lecture_10.pdf - Monotone Comparative Statics Econ 2100...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Monotone Comparative Statics Econ 2100 Fall 2015 Lecture 10, October 5 Outline 1 Comparative Statics Without Calculus 2 Supermodularity 3 Single Crossing 4 Topkis°and Milgrom & Shannon°s Theorems 5 Midterm
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Comparative Statics Without Calculus Remark Using the implicit function theorem, one can show that if there are complementarities between choice variable x and parameter q , the optimum increases in q . First Order Condition : f x ( x ; q ) = 0 . Second Order Condition : f xx < 0 . By IFT x ° q ( q ) = ° f xq ( x ; q ) f xx ( x ; q ) : Then x ° q ( q ) ± 0 if and only if f xq ( x ; q ) ± 0 Here g ± is the derivative of g with respect to ² . Issues with implicit function theorem: 1 IFT needs calculus. 2 The conclusion holds only in a neighborhood of the optimum. 3 The results are dependent on the functional form used for the objective function. 1 In particular, IFT gives cardinal results that depend on the assumptions on f .
Image of page 2
Monotone Comparative Statics Objectives With monotone comparative statics, we seek results about ±changes² that: do not need calculus are not necessarily only local (around the optimum). are ordinal in the sense of being robust to monotonic transformations. The objective is to get a similar result without calculus. The downside is that the results are not as strong. Main Idea: Complementarities We want to generalize the notion of complementarities between endogenous variable and parameters. With calculus, this is the assumption that f xq ( x ; q ) ± 0. We would also like to account for the possibility that the optimum is not unique so that x ° ( q ) is not a function. What does it mean for a correspondence to be increasing?
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Strong Set Order When can we say that one set is larger? De°nition For two sets of real numbers A and B , de³ne the binary relation ± s as follows: A ± s B if for any a 2 A and b 2 B min f a ; b g 2 B and max f a ; b g 2 A A ± s B reads ± A is greater than or equal to B in the strong set order². Generalizes the notion of greater than from numbers to sets of numbers. According to this de³nition f 1 ; 3 g is not greater than or equal to f 0 ; 2 g . This de³nition reduces to the standard de³nition when sets are singletons.
Image of page 4
Non-Decreasing Correspondences De°nition We say a correspondence g : R m ! 2 R is non-decreasing in q if and only if q 0 > q implies g ( q 0 ) ± s g ( q ) This says that q 0 > q implies that for any x 0 2 g ( q 0 ) and x 2 g ( q ) : min f x 0 ; x g 2 g ( q ) and max f x 0 ; x g 2 g ( q 0 ) . Generalizes the notion of increasing function to correspondences. Exercise Prove that if g ( ² ) is non-decreasing and min g ( q ) exists for all q , then min g ( q ) is non-decreasing. Exercise Prove that if g ( ² ) is non-decreasing and max g ( q ) exists for all q , then max g ( q ) is non-decreasing.
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Monotone Comparative Statics: Simplest Case Set up Suppose the function f : R 2 ! R is the objective function; this is not necessarily concave or di/erentiable, and the optimizer could be set valued.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern