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A Remark on Similarity Conditions
Published online by Cambridge University Press: 27 January 2009
Extract
Kramer has shown how singularly restrictive are all ‘similarity’ conditions which guarantee transitivity of majority rule by limiting the family of admissible preference orderings (so-called exclusion restrictions). For suppose, plausibly, that social alternatives are points in an open convex policy space S ⊂ Rn, n ≥ 2, and that voters' preferences, {Rl)1=1…‥ l, are representable by continuously differentiable semi-strictly quasi-concave utility functions ul,. Suppose further that at a single point x ε S, any three voters' utility functions have gradients ∇ul(x), ∇uf(X), ∇uk(x), no one of which can be expressed as a positive linear combination of the other two, and no two of which are linearly dependent. Then all exclusion conditions must fail on S.
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References
1 Kramer, G. H., ‘On a class of equilibrium conditions for majority rule’, Econometrica, XLI (1973). 285–97.CrossRefGoogle Scholar Consult Kramer for the definition of ‘exclusion restrictions’ and of other terms used in this Note.
2 Buck, R. C., Advanced Calculus (New York: McGraw-Hill, 1965).Google Scholar
3 ø1 will be strictly increasing in at least one argument if u i has only a finite number of critical points in N(x 0).
4 Kramer, , ‘On a class of equilibrium conditions’, p. 292.Google Scholar
5 Namely, that x 0 not be a critical point of all / utility functions, and that in any neighbourhood of x 0 no two voters’ indifference surfaces be hyperplanes. This last condition will certainly obtain if all but one u i, is strictly quasi-concave. For details, see P. Wagstaff, ‘Proof of a conjecture in social choice’, forthcoming, International Economic Review.