Ami Pro - LN040611

Ami Pro - LN040611 - Economics 113 Ms Stephanie Fried UCSD...

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Lecture Notes, April 6, 2011 R N , Real N-dimensional Euclidean space Read Starr's General Equilibrium Theory , Chapter 7. R 2 = plane R 3 = 3-dimensional space R N = N-dimensional Euclidean space Definition of R: R = the real line R   +, - ,    closed interval : [a, b] {x| x R, a x b}. R is complete. Nested intervals property: Let x < y and [x  , y  [x , y , = 1, 2, 3, . .. . Then there is z R so that z [x , y , for all . = N-fold Cartesian product of R. R N , x R N x   x 1 , x 2 , , x N x i is the ith co-ordinate of x. x = point (or vector ) in R N Algebra of elements of R N x y   x 1 y 1 , x 2 y 2 , , x N y N 0 = (0, 0, 0, . .., 0) , the origin in N-space = (x 1 -y 1 , x 2 -y 2 , . .., x N -y N ) x y x   y . t R , x R N , then tx   tx 1 , tx 2 , , tx N . If p R N is a price vector and y R N is an economic action, x , y R N , x y N 1 x i y i then p y = is the value of the action y at prices p. n 1 N p n y n Norm in R N , the measure of distance . x x x x i 1 N x i 2 Let . The distance between x and y is . x , y R N x y Economics 113 Prof. R. Starr Ms. Stephanie Fried, UCSD Spring 2011 April 6, 2011 1
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| x - y | = . i x i y i 2 x y 0all x , y R N | x - y | = 0 if and only if x = y. Limits of Sequences x , = 1, 2, 3, . .. , Example: x = 1/ . 1, 1/2, 1/3, 1/4, 1/5, . .. . x 0 . Formally, let . Definition: We say if for any , there is x i R , i 1, 2, x i x 0  0 so that for all . q  q q  , x q x 0   So in the example x = 1/ , q( ) = 1/ Let . We say that if for each co-ordinate x i R N , i 1, 2, x i x 0 n 1, 2, , N , x n i x n 0 Theorem 7.1: Let . Then if and only if for any there x i R N , i 1, 2, x i x 0 is such that for all q  q q  , x q x 0   x o is a cluster point of S R N if there is a sequence x R N so that x x o . Open Sets Let ; X is open if for every there is an so that implies X R N x X   0 x y  y X Open interval in R: (a, b) = { x | x R, a < x < b} are open. and R N Closed Sets Example: Problem - Choose a point x in the closed interval [a, b] (where 0 < a < b) to maximize x 2 . Solution: x = b.
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Ami Pro - LN040611 - Economics 113 Ms Stephanie Fried UCSD...

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