Optimization of Utility Functions in an Admissible Space of Higher Dimension

Optimization of Utility Functions in an Admissible Space of Higher Dimension

German Almanza (University Autonomous of Juarez City, Mexico), Victor M. Carrillo (University Autonomous of Juarez City, Mexico) and Cely C. Ronquillo (University Autonomous of Juarez City, Mexico)
DOI: 10.4018/978-1-4666-9779-9.ch006
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Abstract

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.
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2. Methods

2.1 Utility Function

We assume an economy with l numbers of commodities and the market of each commodity have defined a unity of weight for each good, i.e. each good is quantified and is related with a real number, we will denote the real number set with. We are considering only those commodities that have in the market, that mean we use only positive real numbers. Then we denote the orthant positive of (the real space of dimension ) and to the commodity space, then there exist a homeomorphism of P with an open subset of indeed there is a map

, which is an inclusion.

The coordinate is a commodity bundle that belongs to a consumer, or the consumer wish to choose between all the commodities. Also assume m agents in the market, we denoted to the commodity bundle of the i-agent, then exist bundles for agents. The status of the economy assuming the commodities of the m agents is denoted with the point; i.e. the complete economy of the m agents is represented with a point in the commodity space of dimension. In our economic model we assume exhausted resources and those are quantified, then the next space is a natural definitions.

  • Definition 1: Let the total commodity that agents are allow to choose, we named attainable space to the set

    • with,

In the above space there are interacting the m agents and it represent the whole space of consumption or choice possibilities.

  • Remark 1: The space is identified with an open subset of, with, then we assume to is a -manifold. The manifold has a topological property of compact closed. The Heine-Borel theorem (cf. Apostol, 1983) said: Every compact subset in is closed and bounded. The above theorem has an economic meaning, the relation with the space is: any agent can increase or decrease fairly his commodities, until reach or exhaust any combination of commodity desired. This property is economically relevant, because is important that any agent in the market exhaust or reach their preferences.

  • Definition 2: The utility function of i-agent is defined as

  • ui(x’)>ui(x) means: absolute preference of x’ over x, and

  • ui(x’)≥ui(x) means: that x’ comply equally or more than x

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