Supplementary Appendix for “Weak Monotone Comparative Statics”

Proof. First of all, the nonemptiness of MSpfq follows from Weierstrass’ extreme value theorem. Let us prove that MSpfq is closed, and thus compact. Consider any sequence pxmq with xm P MSpfq, @m, and any limit point x ̊ of the sequence. We must have x ̊ P S since S is compact. Also, the upper semicontinuity of f implies that fpx ̊q ě lim supmÑ8 fpxmq, which in turn implies x ̊ P MSpfq, as desired. By Theorem 2.3 of Li (2014), the compactness of MSpfq implies that MSpfq is chain complete: namely, every chain in MSpfq has a supremum and an infimum in MSpfq. By Zorn’s lemma, it then follows that there are maximal and minimal points in MSpfq. ∗Che: Department of Economics, Columbia University (email: yeonkooche@gmail.com); Kim: Department of Economics, Seoul National University (email: jikim72@gmail.com); Kojima: Department of Economics, Stanford University (email: fuhitokojima1979@gmail.com). Minimal points of S are a set tx P S : x1 ­ă x,@x1 P Su and maximal points of S are a set tx P S : x1 ­ą x,@x1 P Su. Zorn’s lemma states that a partially ordered set S has a maximal element if it satisfies the