Relaxed Optimization: How Close Is a Consumer to Satisfying First-Order Conditions?

We propose relaxing the first-order conditions in optimization to approximate rational consumer choice. We assess the magnitude of departures with a new, axiomatically-founded measure that admits multiple interpretations. Standard inequality tests of rationality for any given reference class of preferences can be conveniently re-purposed to measure goodness-of-fit with that class. Another advantage of our approach is that it is applicable in any context where the first-order approach is meaningful (e.g., convex budget sets arising from progressive taxation). We apply these ideas to shed new light on existing portfolio-choice data.

The Empirical Content of Binary Choice Models

An important goal of empirical demand analysis is choice and welfare prediction on counterfactual budget sets arising from potential policy interventions. Such predictions are more credible when made without arbitrary functional‐form/distributional assumptions, and instead based solely on economic rationality, that is, that choice is consistent with utility maximization by a heterogeneous population. This paper investigates nonparametric economic rationality in the empirically important context of binary choice. We show that under general unobserved heterogeneity, economic rationality is equivalent to a pair of Slutsky‐like shape restrictions on choice‐probability functions. The forms of these restrictions differ from Slutsky inequalities for continuous goods. Unlike McFadden–Richter's stochastic revealed preference, our shape restrictions (a) are global, that is, their forms do not depend on which and how many budget sets are observed, (b) are closed form, hence easy to impose on parametric/semi/nonparametric models in practical applications, and (c) provide computationally simple, theory‐consistent bounds on demand and welfare predictions on counterfactual budge sets.