Walrasian Solutions Without Utility Functions

SUMMARY: This note reviews consumers’ preference orderings in economics and shows that irrationality is a poor explanation for apparent violations of some axioms of order. Apparent violations seem to be better explained by the fact that consumers’ utility functions, if they exist at all, might not even belong to the class of quasi-concave functions. However, the main task of markets is the determination of equilibrium price vectors. The note shows in addition that, in Walrasian structures, quasi-concave utility functions are unnecessary for the determination of equilibrium price vectors.Walrasian structures, preference orderings, irrationality, utility functions, and equilibrium price vectors

Explaining the logic of pure preference in a neurodynamic structure

This paper uses Category Theory to integrate a nonlinear, nonhomogeneous ordinary differential equation system into an input/output representation in an attempt to capture the mechanism behind the formation of pure preference in humans. The model shows that the human brain belongs to the class of functions U ε C2(R3, R). In addition, it shows that there exists an emerging factor, e, which is sine qua non for expressing a preference. The factor, e, may be associated with ‘judgement’ which, in turn, may neatly subsume ‘consciousness’, the arrival of new information, and cases of selection under risks and uncertainty.Input/output; Dynamo; Universal Unfoldings; Emergent factor; Awareness; and Preference

On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes

ABSTRACT: In a recent paper, Dominique (2009) argues that for a Walrasian economy with m consumers and n goods, the equilibrium set of prices becomes a fractal attractor due to continuous destructions and creations of excess demands. The paper also posits that the Hausdorff dimension of the attractor is d = ln (n) / ln (n-1) if there are n copies of sizes (1/(n-1)), but that assumption does not hold. This note revisits the problem, demonstrates that the Walrasian economy is indeed self-similar and recomputes the Hausdorff dimensions of both the attractor and that of a time series of a given market.Fractal Attractors, Contractive Mappings, Self-similarity, Hausdorff Dimension of an Economy,Hausdorff Dimension of Economic Time Series

On the Computation of the Hausdorff Dimension of the Walrasian Economy: Addendum

In a recent paper, Dominique (2009) argues that for a Walrasian economy with m consumers and n goods, the equilibrium set of prices becomes a fractal attractor due to continuous destructions and creations of excess demands. The paper also posits that the Hausdorff dimension of the attractor is d = ln (n) / ln (n-1) if there are n copies of sizes (1/ (n-1)), but that assumption does not hold. A subsequent paper (no 16723) modified that assumption, dealt with the self-similarity of the Walrasian economy, and computed the Hausdorff dimensions of the attractor as if it were a space-filling curve. This paper is an extension of the first two. It shows that the path of the equilibrium price vector within the attractor is rather as close as one can get to a Brownian motion that tends to fill up the whole hyperspace available to it. The end analysis is that the economy obeys a homogeneous power law in the form of f-. Power Spectra and Hausdorff dimensions are then computed for both the attractor and economic time series.Fractal Attractor, Contractive Mappings, Self-similarity, Hausdorff Dimensions of the Walrasian Economy and time series, Brownian Motion, Power Spectra, Hausdorff Dimensions in Higher Dimensions.Fractal Attractor, Contractive Mappings, Self-similarity, Hausdorff Dimensions of the Walrasian Economy and time series, Brownian Motion, Power Spectra, Hausdorff Dimensions in Higher Dimensions.

SHOULD THE UTILITY FUNCTION BE DITCHED?

This note takes a retrospective look at the Utility Construct still in use in economic science and compares it to a new approach based on recent findings in neuroscience. The results show that it is more natural and more compelling to go from the preference order to the price vector. Thus making the non-falsifiable utility apparatus superfluous.Well-ordered Preference Relation, C2 Utility Function, Well-defined Individual Demand Function, and Neuroeconomics

Flows for rectangular matrix models

Several new results on the multicritical behavior of rectangular matrix models are presented. We calculate the free energy in the saddle point approximation, and show that at the triple-scaling point, the result is the same as that derived from the recursion formulae. In the triple-scaling limit, we obtain the string equation and a flow equation for arbitrary multicritical points. Parametric solutions are also examined for the limit of almost-square matrix models. This limit is shown to provide an explicit matrix model realization of the scaling equations proposed to describe open-closed string theory.Comment: 13 pages, LaTeX, McGill/93-2