Afriat and Revealed Preference Theory

Suppose that we can observe a number of decisions xi (where xi is a non-negative N dimensional vector for i = 1, 2, ..., I) which some decision-making unit has made and let us further suppose that each vector of decisions xi satisfies a linear constraint of the form pTxi < 1 for i = 1, 2, ..., I where pi is a given N dimensional vector which has positive components.3 Given the above framework, we may ask the following question: is the observed set of decisions {xi} consistent with the hypothesis that the decision-maker chose decision xi (for i = 1, 2, ..., I) because it maximized a real valued function of N variables, ., subject to the constraint pTx < 1? A related question is how may we use the observed data {pi; xi} i = 1, 2, ..., I in order to construct an approximation to the decision-maker's true 0, assuming that such a b exists. In Section 3 below, we will give an answer to the above two questions by using the observed data {pi; xi} to construct the coefficients of a linear programming problem [4]. If this linear programme has a positive solution, then it turns out that the observed data does not satisfy the hypothesis of consistency. If on the other hand, the objective function of the linear programme has a solution equal to zero, then we may use the solution to the linear programme to construct a real valued function 0 such that for i = 1, 2, ..., I the vector xi is a solution to the following constrained maximization problem: