204Lecture122009 - Economics 204 Lecture 12Tuesday, August...

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Economics 204 Lecture 12–Tuesday, August 11, 2009 Revised 8/12/09, Revisions indicated by ** and Sticky Notes Inverse and Implicit Function Theorems, and Generic Methods: Section 4.3 (Conclusion), Regular and Critical Points and Values: Defnition 1 Suppose X R n is open. Suppose f : X R m is diferentiable at x X ,andle t W = { e 1 ,...,e n } denote the standard basis oF R n .Th en df x L ( R n , R m ), and Rank df x =d imIm( df x ) ∗∗ =d imsp an { df x ( e 1 ) ,...,df x ( e n ) } =d imsp an { Df ( x ) e 1 ,...,Df ( x ) e n } =d imsp an { column 1 oF Df ( x ) ,..., column n oF Df ( x ) } =R an k Df ( x ) Thus, Rank ( df x ) min { m, n } We say x is a regular point oF f iF Rank ( df x )=m in { m, n } . x is a critical point oF f iF Rank ( df x ) < min { m, n } . y is a critical value oF f iF there exists x X , f ( x )= y , x is a critical point oF f . y is a regular value oF f iF
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then y is automatically a regular value of f ). Remark: The deFnition of regular point and critical point in de la ±uente (as well as in Mas-Colell, Whinston, and Green) is dif- ferent: they use m rather than min { m, n } . I think the deFnition I have given is more natural. If m n , the two are equivalent. If m>n , then since Rank ( df x ) min { m, n } ,theneve ry x X will be a critical point in the de la ±uente and MWG deFnitions, and every y f ( X ) will be a critical value. In the deFnition I have given, a point is critical if the rank is smaller than the largest it could possibly be. **The di²erence matters in the theory of general equilibrium with incomplete markets. The two important theorems (Sard’s Theorem and the Transversality Theorem) con- cerning critical values are true with either deFnition. Example: Consider the function f :(0 , 2 π ) R deFned by f ( x )=s in x Then f 0 ( x )=co s x ,so f 0 ( x )=0fo r x = π/ 2and x =3 π/ 2. Df ( x )isthe1 × 1 matrix ( f 0 ( x )), so Rank df x =Rank Df ( x )=1 if and only if f 0 ( x ) 6 = 0. Thus, the critical points of f are π/ 2and 3 π/ 2, so the set of regular points of f is (0 ,π/ 2) ( π/ 2 , 3 π/ 2) (3 π/ 2 , 2 π ) The critical values of f are f ( π/ 2) = sin( π/ 2) = 1 and f (3 π/ 2) = sin(3 π/ 2) = 1; the set of regular values of f is ( −∞ , 1) ( 1 , 1) (1 , ) Notice that 0 is not a critical value. Given α R , consider the perturbed function f α ( x )= f (
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Notice that f 0 α ( x )= f 0 ( x ), so the critical points of f α are the same as those of f .F o r α close to zero, the solution to the equation f α ( x )=0 near x = π moves smoothly with respect to changes in α ;th e direction a solution moves is determined by the sign of f
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at University of California, Berkeley.

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204Lecture122009 - Economics 204 Lecture 12Tuesday, August...

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