# Advances in Mathematical Economics, 1st Edition by C. Castaing, M. Saadoune (auth.), Shigeo Kusuoka, Akira

By C. Castaing, M. Saadoune (auth.), Shigeo Kusuoka, Akira Yamazaki (eds.)

A lot of financial difficulties can formulated as restricted optimizations and equilibration in their ideas. a variety of mathematical theories were offering economists with necessary machineries for those difficulties bobbing up in financial conception. Conversely, mathematicians were encouraged by means of a variety of mathematical problems raised by way of financial theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new difficult stimuli from fiscal theories with these economists who're looking for powerful mathematical instruments for his or her researchers. participants of the editorial board of this sequence includes following famous economists and mathematicians: handling Editors: S. Kusuoka (Univ. Tokyo), A. Yamazaki (Hitotsubashi Univ.) - Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier II), F. H. Clarke (Univ. Lyon I), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Fukuoka Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Hitotsubashi Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), okay. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), ok. Nishimura (Kyoto Univ.), M. okay. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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**Example text**

We first prove that {JC„} is bounded. It is obvious that V(x\, Rx) < V(x, Rx). Suppose that V(xn, Rx) < V(x, Rx) for some n eN. 1 and the convexity of || • ||^, we have V(xn-^\,Rx) = < < < = V{anX + anV(x,Rx) anV(x, Rx) anV(x, Rx) V(x,Rx). Rx) + (l-an)V(yn,Rx) + (1 - an)V(xn, Rx) -h (1 - an)V(x, Rx) Hence, by induction, we have V(xn, Rx) < V(x, Rx) for all n e N. 2), the sequence {xn} is bounded. 1, we have that V(yn,Rx) = V(Jr^Xn,Rx) < V(xn,Rx) for all n e N. So, {yn] is also bounded. We next prove limsup(x - Rx, Jxn - JRx) < 0.

Bij = 0, we have A{w, p) = A(w, p). The Rybczynski and Stolper-Samuelson theorems are equivalent since [dy/dx] = [dw/dpY. However, in presence of externalities, the Rybczynski effects depend on the input coefficients from the private perspective while the Stolper-Samuelson effects depend on the quasi-input coefficients from Capital-labor substitution and indeterminacy 35 the social perspective. The duality between these two effects is thus destroyed. This property is again explained by the same intuition as the one given above.

Since YA^\ ^i — ^ ' ^ ^ ^^^^ D/Sm (^ - a/) = 0 (see Takahashi [21]). Hence, we have limsupy(x„, Rx) = lim sup V(Xfn-\-i-\-\, Rx) < lim sup 2e m-\-l m+l l-[](l-a/) + Yl{l-ai)V{x,n,Rx) = 2e. This implies lim sup„^^ Vix„, Rx) < 0. Hence, we get lim VV{xn,Rx) = 0 . 1, we obtain lim \\x ||jc„ - Rx\\ = 0 . Therefore, {JC„} converges strongly to R(BJ)-^OM' 5. Applications In this section, we study the problem of finding a minimizer of a proper lower semicontinuous convex function; see [2,16,21]. 2, we obtain the following result.