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, The function G is homogeneous of degree one if, for all ? > 0, G (?s) = ?G (s)

, (iii) lim i?? |? x i f (x i )| = +? whenever x 1 , x 2 , . . . is a sequence in int (domf ) converging to a point x ? bd (domf ). A pair (int (domf ) , f ) is a convex function of Legendre type if int (domf ) is an open convex set and f is a strictly convex function on int (domf ) that is essentially smooth

, Let f : R J+1 ? R?{+?} be a proper convex function. Assume that f is continuous and essentially smooth.The following five conditions are equivalent 1. x * = ?f (x), x ? int, 1970.

. Sup-z-{x-*-?-z-?-f,

*. =-sup-z-*-{z-*-?-x-?-f-*,

, Assume that (int(domf ), f ) is a convex function of Legendre type. Then (int(domf * ), f * ) is also a convex function of Legendre type. Furthermore, the gradient mapping ? x f is a continuous bijection between int (domf ) and int (domf * ), with a continuous inverse mapping (? x f ) ?1 = ? x * f * , i.e., (? x f ) ?1 (x * ) = ? x * f * (x * ) for x * ? int (domf * ), Let f : R J+1 ? R ? {+?} be a continuous convex function, 1970.

, This is a consequence of the assumption that ? is fully supported on R J+1 , which, e.g., is satisfied by the logit and nested logit models. The following proposition, due to Daly and Zachary (1979) and restated by, their Theorem 3.1, states that Conditions (DZ i) -(DZ v) are necessary and sufficient for consistency with ARUM maximization, p.20, 1992.

, Proposition 5. The ARUM choice probabilities (19) satisfy Conditions (DZ i) -(DZ v). Conversely, any demand satisfying Conditions (DZ i) -(DZ v) can be derived as ARUM choice probabilities

, See Proofs of Theorem 3.1. in Anderson et al. (1992) or of Theorem 3 in Koning and Ridder, 2003.

, that Slutsky symmetry and positive definiteness is weaker than Slutsky symmetry and positivity. Koning and Ridder (2003) show that Slutsky J p ? symmetry and negative semidefiniteness is weaker than Slutsky symmetry and non-negativity (i.e., Equation (20) holds with weak inequality). As an illustration, they consider the simple J + 1 = 2 case (see their Appendix C), Their illustration can be easily extended to show that Slutsky J ? ? symmetry and positive definiteness is weaker than Slutsky symmetry and positivity, 2002.

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