2PropertiesDemand

# 2PropertiesDemand - Demand Properties: Summary Let u...

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Unformatted text preview: Demand Properties: Summary Let u represent a continuous preference relation % on R L + . In what follows assume that u is continuous and locally nonsatiated. For ( p, w ) >> 0, define d ( p, w ) = { x ∈ B ( p, w ) | u ( x ) ≥ u ( y ) for all y ∈ B ( p, w ) } , where B ( p, w ) = { x ∈ R L + | p · x ≤ w } . For ¯ u ∈ Range ( u ) and p >> 0, define e ( p, ¯ u ) = min x ∈ R L + p · x (1) subject to the constraint that u ( x ) ≥ ¯ u . Theorem 1. For every ¯ u ∈ Range ( u ) • e ( · , ¯ u ) is nonincreasing on R L ++ ; • e ( · , ¯ u ) is homogenous of degree 1 on R L ++ ; • e ( · , ¯ u ) is concave on R L ++ . Let h ( p, ¯ u ) be the set of solutions to (1). The next result follows from Theorem 1 and Shepard’s Lemma. Corollary 1. ( Properties of h ). For every ¯ u ∈ Range ( u ) • h ( · , ¯ u ) is homogeneous of degree 0 on R L ++ ; • If e ( · , ¯ u ) is C 2 (implying h ( p, ¯ u ) is single-valued and C 1 ), then the matrix...
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## This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.

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2PropertiesDemand - Demand Properties: Summary Let u...

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