2 - MS&E 241: ECONOMIC ANALYSIS Thomas A. Weber 2....

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MS&E 241: ECONOMIC ANALYSIS Thomas A. Weber 2. Consumer Choice and Demand Theory Winter 2010 Stanford University Copyright © 2010 T.A. Weber All Rights Reserved -2- MS&E-241-Winter-2010-TAW AGENDA Utility Representation Demand Theory: Basics A Little Refresher on Constrained Optimization Key Concepts to Remember
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-3- MS&E-241-Winter-2010-TAW All consumer preferences assumed to be rational Complete Reflexive Transitive Preferences also assumed to be continuous Preference order does not jump around discontinuously {x: x š y} and {x: y š x} are both closed sets Exclude situation: consumer prefers x(n) to y for sequence of x(n) converging to limit x( 4 ), but strictly prefers y to x( 4 ) (1) Theorem . If preferences are rational and continuous, then there exists a continuous utility function u(x) that describes preferences. THEORY OF THE CONSUMER: PREFERENCES (1) Example: Take lexicographic preferences on the square X = [0,1] x [0,1] (as in the first lecture) and let x(n) = (1/n,1/n), y=(0,1). -4- MS&E-241-Winter-2010-TAW x 1 x 2 u = 7 u = 19 u = 55 u = 295 u = 302 TYPICAL CONSUMPTION SET WITH INDIFFERENCE CURVES
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-5- MS&E-241-Winter-2010-TAW x 1 x 2 u = 7 u = 19 u = 55 u = 295 u = 302 x y z CONVEXITY OF PREFERENCES U x = {x: y š x} “Upper Contour Set (relative to x)” -6- MS&E-241-Winter-2010-TAW CONVEXITY Definition . A rational preference relation on X is convex if the upper contour set is convex for any x in X, i.e., } : { y x y U x ) 1 , 0 ( ) 1 ( , x x U z y U z y Proposition . A utility representation of a convex preference relation is quasi-concave (i.e., single-peaked).
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-7- MS&E-241-Winter-2010-TAW ORDINAL VS. CARDINAL PROPERTIES A utility representation u(x) for a given rational preference relation š on X is generally not unique. The preference relation š fixes only ordering of elements of the choice X, and is therefore called ordinal . Given the utility representation u(x) of š on X, the function v(x) = (u(x)) is also a utility representation of š on X, as long as the (real-valued) transformation is increasing. Each specific utility representation of š on X is called cardinal . Thus, while the ordinal properties of utility functions are invariant with respect to increasing transformations, their cardinal properties are not! -8- MS&E-241-Winter-2010-TAW Theory of Consumers Need assumptions about preferences to ensure utility function exists. Normally only ordinal properties (which express ordering of options) of utility functions are important. For theories of consumer choice under uncertainty, cardinal properties are important. (Cardinal properties express how much better one option is than another.) Theory of Firms Preferences assumed for firms – profit – can always be written as a function, a profit function. Profit function plays the same role as utility function.
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2 - MS&E 241: ECONOMIC ANALYSIS Thomas A. Weber 2....

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