Relaxed Optimization: Ε-Rationalizability and the Foc-Departure Index in Consumer Theory

We propose relaxing the first-order conditions in optimization to approximate rational consumer choice. Departures from the FOCs are assessed using an axiomatically founded measure that is also interpretable in terms of a money-pump multiplier. The framework encompasses measurement errors, information unobservable to the modeler, and consumer misperception. We develop testable implications for demand data, including for subclasses of regular utility functions, and develop the FOC-Departure Index (FDI), which is applicable in all contexts where the first-order approach is meaningful. We extend these ideas to convex budget sets. Our analysis extends to non-convex preferences under a narrower interpretation of price misperception.