New PDF release: Konsum- und Produktionstheorie I

By R. Henn, O. Opitz (auth.)

ISBN-10: 3540049495

ISBN-13: 9783540049494

ISBN-10: 3642872263

ISBN-13: 9783642872266

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Extra info for Konsum- und Produktionstheorie I

Example text

FlI, X. l n X1. = flI. X. l n Xl Ebenso wird Q' = ( a,b) n Q zerlegt: Q' = «a,r ql ) U (r ql } U (r ql ,b}) n Q Betrachtet man x 2 E X", so unterscheidet man drei Falle: A) B) x 2 E X 1; dann setzt man: u' (x 2 ) E (a,r ) n Q. ql > a, also ist das offene Intervall ql ) nicht leer. E(a,r ~ Man wahlt also u' (x 2 ) = ql )nQ , falls r E (a,r ) n Q und r den q2 q2 ql q2 kleinsten Index der im Intervall (a,r ) enthaltenen rationalen ql Zahlen r i besitzt. ; dann setzt man: u'(x 2 ) E (r ,b) n Q. E(r ~ ql ,b)nQ Im folgenden wird angenommen, die Elemente xl' ••• , xm wurden bereits - 43 - betrachtet und es gilt: (I' = u' (xi") = 1, ••.

J nem der auf der Verbindungsstrecke zwischen xi und Xj liegenden GUtervektoren aus X vorzuziehen sein. 2) u(xi ) ~ u(x j ) ~ u(x i ) ~ u(tx i + (l-t)x j ) fUr alle tX i + (l-t)x j E X. 1) bzw. 2) der schwachen Konvexit~t ist fUr eine Nutzenfunktion dann und nur dann erfUllt, wenn die Menge {X EX: (Xi ,x) E P} fUr beliebiges xi E X konvex in X ist (vgl. 18». A. 1) auch (x,tx + (l-t)x') E P fUr alle t E (0,1) mit tx + (l-t)x' E X. , l. tx + (l-t)x') E P fUr aIle t E (0,1) mit tx + (l-t)x' E X und tx + (l-t)x' ist Element von B(X i ), damit ist B(x i ) konvex in X.

IS) S(xi ) und B (~) sind nach Voraussetzung abgeschlossen. 14) und dem Zusammenhang von X) Der Beweis erfolgt nun indirekt. Angenommen, es existiert kein x E XI mit den gewUnschten Eigenschaften (xi,x) E P - J und (x,~) E P - J. Dann gilt fUr aIle x E XI die Enthaltenseinsbezieh~~g x E S(x i ) oder x E B(~), also Damit ist auch die abgeschlossene HUlle XI in der abgeschlossenen HUlle S(xi ) U B(~) von S(xi ) U B(~) als Teilmenge enthalten (vgI. 2S». 17) ein Widerspruch, also ist die Behauptung richtig.

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Konsum- und Produktionstheorie I by R. Henn, O. Opitz (auth.)

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