Following the classic work of Mitjuschin, Polterovich and Milleron, necessary and sufficient as well as sufficient conditions have been developed for when the multicommodity Law of Demand holds. However, far less attention has been focused on the nature and properties of violations. To address these questions, the existing sufficient conditions although simpler in form are of little value unless they are also necessary. We show when the widely cited Mitjuschin and Poterovich sufficient condition also becomes necessary. Using this result, violation regions for the very popular Modified Bergson (or HARA (hyperbolic absolute risk aversion)) class of utility functions, are fully characterized in terms of preference parameters. For a natural extension of the CES member of the Modified Bergson family that is neither homothetic nor quasihomothetic, we create the first simple, explicit example of which we are aware that (i) fully characterizes violation regions in both the preference parameter and commodity spaces and (ii) analyzes the range of relative income and price changes within which violations occur.
Kannai, Yakar, and Larry Selden. "Violation of the Law of Demand." Economic Theory 55, no. 1 (January 2014): 1-28.
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