micro1 - Economics 603 Microeconomics Larry Ausubel Matthew...

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