Learning from Revealed Preference

A sequence of prices and demands are rationalizable if there exists a concave, continuous and monotone utility function such that the demands are the maximizers of the utility function over the budget set corresponding to the price. Afriat (1967) presented necessary and sufficient conditions for a finite sequence to be rationalizable. Varian (1982) and later Blundell et al.(2003, 2005) continued this line of work studying nonparametric methods to forecasts demand. Their methods do not implement any probabilistic model and therefore fall short of giving a general degree of confidence in the forecast. The present paper complements this line of research by introducing a statistical model and a measure of complexity through which we are able to study the learnability of classes of demand functions and derive a degree of confidence in the forecasts. In this paper we develop a framework to study the learnability of real vector valued demand functions through observations on prices and demand. Our results give lower and upper bounds on the sample complexity of PAC learnability and show that the sample complexity of learning a class of vector valued functions with finite fat shattering dimension increases by a linear factor of the dimension. We show that classes of income-Lipschitz demand functions with global bounds on the Lipschitz constant have finite fat shattering dimension.