A New Look at the Symmetry of the Slutsky Matrix

The Slutsky matrix function encodes all the information about local variations in demand corresponding to small (Slutsky) compensated price changes. The Slutsky matrix is symmetric when the demand function results from utility maximization. However, symmetry does not imply rationality. Here, we provide a new necessary and sufficient condition for Slutsky symmetry. The new condition, closely inspired by Slutsky’s original motivation, requires equal monetary compensations after substitution-effect shufflings. Compared to the Ville axiom of revealed preference, which also characterizes symmetry, the new condition focuses on paths where the real income is kept constant in the sense of Slutsky.