mathCompStat3-07-draft - P r e l i m i n a r y d r a f t o...

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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #26: THU, NOV 29, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT3) Contents 7. Foundations of Comparative Statics (cont) 1 7.3. Implicit function Theorem (cont). 1 7.4. Manipulating first order conditions using the Implicit Function Theorem. 6 7. Foundations of Comparative Statics (cont) 7.3. Implicit function Theorem (cont). Implicit function Theorem: intermediate version : As with the other important concepts in the course, the implicit function theorem can be stated in various degrees of generality. We now go up one level and assume that f has n + 1 arguments. Theorem: Given f : R n +1 R 1 continuously differentiable and ( , x ) R n R 1 if f n +1 ( , x ) negationslash = 0, then there exist neighborhoods U of and U x of x and a continuously differentiable function g : U U x such that for all U , f ( ,g ( )) = f ( , x ) i.e., g puts us on the level set of f containing ( , x ) g j ( ) =- f j ( ,g ( )) /f n +1 ( ,g ( )) . In words, implicit function theorem says that if you have one equation in n + 1 unknowns, you can solve for any one of the unknowns in terms of the other n , provided that... 1 2 LECTURE #26: THU, NOV 29, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT3) Proof again is a trivial exercise in differentiation: since f ( ,g ( )) f ( , x ) we can take the partial derivative of f w.r.t. j : d f ( ,g ( )) d j = 0 = f ( ,g ( )) j + f ( ,g ( )) x g ( )) j rearranging yields: g ( )) j =- f ( ,g ( )) j f ( ,g ( )) x An important feature to note is that the domain of f has one more dimension than the domain of g . Reason is that it is the graph of g , i.e., { ,g ( ) : etc } that recovers the level set. That is, the graph of a real valued function is a set that lives in a Euclidean space one dimension higher than the dimension of the domain of the function. In this case, a point ( ,g ( )) is a point on the level set of f . Example: argued that the solution to any economic system can be represented as the level set of some function. Heres a simple economic model: S = S ( t,p ), D = D ( y,p ), S = D , where p denotes market price, t denotes a tax rate paid by the producer and y denotes consumer income level. The solution to this model can be represented as the level set f ( ,x ) 0, where f = S- D , = t,y and x = p . The level set of f corresponding to zero is the set of all ( price , tax , income ) triples such that the price clears the market for the corresponding values of the exogenous variables. ARE211, Fall 2007 3 Explicitly we have the following relationship = ( t,y ) x = p f ( ,x ) = S ( t,p )- D ( y,p ) g ( ) = p ( t,y ) p ( t,y ) tells us how p must change with params to keep us on the level set...
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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mathCompStat3-07-draft - P r e l i m i n a r y d r a f t o...

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