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mathCompStat3-07-draft

# mathCompStat3-07-draft - ARE211 Fall 2007 LECTURE#26 THU...

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Preliminary draft only: please check for final version 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 (¯ α , ¯ x ) R n × R 1 if f n +1 α , ¯ x ) negationslash = 0, then there exist neighborhoods U α of ¯ α and U x of ¯ x and a continuously differentiable function g : U α U x such that for all α U α , f ( α , g ( α )) = f α , ¯ x ) i.e., g puts us on the level set of f containing (¯ α , ¯ x ) g j ( α ) = - f j ( α , g ( α )) /f n +1 ( α , g ( α )) . 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

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2 LECTURE #26: THU, NOV 29, 2007 PRINT DATE: AUGUST 21, 2007 (COMPSTAT3) Proof again is a trivial exercise in differentiation: since f ( α , g ( α )) f α , ¯ x ) we can take the partial derivative of f w.r.t. α j : d f ( α , g ( α )) d α j = 0 = ∂f ( α , g ( α )) ∂α j + ∂f ( α , g ( α )) ∂x ∂g ( α )) ∂α j rearranging yields: ∂g ( α )) ∂α j = - ∂f ( α ,g ( α )) ∂α j ∂f ( α ,g ( α )) ∂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., { α , g ( α ) : α 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 ( α , g ( α )) 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. Here’s 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 ( α , x ) 0, where f = S - D , α = 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 α = ( t, y ) x = p f ( α , x ) = S ( t, p ) - D ( y, p ) g ( α ) = p ( t, y ) p ( t, y ) tells us how p must change with params to keep us on the level set S ( t, p ) - D ( y, p ) = 0.

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mathCompStat3-07-draft - ARE211 Fall 2007 LECTURE#26 THU...

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