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By V.L. Makarov

This publication is dedicated to the mathematical research of versions of financial dynamics and equilibria. those types shape a massive a part of mathemati­ cal economics. versions of monetary dynamics describe the movement of an financial system via time. the fundamental suggestion within the examine of those versions is that of a trajectory, i.e., a series of parts of the section house that describe admissible (possible) improvement of the financial system. From all trajectories, we pick out those who are" desirable," i.e., optimum by way of a definite criterion. The equipment of point-set maps is the right instrument for the research of those versions. The topological features of those maps (particularly, the Kakutani fixed-point theorem) are used to review equilibrium types in addition to n-person video games. to review dynamic types we use a unique category of maps which, during this e-book, are referred to as superlinear maps. the speculation of superlinear point-set maps is, evidently, of curiosity in its personal correct. This conception is defined within the first bankruptcy. Chapters 2-4 are dedicated to types of financial dynamics and current an in depth examine of the homes of optimum trajectories. those houses are defined by way of theorems on features (on the lifestyles of twin costs) and turnpike theorems (theorems on asymptotic trajectories). In bankruptcy five, we nation and learn a version of monetary equilibrium. the elemental notion is to set up a theorem concerning the lifestyles of an equilibrium kingdom for the Arrow-Debreu version and a definite generalization of it.

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If 0 < ~ < 1, then setting y = ~xwe get ~a(x). a(x) Thus, here also, we have = a(! y) = ! a(y) = ! a~x). ~ ~ ~ a~x) = ~(x). 3. Suppose a is a superadditive map and that there exists an element x E Kl such that the set a(x) is bounded. Then a is a Gale map. PROOF. Suppose the assertion is not true. Then there exists an element y of Kl where y -# 0, such that y E a(O). , a(O) is not bounded. Since a(x) = a(x + 0) 2 a(x) + a(O), it follows that a(x) is not bounded. This contradiction proves the proposition.

Without loss of generality, we may assume v = -1. 16) that h E Up. Since, additionally, h E K* it follows that h E Up +. 17) we get (for x = 0): h(xo) ~ w > p'(XO) contradicting the definition of the functional D p'. Later on, we study monotone sublinear functionals defined on normal sets. Such sets are the subject of our next subsection. Normal sets A convex compact subset 0 of a pointed reproducing cone K is said to be normal (with respect to K) if = O. 8 depicts a normal set (with X = 1R2, (0 - K) () K K = IR+ 2, 0 is the shaded set and the boundary of the set 0 - K is indicated by dots).

4: Superlinear maps and their duals Monotone maps The map a: KI -+ II(K2 ) is said to be increasing (decreasing) if the inequalityl2 x' ~ x" where x', x" E K I , implies that a(x') 2 a(x") (respectively a(x') s;;; a(x"». 8. For a map a to be increasing, it is necessary and sufficient that the set a-ley) be Krstablefor any y E a(KI)' The proof is obvious. 9. The map a is decreasing if and only iffor any y E a(KI) the set a-ley) contains the segment <0, x) = {x' E KI I x' ~ x} for any point x in that set.

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