A draw the budget set budget c l 100 l 9 1000 10 l b

Info icon This preview shows pages 2–3. Sign up to view the full content.

View Full Document Right Arrow Icon
a ) Draw the budget set. Answer: 0 50 100 Budget Set 0 50 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c l 100 - l/ 9 1000 - 10 l b ) Does the consumer’s problem have a solution? That is, can utility be maximized over the budget set? Explain. Answer: Yes, the consumer’s problem has a solution. It suffices to show the budget set is compact since the Weierstrass Theorem will imply there is a solution. The budget set is the interesection of four sets: { ( c, l ) : l 0 } , { ( c, l ) : c 0 } , { ( c, l ) : c 1000 - 10 l } , and { ( c, l ) : c 100 - l/ 9 } . Each of these sets is the inverse image of a closed interval under a continuous function, and so each is closed. As the intersection of closed sets, the budget set is closed.
Image of page 2

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

View Full Document Right Arrow Icon
MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 8, 2001 Page 3 Since l 0, c 100 - l/ 9 100. Since c 0, 10 l 1000 - c 1000, so l 100. The budget set is contained in [0 , 100] × [0 , 100] (see the diagram), and thus is bounded. Since it is both closed and bounded, it is compact. By the Weierstrass Theorem, the consumer’s problem has a solution. Note: Polygonal budget sets of this type arise when consumers face progressive taxation, where the number of sides depends on the number of tax brackets. The phase-out of welfare benefits as income increases has a similar effect.
Image of page 3
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