Revealed Preference Theory (Econometric Society Monographs) by Christopher P. Chambers & Federico Echenique

Author:Christopher P. Chambers & Federico Echenique [Chambers, Christopher P.]
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2016-01-04T22:00:00+00:00

The events C(π) form a partition. Define ν({π}) to be the probability of C(π). We can explicitly calculate ν as follows. Let hk = jk+1 − jk − 1 and υkπ = ∑l=1ku(xlπ). Then,

where the next-to-last equality follows by distributing the product ∏ z∈X u(z) appropriately, and using that ∑ l=1|X|u(xlπ) = 1.

Finally, we have already observed that the probability of drawing x before any other alternative in A is equal to u(x)∕∑y∈A u(y) = PA(x). Note that this probability also equals ∑{π:π(x)≥π(A)} μ(C(π)) = ν({π : π(x) ≥ π(A)}). Hence ν rationalizes (X,P).

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