Smooth preferences symmetries and expansion vecto

Authors: Andrea Mantovi

Publish Date: 2016/02/29

Volume: 119, Issue: 2, Pages: 147-169

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Abstract

Tyson J Math Econ 494 266–277 2013 introduces the notion of symmetry vector field for a smooth preference relation and establishes necessary and sufficient conditions for a vector field on consumption space to be a symmetry vector field The structure of a such a condition is discussed on both geometric and economic grounds It is established that symmetry vector fields do commute ie have vanishing Lie bracket for additive and joint separability The marginal utility of money is employed as a normalization of the expansion vector field Mantovi J Econ 1101 83–105 2013 which results in the fundamental expansion symmetry vector field Finally a characterization of symmetry vector fields is given in terms of their action on the distance function and a pattern of complete response is discussed for additive preferences Examples of such constructions are explicitly worked out Potential implications of the results are discussedThe author acknowledges a number of profound remarks by an anomymous referee as well as insightful comments by participants to the Workshop “Taxes Subsidies Regulation in Dynamic Models” Brescia October 2–3 2015 in particular Nir Becker Roberto Cellini Giacomo Corneo Chiara D’Alpaos Luca Di Corato Francesco Menoncin Pierre Pestieau Silvia Tiezzi Andrianos Tsekrekos Sergio Vergalli The author is the sole responsible for errors or misprintsOne can define tangent vectors as a generalization of directional derivatives Let gamma –1 1 rightarrow mathcal Rn be a C1 curve in mathcal Rn and let f mathcal Rn quad rightarrow mathcal R be a C1 function Define the Lie derivative of f at p=gamma 0 as sum nolimits k=1n fracpartial gamma kpartial tvert t=0 fracpartial fpartial xkvert p Such a derivative evidently is linear and satisfies Leibniz rule call it a tangent vector at p call fracpartial gamma kpartial tvert t=0 the components of the vector with respect to the natural coordinates of mathcal Rn A tangent vector at p is identified by an equivalence class of curves tangent at p Abraham and Marsden 1987 p 43 Evidently any linear combination of tangent vectors is again a tangent vector and one is in a position to define a tangent space at each point of mathcal Rn A vector field on an open subset of mathcal Rn is a function which assigns a tangent vector to each point of the spaceOn account of the previous considerations the local coordinate representation of a vector field A reads mathbf A=sum limits k=1n Akxfracpartial partial xk being x1 xn local coordinates Ak the components of A with respect to such coordinates and fracpartial partial xk the coordinate vector fields in that chart which define a basis of the tangent spaces at each point Then the action of the vector field A on the function f which measures the variation of f along the flow of A can be written mathbf Af=sum nolimits k=1n Akfracpartial fpartial xk Correspondingly the coordinate representation of a 1form w reads mathbf w=sum nolimits k=1n w k xdxk so that the pairing between a 1form and a vector field can be written mathbf wAequiv mathbf wA=sum nolimits k=1n w k xAkx Evidently the differential of the function f is the 1form df=sum nolimits k=1n fracpartial fpartial xkxdxk so that one can write mathbf Af=dfmathbf AThe Lie derivative of the vector field B with respect to the vector field A is the vector field mathsfL mathbf A mathbf B which represents so to say the derivative of B along the flow of A Spivak 1999 provides a rigorous account of such a mechanism as well as the proof that the action of mathsfL mathbf A mathbf B on functions f results in the commutator mathbf Amathbf Bfmathbf Bmathbf Af=mathsfL mathbf A mathbf Bf In words the first term on the LHS is obtained by applying B to a function f and then applying A to the function mathbf Bf the second term is obtained by commuting such operations Call Lie bracket the mapping mathbf ABrightarrow mathsfL mathbf A mathbf B which is evidently bilinear and skewsymmetric Spivak 1999 chapter 5 derives the algorithm formula 7 yielding the components of mathsfL mathbf A mathbf B given the components of A and BThe conceptual relevance of the Lie bracket has been long established the Lie bracket represents the geometric synthetic definition of the analytical ‘nucleus’ which rules the integrability problem as tailored by Frobenius theorem Taylor 1996 Spivak 1999 For our purposes it is enough to appreciate the condition mathsfL mathbf A mathbf B as guaranteeing that the flows of the vector fields A and B do commute given any initial point one can follow the flow of A for a parameter value a and then the flow of B for a parameter value b and then reverse the order of such operations and find himself at the same final point see Spivak 1999 p 159 Mantovi 2013 Appendix 2 provides a pair of simple examples for such a pattern one can easily check that coordinate vector fields have vanishing Lie brackets consistently with the commutation of partial derivatives fracpartial 2fpartial xjpartial xk=fracpartial 2fpartial xkpartial xj established by a well known theorem typically named after Schwarz of elementary calculus

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