Macroeconomics Exam Review 95

Macroeconomics Exam Review 95 - Solutions for Foundations...

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

View Full Document Right Arrow Icon
2.142 Analogous to Exercise 2.129, there exists a simplex 𝑇 containing 𝑆 and a retrac- tion of 𝑇 onto 𝑆 , that is a continuous function 𝑔 : 𝑇 𝑆 with 𝑔 ( x ) = x for every x 𝑆 . Then 𝜑 𝑔 : 𝑇 𝑆 𝑇 is closed-valued (Exercise 2.106) and uhc (Exercise 2.103). By the argument in the proof, there exists a point x 𝑇 such that x 𝜑 𝑔 ( x ). However, since 𝜑 𝑔 ( x ) 𝑆 , we must have x 𝑆 and therefore 𝑔 ( x ) = x . This implies x 𝜑 ( x ). That is, x is a fixed point of 𝜑 . 2.143 𝐵 = 𝐵 1 × 𝐵 2 × . . . × 𝐵 𝑛 is the Cartesian product of uhc, compact- and convex- valued correspondences. Therefore 𝐵 is also compact-valued and uhc (Exercise 2.112 and also convex-valued (Exercise 1.165). By Exercise 2.106, 𝐵 is closed. 2.144 Strict quasiconcavity ensures that the best response correspondence is in fact a function 𝐵 : 𝑆 𝑆 . Since the hypotheses of Example 2.96 apply, there exists at least one equilibrium.
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