Abstract
We consider the classical micro-economic foundation of discrete choice, additive random utility models, with conditional utilities depending on expenditure on the numéraire. We show that signs of own- and cross-price effects are identified on the basis of the primal problem only, and Giffen behaviour is ruled out. For the translog specification, we prove that the alternative with highest price behaves as normal good, and the alternative with lowest price behaves as inferior good. We establish conditions for equivalence between the primal and the dual problem. We provide a discrete choice version of the Slutsky equation which, similarly to divisible goods, decomposes the own-price effect into a substitution and an income effect.
Highlights
Classical consumer theory, which deals with divisible goods, does not restrict how demand changes when price changes, since Giffen goods are not ruled out, or how demand changes when income changes, since both normal and inferior goods are contemplated
The aim of this note is to extend to discrete choice, the results on price and income effects that in classical consumer theory are provided by duality
The following proposition shows: 1) that the primal problem is equivalent to a dual problem with a properly defined utility level, and 2) that the dual problem is equivalent to a primal problem with a properly defined income level
Summary
Classical consumer theory, which deals with divisible goods, does not restrict how demand changes when price changes, since Giffen goods are not ruled out, or how demand changes when income changes, since both normal and inferior goods are contemplated. Two key results of duality that reveal properties of price and income effects are the primal-dual equivalence and the Slutsky equation. The aim of this note is to extend to discrete choice, the results on price and income effects that in classical consumer theory are provided by duality. A first presentation of the discrete choice, random utility counterpart of classical consumer theory, was provided by McFadden [6] His approach considers that dependence on economic variables, i.e. prices of the alternatives and income, is through the systematic part of the utilities, and that random terms are independent of observable variables. The latter property defines the class of additive random utility models (ARUMs).
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