathematical functions which demonstrate constant return to scale and thus pose as utilities. An alternative approach to the fitting of a utility function (Eberhard et al. (2007)) allows the raw data to determine the functional form of the utility. This approach is motivated by the desire to allow the data to influence the corresponding price and quantity-price elasticities that are needed to be estimated using real data on consumption preference (Kocoska et al. (2009)) and (Eberhard et al. (2009c)). In econometrics linear regression is used for the computing of elasticities but the approach used in (Kocoska et al. (2009)) is definitely not econometric in this sense as it involves first the fitting of a utility from raw data. However, if we will define econometrics to be the broader activity of fitting economic variables using data then we are definitely employing the econometric approach in the present work. The approach used in (Kocoska et al. (2009)) allows researchers to implement this data fitting procedure by using a very elegant non-linear optimisation algorithm to fit a set of parameters from which the elasticities may be easily deduced using the sensitivity analysis of associated linear programming problems. The purpose of this paper is to provide a theoretical underpinning for the utility fitting or estimation step in this procedure. In practise we only have access to a finite selection of observed consumption data. We do not have access to the hypothesized underlying utility. The problem of fitting a utility function to this finite data set in a way that yields the same preference structure has already been solved by (Afriat (1967)). We observe here that one can fit a concave Afriat utility to any finite sample of consumption data taken from any (not even concavifiable) utility. This provides an approximation paradigm via the indifferent curves so generated. The Afriat utility provides a well defined family of polyhedral indifference curves (irrespective of the arbitrary scaling issue of the utility itself). As we collect more data we may refine our approximation of these level curves and hence the question arises as to whether one can validly discuss some notion of convergence to a limiting preference structure as the data sampled tends to an infinite ”dense” set of points. Indeed is it possible to have a convergence, in some well defined sense, to an underlying utility that rationalises this data set? It is this question we discuss in this paper and provide some concise theory for a positive answer to this question. The critical tool is the adaption of variation limits (Rockafellar and Wets (1998)) to this problem. The reason for the success on this mathematical convergence notion lies in its ability to characterise the set-convergence of indifference curves and to characterise the stability of optimal solutions under objective function perturbations. The main contribution of this theory makes to the literature is that it provides a set of reasonable condition on the demand correspondence that ensures the existence of a utility that rationalises the preference structure and give rise to the observed demand (see Theorem 4). Moreover we can assert that the demand correspondence and the utility can be reconstructed, via the fitting of a sequence of Afriat utilities, using only a countable collection of observations. In this sense this theory provides an existence proof of an underlying utility. We can also give condition that ensure the fitted utility is actually concave improving on the results of (Kannai (2004)).