Smooth preferences, symmetries and expansion vector fields

Abstract

Tyson (J Math Econ 49(4): 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 (i.e. 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 110(1): 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 discussed.

Appendices

Appendix 1

The present appendix is meant to sketch an essential introduction to vector fields and dynamical systems. We refer to Abraham and Marsden (1987), Spivak (1999) and Taylor (1996) for authoritative references; see also Mantovi (2013).

One can define tangent vectors as a generalization of directional derivatives. Let $\gamma$ : (–1, 1) $\rightarrow {\mathcal {R}}^{n}$ be a $C^{1}$ curve in ${\mathcal {R}}^{n}$, and let f : ${\mathcal {R}}^{n} \quad \rightarrow$ ${\mathcal {R}}$ be a $C^{1}$ function. Define the (Lie) derivative of f at $p=\gamma$ (0) as $\sum \nolimits _{k=1}^n { \frac{\partial \gamma ^k}{\partial t}\vert _{t=0} \frac{\partial f}{\partial x^k}\vert _p }$. Such a derivative, evidently, is linear and satisfies Leibniz rule; call it a tangent vector at p; call $\frac{\partial \gamma ^k}{\partial t}\vert _{t=0}$ the components of the vector with respect to the natural coordinates of ${\mathcal {R}}^{n}$. 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 ${\mathcal {R}}^{n}$. A vector field on (an open subset of) ${\mathcal {R}}^{n}$ is a function which assigns a tangent vector to each point of the space.

On account of the previous considerations, the (local) coordinate representation of a vector field A reads ${\mathbf {A}}=\sum \limits _{k=1}^n { A^k(x)\frac{\partial }{\partial x^k}}$, being $x^{1}$, ..., $x^{n}$ (local) coordinates, $A^{k}$ the components of A with respect to such coordinates, and $\frac{\partial }{\partial x^k}$ 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 ${\mathbf {A}}(f)=\sum \nolimits _{k=1}^n {A^k\frac{\partial f}{\partial x^k}}$. Correspondingly, the coordinate representation of a 1-form w reads ${\mathbf {w}}=\sum \nolimits _{k=1}^n {w_k (x)dx^k}$, so that the pairing between a 1-form and a vector field can be written ${\mathbf {w(A)}}\equiv {\mathbf {wA}}=\sum \nolimits _{k=1}^n {w_k (x)A^k(x)}$. Evidently, the differential of the function f is the 1-form $df=\sum \nolimits _{k=1}^n { \frac{\partial f}{\partial x^k}(x)dx^k}$, so that one can write ${\mathbf {A}}(f)=df({\mathbf {A}})$.

The Lie derivative of the vector field B with respect to the vector field A is the vector field $\mathsf{L}_{{\mathbf {A}}} {\mathbf {B}}$ 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 $\mathsf{L}_{{\mathbf {A}}} {\mathbf {B}}$ on functions f results in the commutator ${\mathbf {A}}({\mathbf {B}}(f))-{\mathbf {B}}({\mathbf {A}}(f))=\mathsf{L}_{{\mathbf {A}}} {\mathbf {B}(f)}$. In words, the first term on the LHS is obtained by applying B to a function f and then applying A to the function ${\mathbf {B}}(f)$; the second term is obtained by commuting such operations. Call Lie bracket the mapping $({\mathbf {A,B}})\rightarrow \mathsf{L}_{{\mathbf {A}}} {\mathbf {B}}$, which is, evidently, bilinear and skew-symmetric. (Spivak 1999, chapter 5) derives the algorithm (formula 7) yielding the components of $\mathsf{L}_{{\mathbf {A}}} {\mathbf {B}}$ given the components of A and B.

The 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 $\mathsf{L}_{{\mathbf {A}}} {\mathbf {B}}$ 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 $\frac{\partial ^2f}{\partial x^j\partial x^k}=\frac{\partial ^2f}{\partial x^k\partial x^j}$ established by a well known theorem (typically named after Schwarz) of elementary calculus.

The vector field ${\mathbf {A}}=\sum \nolimits _{k=1}^n { A^k(x)\frac{\partial }{\partial x^k}}$ is equivalent to definition of the first order ordinary differential equation (ODE) system $\dot{x}^k=A^k(x), k=1,\ldots n$; the integral curves of the vector field are the solutions to the system. Thus, a vector field on some space defines a dynamical system in continuous time.

The theory of dynamical systems has long been recognized as a preferred terrain for the cross fertilization of different scientific disciplines. One can think for instance to the physical insights which led Henry Poincaré to envision the geometric approach to dynamical systems, in which it is the properties of flows which drive the development of the theory, and not the analytical form of the differential equations (see the Introduction in Abraham and Marsden 1987). Along such a line of progress, coordinate transformations have become a pivotal element in the study of dynamical systems, for instance in connection with symmetry properties and canonical dynamics. The celebrated address to the 1954 International Congress of Mathematicians set forth by A.N. Kolmogorov (“The General Theory of Dynamical Systems and Classical Mechanics”) is a classical reference for the cross fertilizing effects concerning “the complex question of integrating the systems of differential equations of classical mechanics [...] interwoven with problems of the calculus of variations, many dimensional differential geometry, the theory of analytic functions, and the theory of continuous groups.”

Noticeably, the development of evolutionary game theory in the last few decades can be considered as one more instance of such historical cross fertilization process: the replicator dynamics (Weibull 1995), meant to embody the selection of fitter strategies, is represented naturally in terms of first order ODEs (vector fields) on the space of strategy profiles. Perhaps the most significant economic application of vector fields pertains to the analysis of phase diagrams of growth models (see for instance Barro and Sala-i-Martin 2004).

Our introduction of the expansion vector fields on primal space, meant to represent the class of expansion paths as a flow, aims at capitalizing on the aforementioned cross fertilization potentialities associated with the adoption of vector fields. In fact, the introduction of symmetry vector fields by Tyson (2013) represents a clear example of such general pattern: symmetry vector fields on choice space turn out to provide quite effective a setting for deepening the analysis of separability. Vector fields on consumption space can in principle be employed in order to model any family of smooth curves which fill the space, and which embody significant economic effects (expansion effects, scale effects, etc). Evidently, limits to such approaches can be placed in terms of ‘cost-benefit’ analysis of the analytical description, as gauged by the balance between, on the one hand, the effectiveness of the formalism in establishing sharp conclusions, and on the other hand, the cost of enlarging the analytical toolkit of theoretical microeconomics. True, in the author’s view, employing vector fields in the microeconomics of consumption does not mean enlarging the toolkit, but simply acknowledging that the tangent vectors to such paths may be employed effectively to gauge economic effects.

Appendix 2

The n-dimensional consumption space B = ${\mathcal {R}}^{n}_{++}=(0,\infty )\times \cdots \times (0,\infty )(n$ copies) admits the dual space A of normalized price vectors p, with respect to which the budget constraint can be written pq =1. Despite the fact that B is not a linear space, elements of A are dual to elements of B in the standard algebraic sense, they are linear functionals on B, namely, such that

$$\begin{aligned} {\mathbf {p}}(a{\mathbf {q}}_{1} +b{\mathbf {q}}_{2})=a{\mathbf {pq}}_{1} +b{\mathbf {pq}}_{2} \end{aligned}$$

(provided, evidently, $a{\mathbf {q}}_{1} +b{\mathbf {q}}_{2}$ belongs to B).

Then, let us recall the following well known result (see for instance Spivak 1999, chapter 3). A basis of a finite dimensional real linear space uniquely determines global coordinates on such space, i.e. the components of vectors with respect to such a basis. Then the linear space can be endowed with the structure of real smooth differentiable manifold, and the global chart sets natural isomorphisms between the manifold and each of its tangent spaces. A sketch of the proof goes as follows.

Given any q $\in$ B, the bundles q+t v belong to B for any v $\in$ B, for small enough $\vert \quad t \quad \vert$. Then, the vector V $\in T_{q}$ B tangent to such line at q is uniquely determined by v $\in$ B, and such a mapping is evidently linear and injective. Therefore, the class of isomorphisms B $\rightarrow T_{q}$ B is uniquely determined for any q $\in$ B.

Such a technical result does play a role in our geometric analysis. Tangent vectors to B represent the rate at which economic effects takes place, for instance an expansion effect. A 1-form on B is a linear functional on such vectors: by the previous result, we can consider an element of A (a vector of normalized prices) as a 1-form on B. Given such an interpretation of normalized price vectors as 1-forms, one can consider the Hotelling-Wold identity as defining a “normalization” of the differential dU (a 1-form) of the utility function U, which results in the inverse Marshallian demand (a 1-form), the normalization being performed by the function Z(U). Such geometric insights enable us to deepen the significance of our expansion-symmetry vector field.