Departures from Slutsky Symmetry in Household Demand Models

Maximisation of utility by a single consumer subject to a linear budget constraint is well known to imply strong restrictions on the properties of demand functions. Empirical applications to data on households however frequently reject these restrictions. In particular such data frequently show a failure of Slutsky symmetry - the restriction of symmetry on the matrix of compensated price responses. Browning and Chiappori (1998) show that under assumptions of efficient within-household decision making, the counterpart to the Slutsky matrix for demands from a k member household will be the sum of a symmetric matrix and a matrix of rank k-1. We establish the rank of the departure from Slutsky symmetry for couples under the assumption of Nash equilibrium in individual demands. We show that the Slutsky matrix is the sum of a symmetric matrix and another of rank at most 2. This result implies not only that the Browning-Chiappori assumption of efficiency can be tested against other models within the class of those based on individual optimisation, but also that the hypothesis of Nash equilibrium in demands has testable content against a general alternative.