# micro1 - Economics 603 Microeconomics Larry Ausubel Matthew...

This preview shows pages 1–5. Sign up to view the full content.

Economics 603: Microeconomics Larry Ausubel Matthew Chesnes Updated: Januray 1, 2005

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

View Full Document
1 Lecture 1: August 31, 2004 1.1 Preferences Define the set of possible consumption bundles (an nx 1 vector) as X . X is the “set of alternatives.” Usually all elements of X should be non-negative, X should be closed and convex. Define the following relations: : Strictly Preferred , : Weakly Preferred , : Indifferent . If x y then x y , y x . If x y then x y , y x . Usually we assume a few things in problems involving preferences. Rational Assumptions Completeness: A consumer can rank any 2 consumption bundles, x y and/or y x . Transitivity: If x y , y z , then x z . The lack of this property leads to a money pump. Continuity Assumption is continuous if it is preserved under limits. Suppose: { y i } n i =1 y and { x i } n i =1 x. If for all i , x i y i , then x y and is continuous. The continuity assumption is violated with lexicographic preferences where one good matters much more than the other. Suppose good 1 matters more than good 2 such that you would only consider the relative quantities of good 2 if the quantity of good 1 was the same in both bundles. For example: x n 1 = 1 + 1 n , y n 1 = 1 . x n 2 = 0 y n 2 = 100 . Then, n = 1 = (2 , 0) (1 , 100) , 2
n = 2 = (1 . 5 , 0) (1 , 100) , n = 3 = (1 . 33 , 0) (1 , 100) , . . . limit = (1 , 0) (1 , 100) . So we lost continuity in the limit. Desirability Assumptions is Strongly Monotone if: y x, y = x y x. is Monotone if: y >> x, y x. So strongly monotone is when at least one element of y is greater than x leads to preferring y over x . So in the 2 good case, both goods must matter to the consumer. If you increase one holding the other constant, if your preferences are strongly monotone, you MUST prefer this new bundle. With monotone, you only have to prefer a bundle y over a bundle x if EVERY element in y is greater than x . In the 2 good case, increasing the quantity of one good while leaving the other same may or may not leave the consumer indifferent between the two bundles. See graph in notes. [G-1.1]. exhibits local non-satiation if x X and > 0, y X || y - x || < and y x. See graph in notes [G-1.2]. Thus Strong Monotonicity = Monotonicity = Locally Non-Satiated Preferences. Convexity Assumption is strictly convex if: y x, z x and y = z = αy + (1 - α ) z x. is convex if: y x, z x and y = z = αy + (1 - α ) z x. See graph in notes [G-1.3] Of course if preferences are strictly convex, they are also convex. Proposition 3.c.1 (MWG pg 47). If is rational, continuous, and monotone, then there exists u ( · ) that represents . 3

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

View Full Document
Pf: Let e = (1 , . . . , 1). Define for any x X , u ( x ) = min { α 0 : ae x } . Observe that the set in the definition of u ( · ) is nonempty, since by monotonicity, we can choose α > max { x 1 , . . . , x L } . By continuity, the minimum is attained and has the property αe x . We conclude that u ( · ) can be used as a utility funtion that represents . QED.
This is the end of the preview. Sign up to access the rest of the document.

{[ 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