Concave Rationalization with an Ideal Point: An Afriat Theorem and an Application to Survey Design

This paper develops an Afriat-type characterization of concave rationalization with an unknown ideal point. We show that a system of Afriat inequalities - where the unknown peak enters as a virtual observation with the highest utility - is necessary and sufficient for the existence of a continuous concave utility with an ideal point that rationalizes choices from linear budget sets anchored at different corners of the choice space. A stronger characterization adds the requirement that supergradients at observed choices point coordinatewise toward the peak, a necessary condition for single-peaked rationalizability. The resulting peak-oriented Afriat system provides the basis for a Houtman--Maks consistency index that measures the largest fraction of observations jointly rationalizable with a common ideal point. This characterization provides the theoretical foundation for the Priced Survey Methodology (PSM), in which respondents complete the same survey under different linear constraints. A parametric single-peaked specification then sharpens identification into estimates of ideal answers and importance weights. We apply the PSM to study political preferences in a sample of French respondents.