lecture_10.pdf

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

• Notes
• 20

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

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

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

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 .
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?

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

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.
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.

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

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.
This is the end of the preview. Sign up to access the rest of the document.
• Fall '08
• Board,O

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern