section3_201A_Fall10_03 - Section 3 More on Utility and...

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Section 3 : More on Utility and WalrasianDemand ECON 201A, Fall 2010 GSIs: Omar Nayeem and Aniko Oery These notes were originally prepared by Juan Sebasti´an Lleras during the Fall 2007 semester and have since been revised by Juan Sebasti´an and us. We are grateful for his permission to continue using these resources this semester. Summary In Tuesday’s lecture we discussed some structural properties of utility functions — including quasiconcavity, monotonicity, and quasilinearity — and their analogues for preference relations. In this context, we proved a representation theorem for quasilinear preferences. Moreover, we have shown that some of these imply useful properties of the Walrasian demand function. Strict quasiconcavity guarantees that the demand correspondence is single-valued, local non-satiation (which in turn is implied by monotonicity) implies Walras’s Law, and continuity of preferences results in a nonempty and compact demand correspondence. In Thursday’s lecture the main results were Berge’s Theorem, which assures upper hemicontinuity of the demand correspondence and continuity of the indirect utility function, and the Implicit Function Theorem, which will turn out to be useful for comparative statics of the Walrasian demand function. 3.1 Structural Properties of Utility Here are some definitions that were not really covered in lecture. Throughout this part we are assuming that X = R n . Recall the (partial) vector ordering convention in R n , x y if x i y i for all i = { 1 , 2 , ..., n } . Definition 3.1. C X is convex if, for all x, y C and α [0 , 1], we have αx + (1 α ) y C . Geometrically, this means that any line segment whose two endpoints are in C lies completely inside C . In each of the following definitions, we need C to be a convex subset of X . Think about why this requirement is essential. 3-1
Section 3: More on Utility and Walrasian Demand 3-2 Definition 3.2. Suppose C is a convex subset of X . A function f : C R is: concave if f ( αx + (1 α ) y ) αf ( x ) + (1 α ) f ( y ) for all α [0 , 1] and for all x and y . strictly concave if f ( αx + (1 α ) y ) > αf ( x ) + (1 α ) f ( y ) for all α (0 , 1) and for all x and y with x negationslash = y . convex if f ( αx + (1 α ) y ) αf ( x ) + (1 α ) f ( y ) for all α [0 , 1] and for all x and y . strictly convex if f ( αx + (1 α ) y ) < αf ( x ) + (1 α ) f ( y ) for all α (0 , 1) and for all x and y with x negationslash = y . affine if it is concave and convex, i.e. f ( αx + (1 α ) y ) = αf ( x ) + (1 α ) f ( y ), for all x, y X and all α [0 , 1]. quasiconcave if f ( x ) f ( y ) implies f ( αx + (1 α ) y ) f ( y ), for all α [0 , 1]. strictly quasiconcave if f ( x ) f ( y ) and x negationslash = y implies f ( αx + (1 α ) y ) > f ( y ), for all α (0 , 1). Here are a few exercises to give you some practice working with these definitions.

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