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Economics D10-1: Lecture 6. The Envelope Theorem, Duality, and Consumer Theory: A One Line Proof of the Slutsky Theorem. The logic of the duality to consumer theory:. The Envelope Theorem is used to establish the Derivative Property for value functions in optimizing systems
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Economics D10-1: Lecture 6 The Envelope Theorem, Duality, and Consumer Theory: A One Line Proof of the Slutsky Theorem
The logic of the duality to consumer theory: • The Envelope Theorem is used to establish the Derivative Property for value functions in optimizing systems • The Derivative Property can be used to derive comparative statics results
Corollaries using the Mirrlees’ formulation • The Mirlees’ approach makes it possible to use the SONCs to prove comparative statics results • For objective functions linear in parameters, convexity results are immediate • For objective functions concave in parameters, value functions are concave (convex).
Application: the expenditure function • The expenditure function is the value function from an optimization problem in which the parameters appear only in the objective function. • Its gradient is the vector of Hicksian (compensated) demand functions. • The matrix of cross-partial derivatives of Hicksian demands is negative semi-definite.
Application: Roy’s Law • Differentiate Duality Identity with respect to price. • Use the Derivative Property of the Expenditure Function. • Substitute in another Duality Identity • Solve to obtain Roy’s Law: Walrasian demands are given by the gradient of the indirect utility function divided by the marginal utility of income. • Proof does not require differentiability of Walrasian demand.
Application: The Slutsky equation • Differentiate the Duality Identity for good i with respect to the jth price. • Substitue using the Derivative Property. • Finally, substitute again using the Duality Identities.
