{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Macroeconomics Exam Review 27

Macroeconomics Exam Review 27 - Solutions for Foundations...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
and 𝛼𝑥 + (1 𝛼 ) 𝑦 < 𝛼𝑏 + (1 𝛼 ) 𝑏 Therefore 𝑎 < 𝛼𝑥 +(1 𝛼 ) 𝑦 < 𝑏 and 𝛼𝑥 +(1 𝛼 ) 𝑦 ( 𝑎, 𝑏 ). ( 𝑎, 𝑏 ) is convex. Substituting for < demonstrates that [ 𝑎, 𝑏 ] is convex. Let 𝑆 be an arbitrary convex set in . Assume that 𝑆 is not an interval. This implies that there exist numbers 𝑥, 𝑦, 𝑧 such that 𝑥 < 𝑦 < 𝑧 and 𝑥, 𝑧 𝑆 while 𝑦 / 𝑆 . Define 𝛼 = 𝑧 𝑦 𝑧 𝑥 so that 1 𝛼 = 𝑦 𝑥 𝑧 𝑥 Note that 0 𝛼 1 and that 𝛼𝑥 + (1 𝛼 ) 𝑧 = 𝑧 𝑦 𝑧 𝑥 𝑥 + 𝑦 𝑥 𝑧 𝑥 𝑧 = 𝑦 / 𝑆 which contradicts the assumption that 𝑆 is convex. We conclude that every convex set in is an interval. Note that 𝑆 may be a hybrid interval such ( 𝑎, 𝑏 ] or [ 𝑎, 𝑏 ) as well as an open ( 𝑎, 𝑏 ) or closed [ 𝑎, 𝑏 ] interval. 1.161 Let ( 𝑁, 𝑤 ) be a TP-coalitional game. If core( 𝑁, 𝑤 ) = then it is trivially convex. Otherwise, assume core( 𝑁, 𝑤 ) is nonempty and let x 1 and x 2 belong to core( 𝑁, 𝑤
Image of page 1
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