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Unformatted text preview: Microeconomic Theory Econ 101A Fall 2008 GSI: Eva Vivalt Section Notes 3: The Envelope Theorem, More Identities, and The Slutsky Equation 1 The Envelope Theorem Recall: The Lagrange multiplier was interpreted as the shadow price of loosening the budget constraint: V I = * ( p 1 ,p 2 ,I ) Roys Identity: V p 2 = * ( p 1 ,p 2 ,I ) x * 1 ( p 1 ,p 2 ,I ) Shephards Lemma: e p 2 = x c 1 ( p 1 ,p 2 ,u ) The derivations of each of these are all special cases of what is known as the envelope theorem. To illustrate the envelope theorem, let us turn to our usual constrained maximization problem (from week 1) but instead of focusing solely on the endogenous variables ( x 1 ,x 2 ) we would also like to consider an exogenous variable which can potentially affect the objective function f or the constraint function g . (In the context of the UMP or EMP the exogenous variable could be p 1 ,p 2 or I .) Thus, our maximization problem is: max x 1 ,x 2 f ( x 1 ,x 2 , ) s.t. g ( x 1 ,x 2 , ) = k The Lagrangian is: L ( x 1 ,x 2 ,, ) = f ( x 1 ,x 2 , ) [ g ( x 1 ,x 2 , ) k ] and the FOCs: L 1 ( x * 1 ,x * 2 ,, ) = f 1 ( x * 1 ,x * 2 , ) * g 1 ( x * 1 ,x * 2 , ) = 0 L 2 ( x * 1 ,x * 2 ,, ) = f 2 ( x * 1 ,x * 2 , ) * g 2 ( x * 1 ,x * 2 , ) = 0 L 3 ( x * 1 ,x * 2 ,, ) = g 1 ( x * 1 ,x * 2 , ) k = 0 Note: There is no FOC from the partial derivative of the Lagrangian w.r.t. because is an exogenous variable, i.e. it cannot be manipulated by the person doing the optimization. Solving the maximization 1 leads to optimal values of the choice (or endogenous) variables as functions of the exogenous variable, i.e. x * 1 ( ) and x * 2 ( ). Different values of will imply different solutions. Substituting these solutions into the objective function f gives the value function (or indirect utility function for the UMP): F ( ) = f ( x * 1 ( ) ,x * 2 ( ) , ) which is the maximum (or minimum, if the objective is to minimize) possible value of the objective function given a certain value of the exogenous variable . Now we can take the total derivative of the value function with respect to to get: dF ( ) d = f...
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