04-MS&E-241-Welfare-Demand-Theory-II-Lecture

04-MS&E-241-Welfare-Demand-Theory-II-Lecture -...

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MS&E 241: ECONOMIC ANALYSIS Thomas A. Weber 4. Demand Theory (Cont’d) Winter 2007 Stanford University Copyright © 2007 T.A. Weber All Rights Reserved -2- MS&E-241-Winter-2007-TAW AGENDA Remarks On Comparative Statics and Regularity (for reference only) Demand Theory (Cont’d) Key Concepts to Remember
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-3- MS&E-241-Winter-2007-TAW CONSTRAINED PARAMETRIC OPTIMIZATION Problem : Find the set ) , ( max arg ) ( ) ( t x f t x t X x = Objective function Variable Parameter Constraint set Maximizer ) ,..., , ( 2 1 n x x x x = t ) ( t X t ) , ( t x f Standard assumptions : Variable : n-dimensional vector Parameter : scalar (or from an ordered set) Constraint set : “continuous” in and compact-valued Objective function : continuous in and t x -4- MS&E-241-Winter-2007-TAW EXISTENCE OF A SOLUTION Theorem : Under the standard assumptions , a solution to the parametric constrained optimization problem does exist . Proof (Outline): For any given value of the parameter , continuity of the objective function on the compact constraint set implies that the set is bounded (i.e., its supremum is finite ) Furthermore, by continuity of this set is also closed (i.e., the supremum is achieved and is then referred to as maximum ) Any argument that achieves the maximum lies in the maximizer (i.e., ) The above holds for any parameter , which concludes the proof. ) , ( max arg ) ( ) ( t x f t x t X x = t ) , ( t f ) ( t X {} ) ( : ) , ( ) ( t X x t x f t F = ) , ( t f ) ( t X y ) ( max ) , ( : ) ( ) ( t F t y f t X y t x = = t
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-5- MS&E-241-Winter-2007-TAW REGULARITY OF SOLUTION Maximum Theorem (Berge, 1963): (1) Under the standard assumptions, let and Then 1. Maximizer is compact valued and upper semicontinuous in 2. Value function is continuous in Proof: See Berge (1963, p. 116). Intuition : The maximizer “switches” via indifference points The value function does not “jump” (but it may have kinks at indifference points ) ) , ( max ) ( ) ( t x f t V t X x = ) , ( max arg ) ( ) ( t x f t x t X x = ) ( t x ) ( t V t t (1) Berge, C. (1963) Topological Spaces , Oliver and Boyd, Edinburgh and London, UK. Reprinted by Dover Publications, Mineola, NY, in 1997. -6- MS&E-241-Winter-2007-TAW SENSITIVITY ANALYSIS Consider the parametric constrained optimization problem under the standard assumptions. Assume in addition that the constraint set can be described by an equality constraint where is continuously differentiable in (x,t) The objective function is continuously differentiable in (x,t) Question : How does the value function change as a function of ? Answer: Envelope theorem If the value function is differentiable, then its slope can be computed by partially differentiating the objective function with respect to and evaluating at ) , ( max ) ( ) ( t x f t V t X x = ) ( t X 0 ) , ( = t x g m n g + 1 : t t ) ), ( ( t t x (*)
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-7- MS&E-241-Winter-2007-TAW ENVELOPE THEOREM Theorem : If the value function of the parametric constrained optimization problem (*) (with equality constraints) is differentiable at , then where are the Lagrange multipliers associated with the maximizer .
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04-MS&E-241-Welfare-Demand-Theory-II-Lecture -...

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