RESEARCH

We study nonparametric welfare identification when observed consumer choices may violate transitivity. Our focus is the income compensation function, which measures the minimum expenditure required to attain a given level of well-being and underlies cost-of-living and standard-of-living indexes. Rather than imposing the Generalized Axiom of Revealed Preference (GARP), we work under the Weak Generalized Axiom of Revealed Preference with efficiency slack (e-WGARP), a minimal consistency condition that rules out only direct pairwise contradictions while permitting nonconvex and nontransitive preferences. Using a preference function framework that accommodates such behavior, we introduce the notion of efficient rationalization under e-WGARP. Within this structure, we derive sharp nonparametric bounds on the efficient preferred set and on the income compensation function for any reference bundle. These results yield partial identification of cost-of-living and standard-of-living indexes without imposing transitivity or convexity. When e = (1, . . . , 1), the framework nests exact WGARP; for e < 1, it delivers strictly tighter welfare bounds than existing GARP-based methods.

Traditional revealed-preference methods rely on convexity to provide informative welfare bounds. Recent work has demonstrated that when preferences are nonconvex, these classical bounds are invalid and can lead to erroneous welfare conclusions, suggesting a fallback to bounds based solely on monotonicity. However, such monotonicity-based bounds are often too wide to be empirically useful. We provide a nonparametric characterization of the joint hypothesis of symmetry and homotheticity and derive sharp bounds on indifference curves and expenditure functions. These restrictions propagate revealed preferences across permutations and scales, systematically tightening welfare sets. We prove that these bounds are generically strictly tighter than those obtained under monotonicity alone. Finally, we show that separability further sharpens identification by enforcing cross-block consistency. Our results provide a robust axiomatic toolkit for welfare analysis in discrete-choice or non-convex settings where traditional methods fail.

This paper develops a geometric theory of scalar regulation in multidimensional environments. Agents choose along multiple margins, while regulators are restricted to scalar instruments such as taxes, standards, or intensity benchmarks. The central insight is that efficient regulation is a projection problem: a scalar policy can affect behavior only along the one-dimensional span generated by its policy gradient. Therefore, efficient implementation requires local alignment between the policy gradient and the planner’s wedge. In competitive environments, this wedge reduces to the externality gradient, while under strategic interaction it combines environmental and strategic distortions. The framework yields impossibility theorems for output-only and attribute-only policies whenever externalities depend jointly on multiple behavioral margins. The paper also derives a welfare-loss formula showing that the efficiency cost of scalar regulation is equal to the squared welfare-metric distance between the planner’s wedge and the feasible policy span, isolating the roles of policy misalignment, planner distortions, and local welfare curvature. The welfare-maximizing scalar policy is characterized as the metric projection of the planner’s wedge into the feasible policy span. Under oligopoly, the framework delivers a constrained-optimal tax formula combining the classical markup correction with a novel attribute-correction term reflecting tax-induced changes in product characteristics.

Homotheticity is a property of preferences in which the proportional scaling of all consumption bundles preserves the same preference ranking. This property simplifies economic analysis as a single indifference curve can represent the entire preference profile of a consumer. This paper investigates the empirical implications of consumer behavior with nontransitive and homothetic preference relations. Using datasets consisting of finitely many observations of price vectors and consumption bundles, we offer a rationalization of  the pairwise homothetic axiom of revealed preference (PHARP) proposed by Varian (1982) without requiring  restrictions on the number of goods. Our work presents an exact analog of Varian (1982)'s theorem on nonparametric tests for homothetic utility maximization applied specifically to PHARP. Moreover, we propose a testable condition for PHARP based on the axiomatic approach of bilateral index number theory. We show that compliance with PHARP is equivalent to the conditions where the Laspeyres quantity and price indexes are greater than or equal to the corresponding Paasche indexes. Finally, we outline a method for recovering valid homothetic preferences from data that satisfy PHARP. We show that applying the method of  Knoblauch (1993) to such data when HARP is not satisfied can yield non-sharp lower and upper bounds.