OptExist - Econ 4111 Professor: John Nachbar January 24,...

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Unformatted text preview: Econ 4111 Professor: John Nachbar January 24, 2008 Existence of Optima. 1 Introduction. The mathematics of maximization is the mirror image of the mathematics of min- imization: minimizing a function f is the same thing as maximizing the function- f . Throughout this course, I exploit this basic symmetry and focus most of my discussion on maximization. Consider a non-empty set C and a function f : C R . C is the set of points that are feasible (affordable, physically possible). f is the objective function (utility, profits, social welfare). The goal is to find an x * in C that maximizes f . That is, f ( x * ) f ( x ) for all x C . The basic optimization result, Theorem 2, says that a maximum exists provided C is compact and f is continuous. The contrapositive is that if there is no maximum then either C is not compact or f is not continuous. Suppose, in particular, that C R N . In R N , recall, a set is compact iff it is closed and bounded. In R N , therefore, a maximum can fail to exist for one of only three reasons. 1. C may not be bounded. For example, let C = R and let f ( x ) = x . 2. C may not be closed. For example, let C = [0 , 1) and let f ( x ) = x . 3. f may not be continuous. For example, let C = [0 , 1] and let f ( x ) = ( x if x < 1 , if x = 1 . Theorem 2 gives sufficient, not necessary, conditions for existence of a maximum. For example, suppose C = R and f ( x ) = ( if x 6 = 0 , 1 if x = 0 . Then f is not continuous, C is not compact, but there is a maximum at x * = 1. 2 Existence of a maximum. Theorem 1. Let ( X,d x ) and ( Y,d y ) be metric spaces. For any non-empty set C X , if f : C Y is continuous and C is compact then f ( C ) is compact. 1 Proof. Consider any sequence { y t } in f ( C ). For each y t there is an x t such that f ( x t ) = y t (there may be more than one such x t if f is not one-to-one). Since C is compact, there is a subsequence of { x t } that converges to a point x C : x t k x . Since f is continuous, f ( x t k ) f ( x ). Thus, y t k is a subsequence of...
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OptExist - Econ 4111 Professor: John Nachbar January 24,...

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