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LectureNotes040211

LectureNotes040211 - CB046/Starr LectureNotes040211 10:44 1...

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CB046/Starr LectureNotes040211 April 25, 2011 10:44 1 Economics 113 Prof. R. Starr UCSD Spring 2011 Lecture Notes for May 2 and 4, 2011 A market economy Firms, profits, and household income H , F , α ij R + , i H α ij = 1 , r summationdisplay i H r i . ˜ π j ( p ) sup { p · y | y ∈ Y j } ≡ p · ˜ S j ( p ) Theorem 13.1 Assume P.II, P.III, and P.VI. ˜ π j ( p ) is a well-defined contin- uous function of p for all p R N + , p negationslash = 0. ˜ π j ( p ) is homogeneous of degree 1. ˜ M i ( p ) = p · r i + j F α ij ˜ π j ( p ). P = braceleftBigg p | p R N , p k 0 , k = 1 . . ., N, N summationdisplay k =1 p k = 1 bracerightBigg . Excess demand and Walras’ Law Definition The excess demand function at prices p P is ˜ Z ( p ) = ˜ D ( p ) - ˜ S ( p ) - r = summationdisplay i H ˜ D i ( p ) - summationdisplay j F ˜ S j ( p ) - summationdisplay i H r i . Lemma 13.1 Assume C.I–C.V, C.VI(SC), C.VII, P.II, P.III, P.V, and P.VI. The range of ˜ Z ( p ) is bounded. ˜ Z ( p ) is continuous and well defined for all p P . Proof Apply Theorems 11.1, 12.2, 13.1. The finite sum of bounded sets is bounded. The finite sum of continuous functions is continuous. QED Theorem 13.2 (Weak Walras’ Law) Assume C.I–C.V, C.VI(SC),C.VII, P.II, P.III, P.V, and P.VI. For all p P , p · ˜ Z ( p ) 0. For p such that p · ˜ Z ( p ) < 0, there is k = 1 , 2 , . . ., N so that ˜ Z k ( p ) > 0. Proof of Theorem 13.2 p · ˜ D i ( p ) ˜ M i ( p ) = p · r i + j F α ij ˜ π j ( p ). i H α ij = 1 for each j F .
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CB046/Starr LectureNotes040211 April 25, 2011 10:44 2 p · ˜ Z ( p ) = p · bracketleftBigg summationdisplay i H ˜ D i ( p ) - summationdisplay j F ˜ S j ( p ) - summationdisplay i H r i bracketrightBigg = p · summationdisplay i H ˜ D i ( p ) - p · summationdisplay j F ˜ S j ( p ) - p · summationdisplay i H r i = summationdisplay i H p · ˜ D i ( p ) - summationdisplay j F p · ˜ S j ( p ) - summationdisplay i H p · r i = summationdisplay i H p · ˜ D i ( p ) - summationdisplay j F ˜ π j ( p ) - summationdisplay i H p · r i = summationdisplay i H p · ˜ D i ( p ) - summationdisplay j F bracketleftBigg summationdisplay i H α ij ˜ π j ( p )
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