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Unformatted text preview: Economics 603: Microeconomics Larry Ausubel Matthew Chesnes Updated: Januray 1, 2005 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 nonnegative, 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 6 = 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. [G1.1]. • exhibits local nonsatiation if ∀ x ∈ X and > 0, ∃ y ∈ X 3  y x  < and y x. See graph in notes [G1.2]. • Thus Strong Monotonicity = ⇒ Monotonicity = ⇒ Locally NonSatiated Preferences. Convexity Assumption • is strictly convex if: y x,z x and y 6 = z = ⇒ αy + (1 α ) z x. • is convex if: y x,z x and y 6 = z = ⇒ αy + (1 α ) z x. See graph in notes [G1.3] • 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 ....
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 Spring '11
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