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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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 with978-1-4666-9779-9.ch006.m01. We are considering only those commodities that have in the market, that mean we use only positive real numbers. Then we denote 978-1-4666-9779-9.ch006.m02the orthant positive of 978-1-4666-9779-9.ch006.m03 (the real space of dimension 978-1-4666-9779-9.ch006.m04) and 978-1-4666-9779-9.ch006.m05 to the commodity space, then there exist a homeomorphism of P with an open subset of 978-1-4666-9779-9.ch006.m06 indeed there is a map

978-1-4666-9779-9.ch006.m07, which is an inclusion.

The coordinate 978-1-4666-9779-9.ch006.m08 is a commodity bundle that belongs to a consumer, or the consumer wish to choose between all the 978-1-4666-9779-9.ch006.m09commodities. Also assume m agents in the market, we denoted 978-1-4666-9779-9.ch006.m10 to the commodity bundle of the i-agent, then exist 978-1-4666-9779-9.ch006.m11 bundles for 978-1-4666-9779-9.ch006.m12 agents. The status of the economy assuming the commodities of the m agents is denoted with the point978-1-4666-9779-9.ch006.m13; i.e. the complete economy of the m agents is represented with a point in the commodity space of dimension978-1-4666-9779-9.ch006.m14. In our economic model we assume exhausted resources and those are quantified, then the next space is a natural definitions.

  • Definition 1: Let 978-1-4666-9779-9.ch006.m15 the total commodity that agents are allow to choose, we named attainable space to the set

    • 978-1-4666-9779-9.ch006.m16 with978-1-4666-9779-9.ch006.m17,

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 978-1-4666-9779-9.ch006.m18 is identified with an open subset of978-1-4666-9779-9.ch006.m19, with978-1-4666-9779-9.ch006.m20, then we assume to 978-1-4666-9779-9.ch006.m21 is a 978-1-4666-9779-9.ch006.m22-manifold. The manifold 978-1-4666-9779-9.ch006.m23 has a topological property of compact closed. The Heine-Borel theorem (cf. Apostol, 1983) said: Every compact subset in 978-1-4666-9779-9.ch006.m24is closed and bounded. The above theorem has an economic meaning, the relation with the 978-1-4666-9779-9.ch006.m25 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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