Mathematical Methods for Economic Theory 2 by Professor James C. Moore (auth.)

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By Professor James C. Moore (auth.)

This is the second one of a two-volume paintings meant to operate as a textbook good as a reference paintings for financial for graduate scholars in economics, as students who're both operating in concept, or who've a powerful curiosity in monetary idea. whereas it isn't useful scholar learn the 1st quantity ahead of tackling this one, it may possibly make issues more uncomplicated to have performed so. at least, the coed project a major examine of this quantity will be accustomed to the theories of continuity, convergence and convexity in Euclidean house, and feature had a reasonably refined semester's paintings in Linear Algebra. whereas i've got set forth my purposes for writing those volumes within the preface to quantity 1 of this paintings, it's probably so as to repeat that rationalization the following. i've got undertaken this venture for 3 important purposes. within the first position, i've got amassed a few effects that are often invaluable in economics, yet for which distinct statements and proofs are relatively tricky to discover; for instance, a couple of effects on convex units and their separation via hyperplanes, a few effects on correspondences, and a few effects relating help services and their duals. Secondly, whereas the mathematical best­ ics taken up in those volumes are in most cases taught someplace within the arithmetic curriculum, they're by no means (insofar as i'm conscious) performed in a two-course series as they're prepared here.

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Show that (Rn,~) is a directed system if ~ is the usual weak vector ordering on Rn. 3. 80. 4. 81, respectively, are directed systems. 5. 81, converges to x*. 6. Prove the following. Let S be a subset of a topological space, T, let (Xci) be a net in Sand x* E S Then (Xci) converges to x* in the relative topology for S if, and only if, (Xci) converges to x* in T. 1 • In Relative and Product Topologies In Sections 3 and 5 of the previous chapter, we studied various ways of developing topologies from other concepts or things: arbitrary families of subsets, metrics, and weak orders.

F(Xn-2),f(xn -I)] :::; k 2 . d(xn-2,xn-r} = k 2 . f(Xn-3), f(X n -2)] :::; ... :::; k n . d(xo, xI). Thus, if n > m: :::; d(xo, Xl) L;:~ k i :::; d(xo, xI)k m . ) :::; km. ) where the last inequality is by the fact that 0 :::; k > 1. Thus we see that (xn ) is a Cauchy sequence, and it follows that there exists x* E M such that Xn -> x*. However, since f is a contraction mapping, it is continuous, and therefore: f(x") = f( lim x n ) = lim f(x n ) = lim Xn+l = x*. n~oo n~oo n~oo To prove that x" is unique, suppose there exists f(x).

Show that if f(T) is an interval in R, then f is a continuous function. [Hint. ] b. Is a function representing P necessarily continuous? 0 Exercises 1. 41. 5. METRIC SPACES rv 2. Let T be any collection of topological spaces, and define the relation on T by: S rv 'J <==> S is homeomorphic to 'J. 36). 3. Supposing that S is a topological space, and that I: S -+ Rand g: S -+ R are continuous functions, show that the following functions are all continuous: (a) the function of + /3g, for any real numbers, a and /3, (b) the product function, I·g, (c) the function h defined by: h(x) = min{J(x), g(x)} , for XES, (d) the function h defined by: h(x) = max{J(x),g(x)}, for XES, (e) III, (f) 1/g, in the case where 9 does not vanish anywhere on S.

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