Continuously differentiable preferences | Contaduría y Administración

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Fuente: elaboración propia." ] ] ] "autores" => array:1 [ 0 => array:2 [ "autoresLista" => "Arturo Lorenzo Valdés" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Arturo" "apellidos" => "Lorenzo Valdés" ] ] ] ] ] "idiomaDefecto" => "es" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0186104215001175?idApp=UINPBA00004N" "url" => "/01861042/0000006100000002/v2_201703220239/S0186104215001175/v2_201703220239/es/main.assets" ] "en" => array:19 [ "idiomaDefecto" => true "titulo" => "Continuously differentiable preferences" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "334" "paginaFinal" => "352" ] ] "autores" => array:1 [ 0 => array:3 [ "autoresLista" => "Adolfo García de la Sienra" "autores" => array:1 [ 0 => array:3 [ "nombre" => "Adolfo" "apellidos" => "García de la Sienra" "email" => array:1 [ 0 => "asienrag@gmail.com" ] ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Universidad Veracruzana, Mexico" "identificador" => "aff0005" ] ] ] ] "titulosAlternativos" => array:1 [ "es" => array:1 [ "titulo" => "Preferencias continuamente diferenciables" ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0005" "etiqueta" => "Fig. 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 463 "Ancho" => 500 "Tamanyo" => 14264 ] ] "descripcion" => array:1 [ "en" => "σ is the composition ρ∘ψ:Ω2→Λ." ] ] ] "textoCompleto" => "A ratio is a sort of relation in respect of size between two magnitudes of the same kind.Euclid, ElementsBook V, Definition 3IntroductionJust as classical dynamics proposed to explain the motion of bodies by means of the concept of force, neoclassical consumer theory proposes to explain the behavior of the consumer by means of the concept of preference. This is done by taking, as a starting point, a regular preference structure defined by axioms that actually attribute empirically meaningful (even though idealized) properties to the preference relation. Among these properties, strict convexity, non-satiation or continuity can be mentioned. Restrictions on preference relations translate into restrictions on the form of the utility functions. For instance, if the preference relation is strictly convex, the corresponding utility representation is strictly quasi-concave; if the relation is non-satiated, the corresponding utility representation is monotonically increasing; if the relation is continuous, the corresponding utility representation is also continuous. Certain specializations of the theory require, additionally, that the utility function representing the preference relation be differentiable, in order to apply methods of nonlinear programming to the derivation of the demand functions.Even though some of the aforementioned properties are deemed as “non-substantial” and “technical” by economists of a positivist and instrumentalist philosophical persuasion, nonetheless the tendency has been to formulate them by means of natural and intuitive conditions that depict an idealized consumer described by set-theoretical structures into which the empirical data can be imbedded.<a class="elsevierStyleCrossRef" href="#fn0005">1</a> For the actual meaning of the axioms defining the structures is important: the more idealized they are, the less precise are the empirical consequences of the same, and it is impossible to check intuitively their degree of idealization if their economic meaning is unknown. I think that the reason why it is said (for instance by <a class="elsevierStyleCrossRef" href="#bib0005">Barten & Böhm, 1981</a>, pp. 385–386) that even though “Axioms 1–3 [reflexivity, transitivity and completeness] describe order properties of a preference relation that have intuitive meaning in the context of the theory of choice … [this] is much less so with the topological conditions which are usually assumed as well” is that the language of topology obscures such intuitive meaning altogether because it is not suitable to express the economic meaning of such properties.It is not really difficult to formulate conditions like continuity or convexity in intuitive terms, but the differentiability condition has turned out to be more resilient to such treatment. Certainly, Gerard <a class="elsevierStyleCrossRefs" href="#bib0010">Debreu (1983a, 1983b)</a> and Andreu <a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell (1985)</a> have provided conditions over a preference relation that imply the existence of continuously differentiable utility functions. The problem is that — in contradistinction to the properties I referred to previously — these conditions are admittedly not intuitive. The first aim of the present paper is to propose a language in which all the usual properties attributed to the preference relation, including differentiability, can be formulated in a natural, intuitive way. Even if differentiability is deemed as a mere technical computational convenience, without any actual empirical meaning, the condition presented here is mathematically simpler (once the language has been assimilated) than the ones presented by Debreu and Mas-Colell (which rely upon the heavy machinery of differential topology), and is formulated within the framework of a unified language and conceptual apparatus that clarifies its relationship with the concept of preference strength.After discussing, in the second section, the conditions proposed by Gerard <a class="elsevierStyleCrossRefs" href="#bib0010">Debreu (1983a, 1983b)</a> and Andreu <a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell (1985)</a>, in the third I will motivate and state, in intuitive numerical terms, the required differentiability condition. The fourth section will be devoted to introduce the algebraic theory of difference as a preparation to present, in the fifth section, the conceptual and linguistic apparatus required to provide a geometric theory of preference strength within which differentiability (actually all the usual) conditions can be formulated in an intuitive way. The sixth section contains a development of preference theory within the proposed conceptual apparatus, up to the proof of the existence of a C1 utility function for the preference relation. The seventh section introduces the differentiability condition and the eight and final one discusses the relevance and importance of having a continuously differentiable utility function. The paper ends with a reflection on the convenience of formulating a non-standard version of Hölder's theory in order to formulate the differentiability condition in an even more intuitive way.The conditions of Debreu and Mas-ColellAccording to <a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell et al. (1995, p. 49)</a> “it is possible to give a condition purely in terms of preferences” implying the existence of a C2 utility representation of the same:Intuitively, what is required is that indifference sets be smooth surfaces that fit together nicely so that the rates at which commodities substitute for each other depend differentially of the consumption levels. (<a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell et al., 1995</a>)The problem is that it is not at all clear which empirically meaningful (even if idealized) property must the preference relation of a consumer have so that its indifference sets “fit together nicely”. What is worse, C2 differentiability is restrictive because some demand functions that are derivable do not come from a C2 utility function, as the same authors have noticed (<a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell et al., 1995</a>, p. 95, n. 33). Furthermore, as <a class="elsevierStyleCrossRef" href="#bib0010">Debreu (1983a, p. 201)</a> has pointed out, it is enough for the utility function to be C1 in order to guarantee that the corresponding Walrasian demand function be also C1. Is it possible to find a condition that can be considered sufficiently natural and general for that purpose? My claim is that it is possible, and I intend to substantiate this claim by means of the intuitive discussion motivating <a class="elsevierStyleCrossRef" href="#enun0100">Definition 9</a>.The differentiability condition has been interpreted by Mas-Colell in terms of the concept of a differentiable manifold, giving rise to the following important result.[<a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell, 1985</a>]. Let X be an open set of ℝL and R a locally nonsatiated preference relation over X, with connected indifference sets. Then, for k≥1, R is representable by a x1,x2,x3,x4,x5x6,x1′x2′x3′εΩ: utility function with no critical point iff the frontier of R is a Ck manifold. (Cf. <a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell, 1985</a>, p. 64)In terms of the Gaussian curvature of the indifference curve in each point, Debreu obtained the following result.[Debreu, 1972]. Let X be an open set of ℝL and R a regular preference relation over X which is monotone, continuous, and such that its frontier is a C2 manifold. If the indifference sets of R do not intersect the frontier of X, then there exists a demand function φ of class C1 iff the Gaussian curvature is different from zero in each point of the indifference surfaces. (Cf. <a class="elsevierStyleCrossRef" href="#bib0010">Debreu, 1983a</a>, pp. 194–199)I would like to conclude the present section with a reflection on the meaning of these conditions. In the first place, the definition of Gaussian curvature proposed by Debreu (taken from <a class="elsevierStyleCrossRef" href="#bib0030">Hicks, 1965</a>, Section 2.2) presupposes de facto that the indifference surfaces are already differentiable manifolds (actually, Debreu assumes that they are of class C2), and so the condition only translates the problem to a deeper level. For the question is, precisely, What is the property that must be attributed to the consumer in order to guarantee that the indifference surfaces are differentiable manifolds? In Debreu's definition, the question whether the Gaussian curvature of the manifold is different from zero or not arises once the first problem has been solved. It seems clear that Mas-Colell's condition is just a modified generalization of Debreu's and so analogous considerations apply to it.MotivationThe problem we are concerned with can be formulated thus: Is it possible to find an (idealized) empirically meaningful property over a (cardinal) preference relation that enables a continuously differentiable utility representation of the same?In order to discuss this question let us recall that, according to consumer theory, the satisfaction of a given agent at a certain consumption menu (i.e., when the menu constitutes his current consumption) reaches a certain level. This level normally changes as he moves from that menu to another one (i.e., when he changes his consumption from the previous menu to a new one). If his preferences are continuous, to small changes in his consumption menu there correspond small changes in his satisfaction level. Hence, it makes sense to ask: How fast is his satisfaction changing as he moves from one consumption menu to another nearby? Let be x an interior point of Ω, the nonnegative orthant of vector space ℝL, and notice that, since is x an interior point of Ω, it is possible to move away from x a little in any direction without abandoning Ω.<a class="elsevierStyleCrossRef" href="#fn0010">2</a> As he moves from x to x+¿u (say), where ¿ is a small number and u is a unit vector in a fixed direction, his satisfaction may change at different speeds. If ¿ is infinitesimal and his preferences are continuous, the change Δϕ in his satisfaction level is indeed infinitesimal, but the order of this infinitesimal can be different from that of ¿. Moreover, even if Δϕ is of the same order as ¿, it might be of a different order for a different choice of ¿. Sheer differentiability requires not only that Δϕ be of the same order for any choice of ¿, but that the quotients Δϕ/¿ be all infinitely close to one and the same real number. This real number measures the speed at which the satisfaction level changes at x as the agent changes his consumption slightly in the direction of u. The given condition does not guarantee, however, the continuity of the directional derivative [∂ϕ/∂u](x).Continuous differentiability at a vector x∈Ω in the direction of u means that the rate of satisfaction change along u is continuous. What this means is that the rates [∂ϕ/∂u](x1) and [∂ϕ/∂u](x2) approximate each other as menus x1 and x2 get closer.It is indeed impossible to formulate this condition within the conceptual apparatus of ordinal preference theory, or even within the usual apparatus of cardinal preference theory. This is due to the fact that the notion of differentiability requires the comparison of intervals of the same kind but different interpretation. It requires the comparison of lengths of satisfaction intervals with lengths of geometric intervals; i.e., the comparison of the distance in satisfaction between menus x1,x2 with the geometric distance ||x1−x2|| among them. The problem is that the difference relation R falls short of providing the linguistic and conceptual resources to make this comparison.But there is an indirect way of making claims about R, of attributing (idealized) empirically meaningful properties to R, using the geometric analogy involved in the notion of a satisfaction “interval”. Actually, the very notion of difference comparison is built upon this analogy: When the agent compares the difference in satisfaction (for her) between the pair of menus x1,x2 and the pair x3,x4 she is somehow comparing “distances” between them. That the agent actually thinks or feels that the distance between x1,x2 is at least as long as that between x3,x4 is expressed by the theoretician in economics by means of the formula ‘x1x2Rx3x4’. Hence, it is not far-fetched, but rather natural, for the theoretician to represent this distance by means of a geometric entity of the obvious sort: An interval within a straight line. It seems to me that a fully general theory of cardinal preference must be grounded upon such a representation. I will try to show below how such a theory would be like, but it will be convenient to start considering the axioms required for the usual numerical representation of relation R.The algebraic theory of differenceThere is no doubt that every consumer has an idea of the satisfaction differences between the consumption menus among which she has to make a choice. Hence, the comparison of these differences is only natural. As we already indicated, it is usual to express this comparison by means of a difference relation R among pairs of consumption menus (represented by points in the nonnegative orthant Ω of ℝL). The simplest such relation is defined as follows.Definition 1A difference relation R over Ω is a connected and transitive binary relation over Ω; i.e., a weak order. If R is a difference relation over Ω, we say that 〈Ω×Ω,R〉 is a difference structure.Write x1x2Ex3x4 if x1x2Rx3x4 and x3x4Rx1x2; x1x2Sx3x4 if x1x2Rx3x4 but not x3x4Rx1x2.In the interpretation we are interested in here, formula ‘x1x2Rx3x4’ means that the change from menu x1 to menu x2 is preferred by the agent to the motion from x3 to x4. The change from one to the other can be an improvement or a worsening for the agent. To fix ideas, if we think of the menus as amounts of money, and the agent prefers to have more money to less, a motion from (say) nine thousand (x1) to twelve thousand (x2) dollars is better than one from ten thousand (x3) to eleven thousand (x4). But, if the agent is to lose money, it is preferable for her to fall from eleven (x4) to ten (x3) than from twelve (x2) to nine (x1) thousand dollars. Thus, we shall assume (below) that<elsevierMultimedia ident="eq0005"></elsevierMultimedia>It will be necessary to introduce, also, the operation of composition of motions. For instance, we can compose the motion from nine thousand (x1) to twelve thousand (x2) with the motion from twelve thousand (x2) to eight thousand (x3). The result will be a motion from nine thousand (x2) to eight thousand (x3) a net loss of one thousand dollars. I will define below, in general terms, the required composition operation among intervals.In order to formulate axiomatic conditions over R, say that interval x1x2∈Ω2 is positive (x1x2∈X+) iff x1x2Sxx for any x, which means that moving from x1 to x2 is an improvement for the agent. Interval x1x2 is negative (x1x2∈X−) iff x2x1 is positive. x1x2 is null (x1x2∈X0) iff x1x2 is neither positive nor negative. It is easy to show, out of the axioms that will be introduced below, that all null intervals are equivalent among them selves; i.e., x1x2Ex3x4 for any null intervals x1x2,x3,x4; it can be seen also that if x1x2 is null, then x2x1 is also null.A standard sequence of elements of Ω is a set {xk}k¿K, where K is an initial segment of the set ℤ+ of positive integers (or the whole set), such that xk+1xkEx2x1 for all xk, xk+1 in the sequence, and it is not the case that x2x1Ex1x1. The sequence is strictly bounded if there exist x′,x″∈Ω such that x′,x″Sxkx1Sx′,x″ for all k∈K.I will assume that relation R satisfies the conditions specified in the following definition.Definition 2Difference structure 〈Ω2,R〉 is an algebraic-difference structure<a class="elsevierStyleCrossRef" href="#fn0015">3</a> iff, in addition to being a weak order, it satisfies the following axioms for every x1, x2, x3, x4, x5, x6, x1′, x2′, x3′ εΩ:<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005">(1)If the motion from x1 to x2 is at least as good (bad) as the motion from x3 to x4, then the motion from x4 to x3, is at least as bad (good) as the motion from x2 to x1. In symbols, if x1x2Rx3x4 then x4x3Rx2x1.</li><li class="elsevierStyleListItem" id="lsti0010">(2)If the motion from x1 to x2 is as good (bad) as the motion from x1′ to x2′, and the motion from x2 to x3 is as good (bad) as the motion from x2′ to x3′, then the motion from x1 to x3 is as good (bad) as that from x1′ to x3′; i.e., if x1x2Rx1′x2′ and x2x3Rx2′x3′ then x1x3Rx1′x3′.</li><li class="elsevierStyleListItem" id="lsti0015">(3)If x1x2 and x3x4 are segments such that the motion from x1 to x2 is at least as good (bad) as the motion from x3tox4, it is possible to find a menu x2′ ∈ Ω, slightly less satisfactory than x2, or just as satisfactory, such that the motion from x1 to x2′ matches the motion from x3 to x4. In the same token, it is possible to find a menu x1′ ∈ Ω slightly more satisfactory than x1, or just as satisfactory, such that the difference between x1′ and x2 matches the difference between x3 and x4. In symbols, if x1x2Rx3x4Rxx, then there exist x1′,x2′∈Ω such that x1x2′Ex3x4 and x3x4Ex2′x2.</li><li class="elsevierStyleListItem" id="lsti0020">(4)Every strictly bounded sequence is finite; i.e., for each strictly bounded standard sequence {xk}k¿K of elements of Ω, there exists a number N∈ℤ+ such that k<N for each k∈K.</li></ul>It can be proven<a class="elsevierStyleCrossRef" href="#fn0020">4</a> that, for any algebraic structure 〈Ω2,R〉, there exists a real-valued function φ on Ω such that, for all x1,x2,x3,x4∈Ω<elsevierMultimedia ident="eq0010"></elsevierMultimedia>ϕ, which is a utility function, is unique up to a positive linear transformation; i.e., if ϕ′ is another such utility function, then there are real constants α,β,α>0, such that ϕ′=αu+β This means that ϕ is, indeed, a cardinal utility function.The geometric theory of differenceIt is nearly impossible to formulate differentiability conditions over R within the language and conceptual apparatus of the algebraic theory of difference. What is required is a certain “intermediate” language. In order to introduce this language, let us suppose that a good straight Euclidean line is given in its purity. Following <a class="elsevierStyleCrossRefs" href="#bib0045">Hölder (1996, 1997)</a>, I shall assume that intervals within this straight line are of two kinds, such that any interval is of one and only one kind.Intervals of the same kind are called “of the same direction”,<a class="elsevierStyleCrossRef" href="#fn0025">5</a> and intervals of different kinds are called of opposite direction. The intervals AB and BA are always of opposite direction. Let the intervals of one kind be called “intervals of the first direction” and the fact that AB is an interval of the first direction be expressed as A≻B or B≻A (<a class="elsevierStyleCrossRef" href="#bib0050">Hölder 1997, p. 346</a>).Equality (congruence) of intervals AB and A′B′ will be expressed as AB≐A′B′. Clearly, ≐ is an equivalence relation over the set Λ of all intervals within the straight line.Furthermore, we assume that points and intervals satisfy Hölder's axioms up to the definition of interval numbers (see <a class="elsevierStyleCrossRef" href="#bib0050">1997</a>, §23, p. 351, equations 53 and 54). Hence, we take for granted that there are arbitrarily designated points N and E, with N≺E such that interval NE is taken as unit. We will denote interval NE eventually as 1→.On top of ≐, I will use symbols <elsevierMultimedia ident="201703220242154571"></elsevierMultimedia>, ⋖, or their counterparts <elsevierMultimedia ident="201703220242154572"></elsevierMultimedia> and ⋗, to express the congruence comparisons among intervals. The sum of intervals (for its definition, see <a class="elsevierStyleCrossRef" href="#bib0050">Hölder, 1997</a>, p. 347) will be denoted by ⊕ (Hölder uses symbol +). Notice that what <a class="elsevierStyleCrossRef" href="#bib0045">Hölder (1996)</a> calls ‘magnitudes’ are line intervals in the interpretation intended here. This same interpretation is developed by <a class="elsevierStyleCrossRef" href="#bib0050">Hölder (1997)</a>.It is possible, and it will turn out to be convenient, to express the properties that are attributed to R in terms of relations among geometric intervals within the given Euclidean straight line. What this means is that we, as theoreticians, can represent the comparison of the differences felt by the consumer, expressed by symbol ‘R’, by means of comparisons among intervals in Λ. My proposal is to build the theory of relation R by means of these comparisons, trying to express intuitive, empirical (idealized) properties of R in terms of such comparisons.To that end, let me to introduce the function σ:Ω2→Λ, as an application that assigns to each satisfaction interval x1x2 a line interval whose length is intended to represent the distance that the agent associates to x1x2 (how “far” is x1 from x2 in terms of satisfaction), and whose direction is intended to represent whether the motion from x1 to x2 would be an improvement, a worsening, or indifferent for the agent. In particular, σ will assign to any interval xx in the diagonal the null line interval, which of course does not exist but we can create by a convenient fiat.As I just said, it is not merely the distance among menus what has to be considered, but also the direction of the motions and, moreover, also the composition of motions. If x1x2∈X+, the agent perceives the (actual or potential) change from x1 to x2 as an improvement, and that from x2 to x1 as a worsening. Yet, interval σ(x1x2) is congruent to σ(x2x1), which means that they have the same length, the difference being that they are of opposite directions. The composition of motions can be defined as follows.Definition 3Let 〈Ω2,R〉 be a difference structure. For any menus x1,x2,x3∈Ω, define operation ∘:Λ2→Λ as it is specified in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>.<elsevierMultimedia ident="tbl0005"></elsevierMultimedia>I introduce formally the concept of a geometric representation by means of the following definition.Definition 4Let 〈Ω2,R〉 be a difference structure. A function σ:Ω2→Λ is a geometric representation of R iff it satisfies the following conditions for every x,x1,x2,x2 and x4 in Ω:<ul class="elsevierStyleList" id="lis0010"><li class="elsevierStyleListItem" id="lsti0025">(1)σ(x1x2) is an interval of the first direction iff x1x2 is positive.</li><li class="elsevierStyleListItem" id="lsti0030">(2)σ(x1x2) is an interval of the second direction iff x1x2 is negative.</li><li class="elsevierStyleListItem" id="lsti0035">(3)σ(x1x2) is the null interval iff x1x2 is null.</li><li class="elsevierStyleListItem" id="lsti0040">(4)x1x2Px3x4 iff either both x1x2 and x3x4 are positive or null and σ(x1x2)⋗σ(x3x4); or both x1x2 and x3x4 are negative and σ(x1x2)⋖σ(x3x4); or σ(x1x2) is of the first direction, or null σ(x3x4), and is negative.</li><li class="elsevierStyleListItem" id="lsti0045">(5)x1x2Ex3x4 iff both σ(x1x2) and σ(x3x4) are of the same direction and σ(x1x2)≐σ(x3x4).</li><li class="elsevierStyleListItem" id="lsti0050">(6)σ(x1x3)≐σ(x1x2)∘σ(x2x3).</li><li class="elsevierStyleListItem" id="lsti0055">(7)If AB≐σ(x1x2) for some (x1x2), then, for any interval CD <elsevierMultimedia ident="201703220242154573"></elsevierMultimedia> AB (or CD <elsevierMultimedia ident="201703220242154574"></elsevierMultimedia> AB), there exist x1′,x2′∈Ω such that σx1x2′≐CD≐σx1′x2.</li></ul>It is easy to see that 〈Λ,∘〉 is a group with the null interval as identity element. The following result is immediate, as it is based upon the existence of the numerical representation (<a class="elsevierStyleCrossRef" href="#fig0005">Fig. 1</a>).<elsevierMultimedia ident="fig0005"></elsevierMultimedia>Theorem 1If 〈Ω2,R〉 is an algebraic difference structure then there exists a geometric representation σ:Ω2→Λ of R.ProofConsider any numerical representation ϕ of R. If is x1x2 positive, ψ(x1x2)=ϕ(x2)−ϕ(x1) is a positive real number and so there are points A, B on the line such that A≺B and ψ(x1x2) is equal to the interval number (cut) [AB:1→]. Let ρ be the application mapping [a:1→] into a, and define σ as follows (cf. <a class="elsevierStyleCrossRef" href="#bib0050">Hölder 1997, pp. 351–352</a>):<elsevierMultimedia ident="eq0015"></elsevierMultimedia>Clearly, by construction, σ(x1x2) is of the first direction iff x1x2 is positive; of the second iff it is negative; and null iff it is neither. Since AB≐BA, notice that σ(x1x2)≐σ(x2x1).Suppose that x1x2 and x3x4 are nonnegative and let A, B, C, D, be points such that ψ(x1x2)=[AB:1→] and ψ(x1x2)=[CD:1→]. Then we have<elsevierMultimedia ident="eq0020"></elsevierMultimedia>If both x1x2 and x3x4 are negative, x2x1 and x4x3 are positive and we have<elsevierMultimedia ident="eq0025"></elsevierMultimedia>Given x1x2Px3x4 when the intervals are of opposite signs, the case when x1x2 is negative and x3x4 is nonnegative is excluded because in such a case we would have<elsevierMultimedia ident="eq0030"></elsevierMultimedia>and so x1x2Px3x4. Hence, the only case remaining is when x1x2 is nonnegative and x3x4 is negative.It has to be shown that σ(x1x3)≐σ(x1x2)∘σ(x2x3). I refer the reader to <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>, as I shall consider case by case. Keep in mind that it is always true that<elsevierMultimedia ident="eq0035"></elsevierMultimedia>(cf. <a class="elsevierStyleCrossRef" href="#bib0045">Hölder, 1996</a>, eqn. 19, p. 243). Also, for every x1,x2,x3∈Ω,<elsevierMultimedia ident="eq0040"></elsevierMultimedia>In all cases, let ψ(x1,x2)=[AB:1→] and ψ(x2,x3)=[BC:1→]. It will suffice to show that [AC:1→]=ψ(x1,x3)Case 1: A≺B≺C. We have<elsevierMultimedia ident="eq0045"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0050"></elsevierMultimedia>Case 2: A≺B≺C. We have<elsevierMultimedia ident="eq0055"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0060"></elsevierMultimedia>Case 3: B≺A≺C. We have<elsevierMultimedia ident="eq0065"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0070"></elsevierMultimedia>Case 4: B≺C≺A. We have<elsevierMultimedia ident="eq0075"></elsevierMultimedia>or<elsevierMultimedia ident="eq0080"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0085"></elsevierMultimedia>Case 5: C≺A≺B. We have<elsevierMultimedia ident="eq0090"></elsevierMultimedia>or<elsevierMultimedia ident="eq0095"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0100"></elsevierMultimedia>Case 6: C≺B≺A. We have<elsevierMultimedia ident="eq0105"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0110"></elsevierMultimedia>Hence, at any rate, σ(x1x2)=AC and so axiom (6) of <a class="elsevierStyleCrossRef" href="#enun0020">Definition 4</a> is shown to be satisfied.Finally, assume that x1x2Rx3x4 and let [AB:1→]=ψ(x3x4), so that σ(x3x4)=AB. By axiom 3 of <a class="elsevierStyleCrossRef" href="#enun0010">Definition 2</a>, there exist x1′ and x2′ such that x1x2′Ex3x4Ex1′x2 Setting CD≐σx1x2′≐σx1′x2, condition 7 of <a class="elsevierStyleCrossRef" href="#enun0020">Definition 4</a> is satisfied.□We have shown the existence of a geometric representation of an algebraic difference structure. The point of having this representation is that it provides an adequate conceptual and linguistic apparatus to express the differentiability condition we are looking for. Moreover, it can be shown that the existence of a geometric representation for a difference structure D=〈Ω2,R〉 guarantees that D is an algebraic difference structure. For we can express the properties defining the concept of an algebraic difference structure purely in terms of the geometric representation, and show that the structure has these properties out of the axioms regulating σ. The wages of doing this is that we can also express in terms of the geometric representation all the properties of an algebraic difference structure, plus the required differentiability condition, and establish in this way the existence of a C1 numerical representation of R.I will prove in what follows that the existence of a geometric representation of difference structure D implies that D is an algebraic difference structure. The following four lemmas, all of which share the assumption that such representation exists, are devoted to this end. I will introduce later the differentiability condition. For the sake of brevity, from now on, that an interval is of the first direction will be expressed by saying that “it is I”; and that “it is II” if it is of the second direction. The null interval will be denoted as 0→Lemma 1If x1x2Rx3x4 then x4x3Rx2x1.ProofSuppose that both x1x2 and x3x4 are in X+∪X0. This means that both σ(x1x2) and σ(x3x4) are I or null, with σ(x1x2) <elsevierMultimedia ident="201703220242154575"></elsevierMultimedia>σ(x3x4) Hence, σ(x2x1) and σ(x4x3) are II or null, with σ(x2x1) <elsevierMultimedia ident="201703220242154576"></elsevierMultimedia>σ(x4x3). It follows that x4x3Rx2x1.If both are II, x1x2Rx3x4 implies that x1x2<elsevierMultimedia ident="201703220242154577"></elsevierMultimedia>x3x4 and that x2x1 and x3x4 are I. Hence, again, x4x3Rx2x1.Notice that x1x2Rx3x4 implies that x3x4 cannot be I or null if x1x2 is II. Hence, the only remaining case is when x1x2 is I or null, and x3x4 is II. In this case, x2x1 is II or null and x4x3 is I. It follows that x4x3Px2x1 and so, finally, x4x3Rx2x1.□Lemma 2If x1x2Rx1′x2′ and x2x3Rx2′x3′ then x1x3Rx1′x3′.ProofThe proof of this lemma is easy but laborious, since there are several cases to be considered. Excluding the cases precluded by the hypothesis of the proposition, there are still nine cases to consider. They are given in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>. The proof is interesting because it yields more insight into the meaning of the geometric representation.<elsevierMultimedia ident="tbl0010"></elsevierMultimedia>Case 1 is straightforward because all intervals are I or null and so x1x2Rx1′x2′, and x2x3Rx2′x3′ is tantamount to σ(x1x2) <elsevierMultimedia ident="201703220242154578"></elsevierMultimedia>σx1′x2′ and σ(x2x3) <elsevierMultimedia ident="201703220242154579"></elsevierMultimedia>σx2′x3′. We have<elsevierMultimedia ident="eq0115"></elsevierMultimedia>Hence, by <a class="elsevierStyleCrossRef" href="#bib0045">Hölder's (1996, p. 238)</a> conclusion 2,<elsevierMultimedia ident="eq0120"></elsevierMultimedia>or, equivalently,<elsevierMultimedia ident="eq0125"></elsevierMultimedia>In case 4 there is nothing to prove because σ(x1x3) is I or null and σx1′x3′ is II.In cases 2 and 3, σx1′x3′ can be I or null, or II. When it is II, we are done, because σ(x1x3) is I or null. When σx1′x3′ is or null, we have<elsevierMultimedia ident="eq0130"></elsevierMultimedia>if σx2′x3′0→, or<elsevierMultimedia ident="eq0135"></elsevierMultimedia>if σx1′x2′0→In case 5 we have σ(x1x2) <elsevierMultimedia ident="2017032202421545710"></elsevierMultimedia>σx1′x2′ and σ(x2x2) <elsevierMultimedia ident="2017032202421545711"></elsevierMultimedia>σx2′x3′. We have five subcases, setting B=B′ (see <a class="elsevierStyleCrossRef" href="#fig0010">Fig. 2</a>).<elsevierMultimedia ident="fig0010"></elsevierMultimedia>Subcase (5.1). A≺A′≺C′≺C≺B=B′. In this case we have that both ψ(x1x3) and σx1′x3′ are I or null with<elsevierMultimedia ident="eq0140"></elsevierMultimedia>It follows that x1x3Rx1′x3′.Subcase (5.2). A≺C′≺A′≺C≺B=B′.Subcase (5.3). A≺C′≺C≺A′≺B=B′.Subcase (5.4). C′≺A≺C≺A′≺B=B′.In cases (5.2)–(5.4), AC≐σ(x1x3) is I or null, whereas A′C′≐σx1′x3′ is II. It is immediate that x1x3Rx1′x3′ in these cases.Subcase (5.5). C′≺C≺A≺A′≺B=B′. In this final case, both AC≐σ(x1x3) and A′C′≐σx1′x3′ are II but σ(x1x3) <elsevierMultimedia ident="2017032202421545712"></elsevierMultimedia>σx1′x3′, and so x1x3Rx1′x3′.Case 7 is entirely analogous to case 5.In cases, 6 and 8 σ(x1x3) can be I, null, or II. If it is I or null, we are done. If σ(x1x3) is II, in case 6 we have C≺A≺B and C′≺B′≺A′ with<elsevierMultimedia ident="eq0145"></elsevierMultimedia>Thus, x1x3Rx1′x3′. An analogous argument leads to the same conclusion in case 8.Finally, in case 9, all segments are II and we have both σ(x1x2) <elsevierMultimedia ident="2017032202421545713"></elsevierMultimedia>σx1′x2′ and σ(x2x3) <elsevierMultimedia ident="2017032202421545714"></elsevierMultimedia>σx2′x3′ Hence, σx1′x3′<elsevierMultimedia ident="2017032202421545715"></elsevierMultimedia>σ(x1x3) and so x1x3Rx1′x3′.□Lemma 3if x1x2Rx3x4, then there exist x1′,x2′∈Ω such that x1x2′Ex3x4 and x3x4Ex1′x2.ProofAssume that x1x2Rx3x4. If x1x2Ex3x4, there is nothing to prove, and so we may suppose that x1x2Sx3x4.Let CD≐σ(x3x4). By axiom 7 of <a class="elsevierStyleCrossRef" href="#enun0020">Definition 4</a>, there exist menus x1′,x2′ such that σx1′x2≐CD≐σx1x2′. It follows that x1′x2Ex3x4Ex1x2′.□Lemma 4Every strictly bounded sequence is finite.ProofLet (xk) be a bounded standard sequence and x,x′∈Ω be such that xx′Sx1xk for all xk in the sequence. Let AB=σ(xkxk+1) and CD=σ(xx′). Then CDkAB for every k such that xk is in the sequence. But, (xk) if were not finite, there would be one such positive integer k with kABCD (cf. <a class="elsevierStyleCrossRef" href="#bib0045">Hölder, 1996</a>, p. 239).□Using the previous lemmas and <a class="elsevierStyleCrossRef" href="#enun0025">Theorem 1</a>, we can establish the following proposition.Theorem 2There exists a geometric representation of a difference structure D iff D is an algebraic-difference structure.Hence, the existence of a geometric representation of D is necessary and sufficient for D to be an algebraic-difference structure, and indeed implies the existence of a numerical representation of D. Yet, as the reader shall presently see, the geometric language has more expressive power than the algebraic one.The theory of preferenceA preference relation among consumption menus in Ω can be defined out of the difference relation. As we said, if x1x2 is I or null, x2 is weakly preferred by the agent to x1. We may express this preference relation by means of symbol ≻˜.Definition 5For consumption menus x1,x2 in Ω, say that x1 is weakly preferred to x2, and write x1≻˜x2, iff x2x1 is I or null.The kin notions of strict preference and indifference, denoted by symbols ≻ and ∼, are defined as usual. Clearly, ϕ as characterized by the numerical representation of the difference relation is a utility function representing ≻˜, for we have<elsevierMultimedia ident="eq0150"></elsevierMultimedia>Actually, all the properties that have been attributed to the preference relation in microeconomic textbooks can be defined in terms of relations among geometric intervals within the given Euclidean straight line, just as we did with the properties of the difference relation. This shows that the language of intervals, being more powerful than the usual language used in economic theory, is a suitable way of expressing the theories of difference and preference. I hope the reader will find natural this way of expressing the properties of R (some usual ones are given below), particularly the one implying differentiability. We keep assuming that 〈Ω2,R〉 is an algebraic difference structure.Definition 6≻˜ is monotonic iff, for all x1x2∈Ω, x1≥x2 implies that σ(x2x1) is an interval of the first direction.Definition 7≻˜ is continuous at x0∈Ω iff, for every interval a∈Λ, as small as you wish, there is a δ>0 such that x0+h∈Ω and σ(x0(x0+h))a whenever h<δ.Definition 8≻˜ is strictly convex iff, for every α∈[0,1] and menus x1,x2∈Ω,x1≠x2,αx1+(1−α)x2∈Ω, and σ(αx1+(1−α)x2) is of the first direction whenever σ(x1x2) is of the first direction.The great advantage of the language of intervals over the languages typically used to formulate preference theories is that it provides resources by means of which we can also express natural, intuitive differentiability conditions for the preference relation. We turn now to these.The empirical meaning of differentiabilityDerivates are, and cannot be, but ratios between homogeneous magnitudes. That is why it is necessary to represent satisfaction differences by means of intervals within the same geometric space in which distances among consumption vectors are represented. Notice that there is a natural mapping τ from the segments within Ω into Λ, namely τ(x1x2) is (the equivalence class of) that interval in Λ whose Euclidean norm is x1−x2. Notice that the cut [τ(x1x2):1→] is precisely x1−x2. Using Hölder's interval numbers (cuts) or measure-numbers (<a class="elsevierStyleCrossRef" href="#bib0045">Hölder, 1996</a>, p. 242), as we did in the proof of <a class="elsevierStyleCrossRef" href="#enun0025">Theorem 1</a>, it is possible to map the Ω-segments into Λ. In particular all segments of length 1 are mapped by τ into segment 1→≐NE in Λ.Recall that a real-valued function defined on an open subset D of ℝL is continuously differentiable at x∈D iff all its partial derivatives exist throughout a neighborhood of x and are continuous at x. Hence, our aim is to find conditions over the satisfaction differences (or their proxies in Λ) implying the existence of a function ϕ fulfilling these requirements.The empirical meaning of the differentiability condition is that the agent's tastes have a certain sort of stability. That is to say, the rate of change of the agent's satisfaction is almost constant within a small vicinity of any consumption menu x and, at any rate, it varies continuously in any given direction. This means that, within the infinitesimal neighborhood of x (within the “halo” or “monad” of x) the rate of change of satisfaction of the agent is “almost” constant. This implies, in particular, that in an arbitrarily given direction, determined by the unit vector u, the ratio of the satisfaction difference between any two menus to their physical difference is “almost” constant. How can this condition be expressed in a formal way?Let x be an arbitrary point in the interior of Ω and u a unit vector in a given fixed direction. Following Newton's conception of the theory of proportions, the ratio of one magnitude to another of the same kind is to be expressed as a real positive number,<a class="elsevierStyleCrossRef" href="#fn0030">6</a> and so the required condition is that, for any infinitesimal number ¿, the ratio of the satisfaction segment σ(x(x+¿u)) to the quantity segment τ(x(x+¿u)) be infinitely close to a certain positive real number which we can conveniently identify with the cut [Δ(x):1→]. Naturally, we want to identify the cut [Δ(x):1→] with a certain directional derivative. Hence, the condition we are looking for can be formulated, in the language of intervals, in the following way.If x1,x2 are menus in Ω, let us denote with (x1,x2) the set<elsevierMultimedia ident="eq0155"></elsevierMultimedia>Notice that the points in (x1,x2) are interior points of Ω whenever at least one of the two points x1,x2 is an interior point (see <a class="elsevierStyleCrossRef" href="#fig0015">Fig. 3</a>).Definition 9≻˜ is uniformly differentiable or smooth on (x1, x2) if there is a function<elsevierMultimedia ident="eq0160"></elsevierMultimedia>such that, for any menus x3,x4∈(x1,x2), any x∈(x3,x4) and ¿>0, there exist positive integers μ,v,μ′,v′,μ″,v″ and δ>0 such that, whenever h<δ with x+h∈(x1,x2), |μ″/v″−μ/v|<ε and<elsevierMultimedia ident="eq0165"></elsevierMultimedia>if<elsevierMultimedia ident="eq0170"></elsevierMultimedia>or<elsevierMultimedia ident="eq0175"></elsevierMultimedia>if<elsevierMultimedia ident="eq0180"></elsevierMultimedia>We say that ≻˜ is uniformly differentiable iff it is uniformly differentiable on every set (x1,x2) with at least one of x1,x2 being an interior point of Ω.Theorem 3If ≻˜ is uniformly differentiable, then there exists a utility function representing ≻˜ which is continuously differentiable in the interior of Ω.ProofWe will show, first, that there is a uniformly differentiable function ϕˆ on any open interval. Let (x1,x2) be any such interval, x3,x4 any menus in (x1,x2) with x3≠x4, x any menu in (x3,x4) and ¿ any positive number, as small as you wish.<elsevierMultimedia ident="fig0015"></elsevierMultimedia>It follows that there exist positive integers μ,v,μ′,v′,μ″,v″ and δ>0 such that, whenever h<δ with x|+h∈(x1,x2), |μ″/v″−μ/v|<ε and<elsevierMultimedia ident="eq0185"></elsevierMultimedia>if<elsevierMultimedia ident="eq0190"></elsevierMultimedia>or<elsevierMultimedia ident="eq0195"></elsevierMultimedia>if<elsevierMultimedia ident="eq0200"></elsevierMultimedia>Assume that<elsevierMultimedia ident="eq0205"></elsevierMultimedia>let ψ(x1x2)=[σ(x1x2):1→], and choose ϕ with ϕ(x2)−ϕ(x1)=ψ(x1x2). Since<elsevierMultimedia ident="eq0210"></elsevierMultimedia>for any segment a∈Λ (cf. <a class="elsevierStyleCrossRef" href="#bib0045">Hölder, 1996</a>, p. 244),<elsevierMultimedia ident="eq0215"></elsevierMultimedia>If we let ϕˆ be the function defined by condition<elsevierMultimedia ident="eq0220"></elsevierMultimedia>it follows that<elsevierMultimedia ident="eq0225"></elsevierMultimedia>with<elsevierMultimedia ident="eq0230"></elsevierMultimedia>Hence,<elsevierMultimedia ident="eq0235"></elsevierMultimedia>The assumption that<elsevierMultimedia ident="eq0240"></elsevierMultimedia>leads, by an analogous argument, to<elsevierMultimedia ident="eq0245"></elsevierMultimedia>Thus, at any rate,<elsevierMultimedia ident="eq0250"></elsevierMultimedia>This establishes that ϕ is uniformly differentiable on the interval (x1,x2) (with derivative ϕˆ(x) at point x) and, therefore, the derivative of any point within the interval is continuous.Consider now, for any interior point x0 of Ω, in particular, intervals of the form (x0−αel,x0+αel)⊂Ω where el is the canonical vector in direction l. The derivative of ϕ at x0 is then nothing but the partial derivative of ϕ with respect to xl evaluated at x0:<elsevierMultimedia ident="eq0255"></elsevierMultimedia>Since this derivative is continuous for every l (l=1,…,L), we may conclude that ϕ is continuously differentiable at x0. As x0 was arbitrarily chosen, we may conclude that ϕ is continuously differentiable in the interior of Ω.□New foundations of preference theoryThe previous argument shows that we can provide a new conceptual apparatus for preference theory by means of which all the usual properties of the preference relation, as well as the differentiability condition, can be expressed. Is it possible to find another conceptual apparatus that allows the expression of some differentiability condition? That is unlikely because differentiability requires the comparison of satisfaction distances with quantity distances within the same space. At any rate, it is incumbent upon those who believe that it is feasible to do so to produce such an apparatus.By <a class="elsevierStyleCrossRef" href="#enun0075">Theorem 2</a>, a difference structure D is an algebraic-difference structure iff D admits a geometric representation. Hence, the new conceptual apparatus for preference theory can be briefly summarized as follows. I use the term ‘geometric-difference’ to avoid confusion.Definition 10D is a geometric-difference structure iff there exist Ω, R and σ such that<ul class="elsevierStyleList" id="lis0015"><li class="elsevierStyleListItem" id="lsti0060">(1)D=〈Ω2,R,σ,Λ〉;</li><li class="elsevierStyleListItem" id="lsti0065">(2)〈Ω2,R〉 is a weak order;</li><li class="elsevierStyleListItem" id="lsti0070">(3)σ:Ω2→Λ is a geometric representation of R.</li></ul>As pointed out in section ‘The theory of preference’, the preference relation ≻˜ can be defined in terms of σ and all the properties usually attributed to it can be expressed using the language of geometric intervals (see the examples there). The novelty — as I have just shown — is that the smoothness condition can also be so expressed. Thus, using the notion introduced in <a class="elsevierStyleCrossRef" href="#enun0100">Definition 9</a> we can define the concept of a smooth preference structure.Definition 11<elsevierMultimedia ident="2017032202421545716"></elsevierMultimedia> is a smooth preference structure iff there exist Ω and ≻˜ such that<ul class="elsevierStyleList" id="lis0020"><li class="elsevierStyleListItem" id="lsti0075">(1)<elsevierMultimedia ident="2017032202421545717"></elsevierMultimedia></li><li class="elsevierStyleListItem" id="lsti0080">(2)≻˜ is a preference relation induced by a geometric-difference structure;</li><li class="elsevierStyleListItem" id="lsti0085">(3)≻˜ is smooth.</li></ul>By virtue of <a class="elsevierStyleCrossRef" href="#enun0105">Theorem 3</a>, there is a continuously differentiable utility function ϕ:Ω→ℝ representing ≻˜. The relevance and usefulness of having such a function lies in that it allows the application of non-linear programming techniques in order to find the optimal points.It would be desirable, because the empirical condition would be even more intuitive, to formulate entirely the smoothness condition in non-standard language, by means of the notion of an infinitesimal segment, as intimated in the informal discussion preceding the formal introduction of the condition. That is entirely feasible because Euclid's Archimedian axiom is logically independent of the rest,<a class="elsevierStyleCrossRef" href="#fn0035">7</a> but it would require a complete reformulation of Hölder's theory, as well as of the theory of algebraic-difference measurement." "textoCompletoSecciones" => array:1 [ "secciones" => array:15 [ 0 => array:3 [ "identificador" => "xres818186" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec815215" "titulo" => "Keywords" ] 2 => array:2 [ "identificador" => "xpalclavsec815212" "titulo" => "JEL classification" ] 3 => array:3 [ "identificador" => "xres818185" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 4 => array:2 [ "identificador" => "xpalclavsec815214" "titulo" => "Palabras clave" ] 5 => array:2 [ "identificador" => "xpalclavsec815213" "titulo" => "Códigos JEL" ] 6 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 7 => array:2 [ "identificador" => "sec0010" "titulo" => "The conditions of Debreu and Mas-Colell" ] 8 => array:2 [ "identificador" => "sec0015" "titulo" => "Motivation" ] 9 => array:2 [ "identificador" => "sec0020" "titulo" => "The algebraic theory of difference" ] 10 => array:2 [ "identificador" => "sec0025" "titulo" => "The geometric theory of difference" ] 11 => array:2 [ "identificador" => "sec0030" "titulo" => "The theory of preference" ] 12 => array:2 [ "identificador" => "sec0035" "titulo" => "The empirical meaning of differentiability" ] 13 => array:2 [ "identificador" => "sec0040" "titulo" => "New foundations of preference theory" ] 14 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2014-11-21" "fechaAceptado" => "2015-06-17" "PalabrasClave" => array:2 [ "en" => array:2 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Keywords" "identificador" => "xpalclavsec815215" "palabras" => array:4 [ 0 => "Differentiable preferences" 1 => "Utility theory" 2 => "Rational choice" 3 => "Foundations of consumer theory" ] ] 1 => array:4 [ "clase" => "jel" "titulo" => "JEL classification" "identificador" => "xpalclavsec815212" "palabras" => array:2 [ 0 => "B400" 1 => "D010" ] ] ] "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Palabras clave" "identificador" => "xpalclavsec815214" "palabras" => array:4 [ 0 => "Preferencias diferenciables" 1 => "Teoría de la utilidad" 2 => "Elección racional" 3 => "Fundamentos de la teoría del consumidor" ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "en" => array:2 [ "titulo" => "Abstract" "resumen" => "This is a paper on the foundations of individual rational choice, specifically on the foundations of consumer theory. Neoclassical consumer theory requires that the behavior of the consumer be explained by means of a preference relation, and that all the required properties of the corresponding utility representation be derived from the properties of this relation. Yet, it is not clear what is the meaning of the property of the preference relation required in order to show that it is representable by means of a continuously differentiable C1 utility function. The aim of the present paper is to propose an explanation of such property and to prove the existence of a C1 representation." ] "es" => array:2 [ "titulo" => "Resumen" "resumen" => "Este es un artículo sobre los fundamentos de la teoría de la elección racional individual, específicamente sobre los fundamentos de la teoría del consumidor. La teoría neoclásica del consumidor requiere que el comportamiento del consumidor sea explicado mediante una relación de preferencia y que todas las propiedades requeridas de la correspondiente función de utilidad que la represente sean derivadas de las propiedades de esta relación. No obstante, no está claro cuál es el significado de la propiedad de la relación de preferencia requerida para mostrar que la misma es representable mediante una función continuamente diferenciable (C1). El propósito del presente artículo es proponer una explicación de tal propiedad y demostrar la existencia de una representación." ] ] "NotaPie" => array:8 [ 0 => array:1 [ "nota" => "Peer Review under the responsibility of Universidad Nacional Autónoma de México." ] 1 => array:3 [ "etiqueta" => "1" "nota" => "See, for instance, <a class="elsevierStyleCrossRef" href="#bib0055">Katzner (1970)</a>, <a class="elsevierStyleCrossRef" href="#bib0005">Barten and Böhm (1981)</a>, and <a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell, Whinston and Green (1995)</a>." "identificador" => "fn0005" ] 2 => array:3 [ "etiqueta" => "2" "nota" => "Because the dimension of Ω is, which is the same as that of ℝL. This implies that the relative interior of Ω with respect to the linear space ℝL is nonempty and so, for sufficiently small ¿>0, the open ball Be(x) centered in x is contained in Ω. For a discussion of the notion of relative interior that relates it to economic theory, see <a class="elsevierStyleCrossRef" href="#bib0060">Koopmans (1951)</a>, especially p. 45." "identificador" => "fn0010" ] 3 => array:3 [ "etiqueta" => "3" "nota" => "Precisely in the sense of Definition 3 in <a class="elsevierStyleCrossRef" href="#bib0065">Krantz, Luce, Suppes and Tversky (1971, p. 151)</a>." "identificador" => "fn0015" ] 4 => array:3 [ "etiqueta" => "4" "nota" => "See Theorem 2 in <a class="elsevierStyleCrossRef" href="#bib0065">Krantz et al. (1971, p. 151)</a>; see p. 158 for a proof." "identificador" => "fn0020" ] 5 => array:3 [ "etiqueta" => "5" "nota" => "Von gleicher Richtung in the original." "identificador" => "fn0025" ] 6 => array:3 [ "etiqueta" => "6" "nota" => "Cf. <a class="elsevierStyleCrossRef" href="#bib0045">Hölder (1996, p. 241, 8)</a>." "identificador" => "fn0030" ] 7 => array:3 [ "etiqueta" => "7" "nota" => "Cf. <a class="elsevierStyleCrossRefs" href="#bib0035">Hilbert (1950, pp. 21–22); Fleuriot (2001, chap. 4)</a>." "identificador" => "fn0035" ] ] "multimedia" => array:73 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Fig. 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 463 "Ancho" => 500 "Tamanyo" => 14264 ] ] "descripcion" => array:1 [ "en" => "σ is the composition ρ∘ψ:Ω2→Λ." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Fig. 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 95 "Ancho" => 1971 "Tamanyo" => 10096 ] ] "descripcion" => array:1 [ "en" => "Both AB and A′B′ are I (or null): A≺A′≺B=B′." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Fig. 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 1251 "Ancho" => 1219 "Tamanyo" => 25380 ] ] "descripcion" => array:1 [ "en" => "The segment (x1, x2) and a point x within it. Notice that the extremes, x1, x2, do not belong to the segment." ] ] 3 => array:7 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " colspan="5" align="center" valign="top" scope="col" style="border-bottom: 2px solid black">Table of motion compositions</th></tr><tr title="table-row"><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Case \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σ(x1x2) \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σ(x2x3) \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Directions \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σ(x1x2)∘σ(x1x3) \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">AB \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">BC \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">A≺B≺C \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">AC \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">2 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">AB \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">CB \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">A≺C≺B \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">AC \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">BA \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">BC \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">B≺A≺C \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">AC \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">BA \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">BC \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">B≺C≺A \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">CA \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">AB \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">CB \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">C≺A≺B \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">CA \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">BA \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">CB \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">C≺B≺A \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">CA \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab1375218.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Definition of composition of motions." ] ] 4 => array:7 [ "identificador" => "tbl0010" "etiqueta" => "Table 2" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " colspan="7" align="center" valign="top" scope="col" style="border-bottom: 2px solid black">Feasible cases in <a class="elsevierStyleCrossRef" href="#enun0055">Lemma 3</a></th></tr><tr title="table-row"><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Case \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σ(x1x2) \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σ(x2x3) \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σx1′x2′ \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σx2′x3′ \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σ(x1x3) \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">σx1′x3′ \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">2 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I or null \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">any \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">I \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">II \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab1375217.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Feasible cases in <a class="elsevierStyleCrossRef" href="#enun0055">Lemma 3</a>." ] ] 5 => array:5 [ "identificador" => "201703220242154571" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx1.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 974 ] ] ] 6 => array:5 [ "identificador" => 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=> "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx2.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 984 ] ] ] 11 => array:5 [ "identificador" => "201703220242154577" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx1.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 974 ] ] ] 12 => array:5 [ "identificador" => "201703220242154578" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx2.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 984 ] ] ] 13 => array:5 [ "identificador" => "201703220242154579" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx2.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 984 ] ] ] 14 => array:5 [ "identificador" => "2017032202421545710" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx2.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 984 ] ] ] 15 => array:5 [ "identificador" => "2017032202421545711" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx1.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 974 ] ] ] 16 => array:5 [ "identificador" => "2017032202421545712" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx1.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 974 ] ] ] 17 => array:5 [ "identificador" => "2017032202421545713" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx1.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 974 ] ] ] 18 => array:5 [ "identificador" => "2017032202421545714" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx1.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 974 ] ] ] 19 => array:5 [ "identificador" => "2017032202421545715" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx2.jpeg" "Alto" => 50 "Ancho" => 32 "Tamanyo" => 984 ] ] ] 20 => array:5 [ "identificador" => "2017032202421545716" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx7.jpeg" "Alto" => 100 "Ancho" => 87 "Tamanyo" => 2970 ] ] ] 21 => array:5 [ "identificador" => "2017032202421545717" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "fx8.jpeg" "Alto" => 153 "Ancho" => 730 "Tamanyo" => 15520 ] ] ] 22 => array:5 [ "identificador" => "eq0005" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x1x2Rx3x4 iff x4x3Rx2x1" "Fichero" => "STRIPIN_si24.jpeg" "Tamanyo" => 1638 "Alto" => 14 "Ancho" => 191 ] ] 23 => array:5 [ "identificador" => "eq0010" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x1x2Rx3x4 iff ϕ(x2)−ϕ(x1)≥ϕ(x4)−ϕ(x3)." "Fichero" => "STRIPIN_si42.jpeg" "Tamanyo" => 2662 "Alto" => 16 "Ancho" => 327 ] ] 24 => array:5 [ "identificador" => "eq0015" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "σ(x1x2)=ABif[AB:1→]=ψ(x1x2)>0AAifψ(x1x2)=0BAif[AB:1→]=ψ(x2x1)>0" "Fichero" => "STRIPIN_si67.jpeg" "Tamanyo" => 5908 "Alto" => 71 "Ancho" => 315 ] ] 25 => array:5 [ "identificador" => "eq0020" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x1x2Px3x4⇔ψ(x1x2)>ψ(x3x4) ⇔[AB:1→]>[CD:1→] ⇔AB⋗CD ⇔σ(x1x2)⋗(x3x4)" "Fichero" => "STRIPIN_si72.jpeg" "Tamanyo" => 5600 "Alto" => 90 "Ancho" => 244 ] ] 26 => array:5 [ "identificador" => "eq0025" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x1x2Px3x4⇔x4x3Px2x1 ⇔σ(x4x3)σ(x2x1) ⇔σ(x1x2)σ(x3x4)" "Fichero" => "STRIPIN_si73.jpeg" "Tamanyo" => 4016 "Alto" => 63 "Ancho" => 212 ] ] 27 => array:5 [ "identificador" => "eq0030" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x3x4Rxx∧xxPx1x2," "Fichero" => "STRIPIN_si74.jpeg" "Tamanyo" => 1394 "Alto" => 13 "Ancho" => 140 ] ] 28 => array:5 [ "identificador" => "eq0035" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[a:1→]+[a′:1→]=[aa′:1→]" "Fichero" => "STRIPIN_si76.jpeg" "Tamanyo" => 1670 "Alto" => 20 "Ancho" => 179 ] ] 29 => array:5 [ "identificador" => "eq0040" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "ψ(x1x2)+ψ(x2,x3)=ϕ(x2)−ϕ(x1)+ϕ(x3)−ϕ(x2) ϕ(x3)−ϕ(x1) ψ(x1,x3)" "Fichero" => "STRIPIN_si77.jpeg" "Tamanyo" => 4565 "Alto" => 64 "Ancho" => 369 ] ] 30 => array:5 [ "identificador" => "eq0045" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[AB:1→]+[BC:1→]" "Fichero" => "STRIPIN_si81.jpeg" "Tamanyo" => 1969 "Alto" => 20 "Ancho" => 207 ] ] 31 => array:5 [ "identificador" => "eq0050" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[AB:1→]+[BC:1→] =ψ(x1x2)+ψ(x1x2) =ψ(x1x3)" "Fichero" => "STRIPIN_si82.jpeg" "Tamanyo" => 3933 "Alto" => 69 "Ancho" => 219 ] ] 32 => array:5 [ "identificador" => "eq0055" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]+[CB:1→]=[AB:1→]" "Fichero" => "STRIPIN_si83.jpeg" "Tamanyo" => 2029 "Alto" => 20 "Ancho" => 207 ] ] 33 => array:5 [ "identificador" => "eq0060" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[AB:1→]−[CB:1→] =[AB:1→]−[BC:1→] =ψ(x1x2)+ψ(x1x2) =ψ(x1x3)" "Fichero" => "STRIPIN_si84.jpeg" "Tamanyo" => 5137 "Alto" => 95 "Ancho" => 219 ] ] 34 => array:5 [ "identificador" => "eq0065" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[BA:1→]+[AC:1→]=[BC:1→]" "Fichero" => "STRIPIN_si85.jpeg" "Tamanyo" => 1993 "Alto" => 20 "Ancho" => 207 ] ] 35 => array:5 [ "identificador" => "eq0070" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[BC:1→]−[BA:1→] =[AB:1→]+[BC:1→] =ψ(x1x2)+ψ(x1x2) =ψ(x1x3)" "Fichero" => "STRIPIN_si86.jpeg" "Tamanyo" => 5158 "Alto" => 95 "Ancho" => 219 ] ] 36 => array:5 [ "identificador" => "eq0075" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[BC:1→]+[CA:1→]=[BA:1→]" "Fichero" => "STRIPIN_si87.jpeg" "Tamanyo" => 1996 "Alto" => 20 "Ancho" => 206 ] ] 37 => array:5 [ "identificador" => "eq0080" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[CA:1→]=[BA:1→]−[BC:1→]" "Fichero" => "STRIPIN_si88.jpeg" "Tamanyo" => 1983 "Alto" => 20 "Ancho" => 206 ] ] 38 => array:5 [ "identificador" => "eq0085" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[BC:1→]−[BA:1→] =[AB:1→]+[BC:1→] =ψ(x1x2)+ψ(x1x2) =ψ(x1x3)" "Fichero" => "STRIPIN_si86.jpeg" "Tamanyo" => 5158 "Alto" => 95 "Ancho" => 219 ] ] 39 => array:5 [ "identificador" => "eq0090" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[CA:1→]+[AB:1→]−[CB:1→]" "Fichero" => "STRIPIN_si89.jpeg" "Tamanyo" => 1984 "Alto" => 20 "Ancho" => 205 ] ] 40 => array:5 [ "identificador" => "eq0095" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[CA:1→]=[CB:1→]−[AB:1→]" "Fichero" => "STRIPIN_si90.jpeg" "Tamanyo" => 1954 "Alto" => 20 "Ancho" => 206 ] ] 41 => array:5 [ "identificador" => "eq0100" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[AB:1→]−[CB:1→][AC:1→]=[AB:1→]−[CB:1→] =ψ(x1x2)+ψ(x1x2) =ψ(x1x3)" "Fichero" => "STRIPIN_si91.jpeg" "Tamanyo" => 5686 "Alto" => 95 "Ancho" => 218 ] ] 42 => array:5 [ "identificador" => "eq0105" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[CB:1→]+[BA:1→]=[CA:1→]" "Fichero" => "STRIPIN_si92.jpeg" "Tamanyo" => 1990 "Alto" => 20 "Ancho" => 206 ] ] 43 => array:5 [ "identificador" => "eq0110" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "[AC:1→]=[AB:1→]+[BC:1→] =ψ(x1x2)+ψ(x1x2) =ψ(x1x3)" "Fichero" => "STRIPIN_si82.jpeg" "Tamanyo" => 3933 "Alto" => 69 "Ancho" => 219 ] ] 44 => array:5 [ "identificador" => "eq0115" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "σ(x1x3)≐σ(x1x2)⊕σ(x2x3) and σx1′x3′≐σx1′x2′⊕σx2′x3′." 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Continuously differentiable preferences

Preferencias continuamente diferenciables

Adolfo García de la Sienra

Universidad Veracruzana, Mexico

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    "textoCompleto" => "<span class="elsevierStyleSections"><p id="par0005" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleDisplayedQuote" id="dsq0005"><p id="spar0015" class="elsevierStyleSimplePara elsevierViewall">A <span class="elsevierStyleBold">ratio</span> is a sort of relation in respect of size between two magnitudes of the same kind&#46;</p><p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">Euclid&#44; Elements</p><p id="spar0025" class="elsevierStyleSimplePara elsevierViewall">Book V&#44; Definition 3</p></span></p><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Introduction</span><p id="par0010" class="elsevierStylePara elsevierViewall">Just as classical dynamics proposed to explain the motion of bodies by means of the concept of force&#44; neoclassical consumer theory proposes to explain the behavior of the consumer by means of the concept of preference&#46; This is done by taking&#44; as a starting point&#44; a regular preference structure defined by axioms that actually attribute empirically meaningful &#40;even though idealized&#41; properties to the preference relation&#46; Among these properties&#44; strict convexity&#44; non-satiation or continuity can be mentioned&#46; Restrictions on preference relations translate into restrictions on the form of the utility functions&#46; For instance&#44; if the preference relation is strictly convex&#44; the corresponding utility representation is strictly quasi-concave&#59; if the relation is non-satiated&#44; the corresponding utility representation is monotonically increasing&#59; if the relation is continuous&#44; the corresponding utility representation is also continuous&#46; Certain specializations of the theory require&#44; additionally&#44; that the utility function representing the preference relation be differentiable&#44; in order to apply methods of nonlinear programming to the derivation of the demand functions&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">Even though some of the aforementioned properties are deemed as &#8220;non-substantial&#8221; and &#8220;technical&#8221; by economists of a positivist and instrumentalist philosophical persuasion&#44; nonetheless the tendency has been to formulate them by means of natural and intuitive conditions that depict an idealized consumer described by set-theoretical structures into which the empirical data can be imbedded&#46;<a class="elsevierStyleCrossRef" href="#fn0005"><span class="elsevierStyleSup">1</span></a> For the actual meaning of the axioms defining the structures is important&#58; the more idealized they are&#44; the less precise are the empirical consequences of the same&#44; and it is impossible to check intuitively their degree of idealization if their economic meaning is unknown&#46; I think that the reason why it is said &#40;for instance by <a class="elsevierStyleCrossRef" href="#bib0005">Barten &#38; B&#246;hm&#44; 1981</a>&#44; pp&#46; 385&#8211;386&#41; that even though &#8220;Axioms 1&#8211;3 &#91;reflexivity&#44; transitivity and completeness&#93; describe order properties of a preference relation that have intuitive meaning in the context of the theory of choice &#8230; &#91;this&#93; is much less so with the topological conditions which are usually assumed as well&#8221; is that the language of topology obscures such intuitive meaning altogether because it is not suitable to express the economic meaning of such properties&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">It is not really difficult to formulate conditions like continuity or convexity in intuitive terms&#44; but the differentiability condition has turned out to be more resilient to such treatment&#46; Certainly&#44; Gerard <a class="elsevierStyleCrossRefs" href="#bib0010">Debreu &#40;1983a&#44; 1983b&#41;</a> and Andreu <a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell &#40;1985&#41;</a> have provided conditions over a preference relation that imply the existence of continuously differentiable utility functions&#46; The problem is that &#8212; in contradistinction to the properties I referred to previously &#8212; these conditions are admittedly not intuitive&#46; The first aim of the present paper is to propose a language in which all the usual properties attributed to the preference relation&#44; including differentiability&#44; can be formulated in a natural&#44; intuitive way&#46; Even if differentiability is deemed as a mere technical computational convenience&#44; without any actual empirical meaning&#44; the condition presented here is mathematically simpler &#40;once the language has been assimilated&#41; than the ones presented by Debreu and Mas-Colell &#40;which rely upon the heavy machinery of differential topology&#41;&#44; and is formulated within the framework of a unified language and conceptual apparatus that clarifies its relationship with the concept of preference strength&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">After discussing&#44; in the second section&#44; the conditions proposed by Gerard <a class="elsevierStyleCrossRefs" href="#bib0010">Debreu &#40;1983a&#44; 1983b&#41;</a> and Andreu <a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell &#40;1985&#41;</a>&#44; in the third I will motivate and state&#44; in intuitive numerical terms&#44; the required differentiability condition&#46; The fourth section will be devoted to introduce the algebraic theory of difference as a preparation to present&#44; in the fifth section&#44; the conceptual and linguistic apparatus required to provide a geometric theory of preference strength within which differentiability &#40;actually all the usual&#41; conditions can be formulated in an intuitive way&#46; The sixth section contains a development of preference theory within the proposed conceptual apparatus&#44; up to the proof of the existence of a <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span> utility function for the preference relation&#46; The seventh section introduces the differentiability condition and the eight and final one discusses the relevance and importance of having a continuously differentiable utility function&#46; The paper ends with a reflection on the convenience of formulating a non-standard version of H&#246;lder&#39;s theory in order to formulate the differentiability condition in an even more intuitive way&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">The conditions of Debreu and Mas-Colell</span><p id="par0030" class="elsevierStylePara elsevierViewall">According to <a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell et al&#46; &#40;1995&#44; p&#46; 49&#41;</a> &#8220;it is possible to give a condition purely in terms of preferences&#8221; implying the existence of a <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">2</span> utility representation of the same&#58;<span class="elsevierStyleDisplayedQuote" id="dsq0010"><p id="spar0030" class="elsevierStyleSimplePara elsevierViewall">Intuitively&#44; what is required is that indifference sets be smooth surfaces that fit together nicely so that the rates at which commodities substitute for each other depend differentially of the consumption levels&#46; &#40;<a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell et al&#46;&#44; 1995</a>&#41;</p></span></p><p id="par0035" class="elsevierStylePara elsevierViewall">The problem is that it is not at all clear which empirically meaningful &#40;even if idealized&#41; property must the preference relation of a consumer have so that its indifference sets &#8220;fit together nicely&#8221;&#46; What is worse&#44; <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">2</span> differentiability is restrictive because some demand functions that are derivable do not come from a <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">2</span> utility function&#44; as the same authors have noticed &#40;<a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell et al&#46;&#44; 1995</a>&#44; p&#46; 95&#44; n&#46; 33&#41;&#46; Furthermore&#44; as <a class="elsevierStyleCrossRef" href="#bib0010">Debreu &#40;1983a&#44; p&#46; 201&#41;</a> has pointed out&#44; it is enough for the utility function to be <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span> in order to guarantee that the corresponding Walrasian demand function be also <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span>&#46; Is it possible to find a condition that can be considered sufficiently natural and general for that purpose&#63; My claim is that it is possible&#44; and I intend to substantiate this claim by means of the intuitive discussion motivating <a class="elsevierStyleCrossRef" href="#enun0100">Definition 9</a>&#46;</p><p id="par0040" class="elsevierStylePara elsevierViewall">The differentiability condition has been interpreted by Mas-Colell in terms of the concept of a differentiable manifold&#44; giving rise to the following important result&#46;<span class="elsevierStyleDisplayedQuote" id="dsq0015"><p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">&#91;<a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell&#44; 1985</a>&#93;&#46; Let X be an open set of &#8477;L and R a locally nonsatiated preference relation over X&#44; with connected indifference sets&#46; Then&#44; for k&#8805;1&#44; R is representable by a x1&#44;x2&#44;x3&#44;x4&#44;x5x6&#44;x1&#8242;x2&#8242;x3&#8242;&#949;&#937;&#58; utility function with no critical point iff the frontier of R is a Ck manifold&#46; &#40;Cf&#46; <a class="elsevierStyleCrossRef" href="#bib0070">Mas-Colell&#44; 1985</a>&#44; p&#46; 64&#41;</p></span></p><p id="par0045" class="elsevierStylePara elsevierViewall">In terms of the Gaussian curvature of the indifference curve in each point&#44; Debreu obtained the following result&#46;<span class="elsevierStyleDisplayedQuote" id="dsq0020"><p id="spar0040" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">&#91;Debreu&#44; 1972&#93;</span>&#46; Let X be an open set of &#8477;L and R a regular preference relation over X which is monotone&#44; continuous&#44; and such that its frontier is a C2 manifold&#46; If the indifference sets of R do not intersect the frontier of X&#44; then there exists a demand function &#966; of class C1 iff the Gaussian curvature is different from zero in each point of the indifference surfaces&#46; &#40;Cf&#46; <a class="elsevierStyleCrossRef" href="#bib0010">Debreu&#44; 1983a</a>&#44; pp&#46; 194&#8211;199&#41;</p></span></p><p id="par0050" class="elsevierStylePara elsevierViewall">I would like to conclude the present section with a reflection on the meaning of these conditions&#46; In the first place&#44; the definition of Gaussian curvature proposed by Debreu &#40;taken from <a class="elsevierStyleCrossRef" href="#bib0030">Hicks&#44; 1965</a>&#44; Section 2&#46;2&#41; presupposes de facto that the indifference surfaces are already differentiable manifolds &#40;actually&#44; Debreu assumes that they are of class <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">2</span>&#41;&#44; and so the condition only translates the problem to a deeper level&#46; For the question is&#44; precisely&#44; What is the property that must be attributed to the consumer in order to guarantee that the indifference surfaces are differentiable manifolds&#63; In Debreu&#39;s definition&#44; the question whether the Gaussian curvature of the manifold is different from zero or not arises once the first problem has been solved&#46; It seems clear that Mas-Colell&#39;s condition is just a modified generalization of Debreu&#39;s and so analogous considerations apply to it&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Motivation</span><p id="par0055" class="elsevierStylePara elsevierViewall">The problem we are concerned with can be formulated thus&#58; Is it possible to find an &#40;idealized&#41; empirically meaningful property over a &#40;cardinal&#41; preference relation that enables a continuously differentiable utility representation of the same&#63;</p><p id="par0060" class="elsevierStylePara elsevierViewall">In order to discuss this question let us recall that&#44; according to consumer theory&#44; the satisfaction of a given agent at a certain consumption menu &#40;i&#46;e&#46;&#44; when the menu constitutes his current consumption&#41; reaches a certain level&#46; This level normally changes as he moves from that menu to another one &#40;i&#46;e&#46;&#44; when he changes his consumption from the previous menu to a new one&#41;&#46; If his preferences are continuous&#44; to small changes in his consumption menu there correspond small changes in his satisfaction level&#46; Hence&#44; it makes sense to ask&#58; How fast is his satisfaction changing as he moves from one consumption menu to another nearby&#63; Let be <span class="elsevierStyleBold">x</span> an interior point of &#937;&#44; the nonnegative orthant of vector space &#8477;L&#44; and notice that&#44; since is <span class="elsevierStyleBold">x</span> an interior point of &#937;&#44; it is possible to move away from <span class="elsevierStyleBold">x</span> a little in any direction without abandoning &#937;&#46;<a class="elsevierStyleCrossRef" href="#fn0010"><span class="elsevierStyleSup">2</span></a> As he moves from <span class="elsevierStyleBold">x</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">¿</span><span class="elsevierStyleBold">u</span> &#40;say&#41;&#44; where <span class="elsevierStyleItalic">¿</span> is a small number and <span class="elsevierStyleBold">u</span> is a unit vector in a fixed direction&#44; his satisfaction may change at different speeds&#46; If <span class="elsevierStyleItalic">¿</span> is infinitesimal and his preferences are continuous&#44; the change &#916;<span class="elsevierStyleItalic">&#981;</span> in his satisfaction level is indeed infinitesimal&#44; but the order of this infinitesimal can be different from that of <span class="elsevierStyleItalic">¿</span>&#46; Moreover&#44; even if &#916;<span class="elsevierStyleItalic">&#981;</span> is of the same order as <span class="elsevierStyleItalic">¿</span>&#44; it might be of a different order for a different choice of <span class="elsevierStyleItalic">¿</span>&#46; Sheer differentiability requires not only that &#916;<span class="elsevierStyleItalic">&#981;</span> be of the same order for any choice of <span class="elsevierStyleItalic">¿</span>&#44; but that the quotients &#916;<span class="elsevierStyleItalic">&#981;</span>&#47;<span class="elsevierStyleItalic">¿</span> be all infinitely close to one and the same real number&#46; This real number measures the speed at which the satisfaction level changes at <span class="elsevierStyleBold">x</span> as the agent changes his consumption slightly in the direction of <span class="elsevierStyleBold">u</span>&#46; The given condition does not guarantee&#44; however&#44; the continuity of the directional derivative &#91;&#8706;&#981;&#47;&#8706;u&#93;&#40;x&#41;&#46;</p><p id="par0065" class="elsevierStylePara elsevierViewall">Continuous differentiability at a vector <span class="elsevierStyleBold">x</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937; in the direction of <span class="elsevierStyleBold">u</span> means that the rate of satisfaction change along <span class="elsevierStyleBold">u</span> is continuous&#46; What this means is that the rates &#91;&#8706;&#981;&#47;&#8706;u&#93;&#40;x1&#41; and &#91;&#8706;&#981;&#47;&#8706;u&#93;&#40;x2&#41; approximate each other as menus <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> get closer&#46;</p><p id="par0070" class="elsevierStylePara elsevierViewall">It is indeed impossible to formulate this condition within the conceptual apparatus of ordinal preference theory&#44; or even within the usual apparatus of cardinal preference theory&#46; This is due to the fact that the notion of differentiability requires the comparison of intervals of the same kind but different interpretation&#46; It requires the comparison of lengths of satisfaction intervals with lengths of geometric intervals&#59; i&#46;e&#46;&#44; the comparison of the distance in satisfaction between menus <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> with the geometric distance &#124;&#124;x1&#8722;x2&#124;&#124; among them&#46; The problem is that the difference relation <span class="elsevierStyleItalic">R</span> falls short of providing the linguistic and conceptual resources to make this comparison&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall">But there is an indirect way of making claims about <span class="elsevierStyleItalic">R</span>&#44; of attributing &#40;idealized&#41; empirically meaningful properties to <span class="elsevierStyleItalic">R</span>&#44; using the geometric analogy involved in the notion of a satisfaction &#8220;interval&#8221;&#46; Actually&#44; the very notion of difference comparison is built upon this analogy&#58; When the agent compares the difference in satisfaction &#40;for her&#41; between the pair of menus <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and the pair <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> she is somehow comparing &#8220;distances&#8221; between them&#46; That the agent actually thinks or feels that the distance between <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is at least as long as that between <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> is expressed by the theoretician in economics by means of the formula &#8216;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#8217;&#46; Hence&#44; it is not far-fetched&#44; but rather natural&#44; for the theoretician to represent this distance by means of a geometric entity of the obvious sort&#58; An interval within a straight line&#46; It seems to me that a fully general theory of cardinal preference must be grounded upon such a representation&#46; I will try to show below how such a theory would be like&#44; but it will be convenient to start considering the axioms required for the usual numerical representation of relation <span class="elsevierStyleItalic">R</span>&#46;</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">The algebraic theory of difference</span><p id="par0080" class="elsevierStylePara elsevierViewall">There is no doubt that every consumer has an idea of the satisfaction differences between the consumption menus among which she has to make a choice&#46; Hence&#44; the comparison of these differences is only natural&#46; As we already indicated&#44; it is usual to express this comparison by means of a difference relation <span class="elsevierStyleItalic">R</span> among pairs of consumption menus &#40;represented by points in the nonnegative orthant &#937; of &#8477;L&#41;&#46; The simplest such relation is defined as follows&#46;<span class="elsevierStyleEnunciation" id="enun0005"><span class="elsevierStyleLabel">Definition 1</span><p id="par0085" class="elsevierStylePara elsevierViewall">A difference relation R over &#937; is a connected and transitive binary relation over &#937;&#59; i&#46;e&#46;&#44; a weak order&#46; If R is a difference relation over &#937;&#44; we say that &#9001;&#937;<span class="elsevierStyleHsp" style=""></span>&#215;<span class="elsevierStyleHsp" style=""></span>&#937;&#44;<span class="elsevierStyleItalic">R</span>&#9002; is a difference structure&#46;</p><p id="par0090" class="elsevierStylePara elsevierViewall">Write <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">E</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#59; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> but not <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#46;</p><p id="par0095" class="elsevierStylePara elsevierViewall">In the interpretation we are interested in here&#44; formula &#8216;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#8217; means that the change from menu <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to menu <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is preferred by the agent to the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#46; The change from one to the other can be an improvement or a worsening for the agent&#46; To fix ideas&#44; if we think of the menus as amounts of money&#44; and the agent prefers to have more money to less&#44; a motion from &#40;say&#41; nine thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41; to twelve thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; dollars is better than one from ten thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; to eleven thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41;&#46; But&#44; if the agent is to lose money&#44; it is preferable for her to fall from eleven &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41; to ten &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; than from twelve &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; to nine &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41; thousand dollars&#46; Thus&#44; we shall assume &#40;below&#41; that<elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><p id="par0100" class="elsevierStylePara elsevierViewall">It will be necessary to introduce&#44; also&#44; the operation of composition of motions&#46; For instance&#44; we can compose the motion from nine thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41; to twelve thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; with the motion from twelve thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; to eight thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41;&#46; The result will be a motion from nine thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; to eight thousand &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; a net loss of one thousand dollars&#46; I will define below&#44; in general terms&#44; the required composition operation among intervals&#46;</p><p id="par0105" class="elsevierStylePara elsevierViewall">In order to formulate axiomatic conditions over <span class="elsevierStyleItalic">R</span>&#44; say that interval <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937;<span class="elsevierStyleSup">2</span> is <span class="elsevierStyleItalic">positive</span> &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X</span><span class="elsevierStyleSup">&#43;</span>&#41; iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleBold">xx</span> for any <span class="elsevierStyleBold">x</span>&#44; which means that moving from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is an improvement for the agent&#46; Interval <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is negative &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X</span><span class="elsevierStyleSup">&#8722;</span>&#41; iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> is positive&#46; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is null &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X</span><span class="elsevierStyleSup">0</span>&#41; iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is neither positive nor negative&#46; It is easy to show&#44; out of the axioms that will be introduced below&#44; that all null intervals are equivalent among them selves&#59; i&#46;e&#46;&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">E</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> for any null intervals <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#59; it can be seen also that if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is null&#44; then <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> is also null&#46;</p><p id="par0110" class="elsevierStylePara elsevierViewall">A <span class="elsevierStyleItalic">standard sequence</span> of elements of &#937; is a set &#123;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span>&#125;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">k¿K</span></span>&#44; where <span class="elsevierStyleItalic">K</span> is an initial segment of the set &#8484;&#43; of positive integers &#40;or the whole set&#41;&#44; such that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span>&#43;1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span><span class="elsevierStyleItalic">E</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> for all <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span>&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span>&#43;1</span> in the sequence&#44; and it is not the case that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">E</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46; The sequence is <span class="elsevierStyleItalic">strictly bounded</span> if there exist x&#8242;&#44;x&#8243;&#8712;&#937; such that x&#8242;&#44;x&#8243;Sxkx1Sx&#8242;&#44;x&#8243; for all <span class="elsevierStyleItalic">k</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">K</span>&#46;</p><p id="par0115" class="elsevierStylePara elsevierViewall">I will assume that relation <span class="elsevierStyleItalic">R</span> satisfies the conditions specified in the following definition&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0010"><span class="elsevierStyleLabel">Definition 2</span><p id="par0120" class="elsevierStylePara elsevierViewall">Difference structure &#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002; is an <span class="elsevierStyleItalic">algebraic-difference structure</span><a class="elsevierStyleCrossRef" href="#fn0015"><span class="elsevierStyleSup">3</span></a> iff&#44; in addition to being a weak order&#44; it satisfies the following axioms for every x1&#44;&#8201;x2&#44;&#8201;x3&#44;&#8201;x4&#44;&#8201;x5&#44;&#8201;x6&#44;&#8201;x1&#8242;&#44;&#8201;x2&#8242;&#44;&#8201;x3&#8242;&#8201;&#949;&#937;&#58;<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005"><span class="elsevierStyleLabel">&#40;1&#41;</span><p id="par0125" class="elsevierStylePara elsevierViewall">If the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is at least as good &#40;bad&#41; as the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#44; then the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44; is at least as bad &#40;good&#41; as the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46; In symbols&#44; if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> then <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46;</p></li><li class="elsevierStyleListItem" id="lsti0010"><span class="elsevierStyleLabel">&#40;2&#41;</span><p id="par0130" class="elsevierStylePara elsevierViewall">If the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is as good &#40;bad&#41; as the motion from x1&#8242; to x2&#8242;&#44; and the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> is as good &#40;bad&#41; as the motion from x2&#8242; to x3&#8242;&#44; then the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> is as good &#40;bad&#41; as that from x1&#8242; to x3&#8242;&#59; i&#46;e&#46;&#44; if x1x2Rx1&#8242;x2&#8242; and x2x3Rx2&#8242;x3&#8242; then x1x3Rx1&#8242;x3&#8242;&#46;</p></li><li class="elsevierStyleListItem" id="lsti0015"><span class="elsevierStyleLabel">&#40;3&#41;</span><p id="par0135" class="elsevierStylePara elsevierViewall">If <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are segments such that the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is at least as good &#40;bad&#41; as the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>to<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#44; it is possible to find a menu x2&#8242;&#8201;&#8712;&#8201;&#937;&#44; slightly less satisfactory than <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44; or just as satisfactory&#44; such that the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to x2&#8242; matches the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#46; In the same token&#44; it is possible to find a menu x1&#8242;&#8201;&#8712;&#8201;&#937; slightly more satisfactory than <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44; or just as satisfactory&#44; such that the difference between x1&#8242; and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> matches the difference between <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#46; In symbols&#44; if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">xx</span>&#44; then there exist x1&#8242;&#44;x2&#8242;&#8712;&#937; such that x1x2&#8242;Ex3x4 and x3x4Ex2&#8242;x2&#46;</p></li><li class="elsevierStyleListItem" id="lsti0020"><span class="elsevierStyleLabel">&#40;4&#41;</span><p id="par0140" class="elsevierStylePara elsevierViewall">Every strictly bounded sequence is finite&#59; i&#46;e&#46;&#44; for each strictly bounded standard sequence &#123;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span>&#125;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">k¿K</span></span> of elements of &#937;&#44; there exists a number N&#8712;&#8484;&#43; such that <span class="elsevierStyleItalic">k</span><span class="elsevierStyleHsp" style=""></span>&#60;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">N</span> for each <span class="elsevierStyleItalic">k</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">K</span>&#46;</p></li></ul></p><p id="par0145" class="elsevierStylePara elsevierViewall">It can be proven<a class="elsevierStyleCrossRef" href="#fn0020"><span class="elsevierStyleSup">4</span></a> that&#44; for any algebraic structure &#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002;&#44; there exists a real-valued function <span class="elsevierStyleItalic">&#966;</span> on &#937; such that&#44; for all <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937;<elsevierMultimedia ident="eq0010"></elsevierMultimedia><span class="elsevierStyleItalic">&#981;</span>&#44; which is a utility function&#44; is unique up to a positive linear transformation&#59; i&#46;e&#46;&#44; if <span class="elsevierStyleItalic">&#981;</span>&#8242; is another such utility function&#44; then there are real constants <span class="elsevierStyleItalic">&#945;</span>&#44;<span class="elsevierStyleItalic">&#946;</span>&#44;<span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0&#44; such that <span class="elsevierStyleItalic">&#981;</span>&#8242;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#945;u</span>&#43;<span class="elsevierStyleItalic">&#946;</span> This means that <span class="elsevierStyleItalic">&#981;</span> is&#44; indeed&#44; a cardinal utility function&#46;</p></span></p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">The geometric theory of difference</span><p id="par0150" class="elsevierStylePara elsevierViewall">It is nearly impossible to formulate differentiability conditions over <span class="elsevierStyleItalic">R</span> within the language and conceptual apparatus of the algebraic theory of difference&#46; What is required is a certain &#8220;intermediate&#8221; language&#46; In order to introduce this language&#44; let us suppose that a good straight Euclidean line is given in its purity&#46; Following <a class="elsevierStyleCrossRefs" href="#bib0045">H&#246;lder &#40;1996&#44; 1997&#41;</a>&#44; I shall assume that intervals within this straight line are of two kinds&#44; such that any interval is of one and only one kind&#46;<span class="elsevierStyleDisplayedQuote" id="dsq0025"><p id="spar0045" class="elsevierStyleSimplePara elsevierViewall">Intervals of the same kind are called &#8220;of the same direction&#8221;&#44;<a class="elsevierStyleCrossRef" href="#fn0025"><span class="elsevierStyleSup">5</span></a> and intervals of different kinds are called of opposite direction&#46; The intervals <span class="elsevierStyleItalic">AB</span> and <span class="elsevierStyleItalic">BA</span> are always of opposite direction&#46; Let the intervals of one kind be called &#8220;intervals of the first direction&#8221; and the fact that <span class="elsevierStyleItalic">AB</span> is an interval of the first direction be expressed as A&#8827;B or B&#8827;A &#40;<a class="elsevierStyleCrossRef" href="#bib0050">H&#246;lder 1997&#44; p&#46; 346</a>&#41;&#46;</p></span></p><p id="par0155" class="elsevierStylePara elsevierViewall">Equality &#40;congruence&#41; of intervals <span class="elsevierStyleItalic">AB</span> and <span class="elsevierStyleItalic">A</span>&#8242;<span class="elsevierStyleItalic">B</span>&#8242; will be expressed as AB&#8784;A&#8242;B&#8242;&#46; Clearly&#44; &#8784; is an equivalence relation over the set &#923; of all intervals within the straight line&#46;</p><p id="par0160" class="elsevierStylePara elsevierViewall">Furthermore&#44; we assume that points and intervals satisfy H&#246;lder&#39;s axioms up to the definition of interval numbers &#40;see <a class="elsevierStyleCrossRef" href="#bib0050">1997</a>&#44; &#167;23&#44; p&#46; 351&#44; equations 53 and 54&#41;&#46; Hence&#44; we take for granted that there are arbitrarily designated points <span class="elsevierStyleItalic">N</span> and <span class="elsevierStyleItalic">E</span>&#44; with N&#8826;E such that interval <span class="elsevierStyleItalic">NE</span> is taken as unit&#46; We will denote interval <span class="elsevierStyleItalic">NE</span> eventually as 1&#8594;&#46;</p><p id="par0165" class="elsevierStylePara elsevierViewall">On top of &#8784;&#44; I will use symbols <elsevierMultimedia ident="201703220242154571"></elsevierMultimedia>&#44; &#8918;&#44; or their counterparts <elsevierMultimedia ident="201703220242154572"></elsevierMultimedia> and &#8919;&#44; to express the congruence comparisons among intervals&#46; The sum of intervals &#40;for its definition&#44; see <a class="elsevierStyleCrossRef" href="#bib0050">H&#246;lder&#44; 1997</a>&#44; p&#46; 347&#41; will be denoted by &#8853; &#40;H&#246;lder uses symbol &#43;&#41;&#46; Notice that what <a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder &#40;1996&#41;</a> calls &#8216;magnitudes&#8217; are line intervals in the interpretation intended here&#46; This same interpretation is developed by <a class="elsevierStyleCrossRef" href="#bib0050">H&#246;lder &#40;1997&#41;</a>&#46;</p><p id="par0170" class="elsevierStylePara elsevierViewall">It is possible&#44; and it will turn out to be convenient&#44; to express the properties that are attributed to <span class="elsevierStyleItalic">R</span> in terms of relations among geometric intervals within the given Euclidean straight line&#46; What this means is that <span class="elsevierStyleItalic">we</span>&#44; <span class="elsevierStyleItalic">as theoreticians</span>&#44; can represent the comparison of the differences felt by the consumer&#44; expressed by symbol &#8216;<span class="elsevierStyleItalic">R</span>&#8217;&#44; by means of comparisons among intervals in &#923;&#46; My proposal is to build the theory of relation <span class="elsevierStyleItalic">R</span> by means of these comparisons&#44; trying to express intuitive&#44; empirical &#40;idealized&#41; properties of <span class="elsevierStyleItalic">R</span> in terms of such comparisons&#46;</p><p id="par0175" class="elsevierStylePara elsevierViewall">To that end&#44; let me to introduce the function &#963;&#58;&#937;2&#8594;&#923;&#44; as an application that assigns to each satisfaction interval <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> a line interval whose length is intended to represent the distance that the agent associates to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> &#40;how &#8220;far&#8221; is <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> in terms of satisfaction&#41;&#44; and whose direction is intended to represent whether the motion from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> would be an improvement&#44; a worsening&#44; or indifferent for the agent&#46; In particular&#44; <span class="elsevierStyleItalic">&#963;</span> will assign to any interval <span class="elsevierStyleBold">xx</span> in the diagonal the null line interval&#44; which of course does not exist but we can create by a convenient <span class="elsevierStyleItalic">fiat</span>&#46;</p><p id="par0180" class="elsevierStylePara elsevierViewall">As I just said&#44; it is not merely the <span class="elsevierStyleItalic">distance</span> among menus what has to be considered&#44; but also the <span class="elsevierStyleItalic">direction</span> of the motions and&#44; moreover&#44; also the <span class="elsevierStyleItalic">composition</span> of motions&#46; If <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X</span><span class="elsevierStyleSup">&#43;</span>&#44; the agent perceives the &#40;actual or potential&#41; change from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> as an improvement&#44; and that from <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> as a worsening&#46; Yet&#44; interval <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is congruent to <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41;&#44; which means that they have the same length&#44; the difference being that they are of opposite directions&#46; The composition of motions can be defined as follows&#46;<span class="elsevierStyleEnunciation" id="enun0015"><span class="elsevierStyleLabel">Definition 3</span><p id="par0185" class="elsevierStylePara elsevierViewall">Let &#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002; be a difference structure&#46; For any menus <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937;&#44; define operation &#8728;&#58;&#923;2&#8594;&#923; as it is specified in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#46;</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><p id="par0190" class="elsevierStylePara elsevierViewall">I introduce formally the concept of a geometric representation by means of the following definition&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0020"><span class="elsevierStyleLabel">Definition 4</span><p id="par0195" class="elsevierStylePara elsevierViewall">Let &#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002; be a difference structure&#46; A function &#963;&#58;&#937;2&#8594;&#923; is a <span class="elsevierStyleItalic">geometric representation</span> of <span class="elsevierStyleItalic">R</span> iff it satisfies the following conditions for every <span class="elsevierStyleBold">x</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> in &#937;&#58;<ul class="elsevierStyleList" id="lis0010"><li class="elsevierStyleListItem" id="lsti0025"><span class="elsevierStyleLabel">&#40;1&#41;</span><p id="par0200" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is an interval of the first direction iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is positive&#46;</p></li><li class="elsevierStyleListItem" id="lsti0030"><span class="elsevierStyleLabel">&#40;2&#41;</span><p id="par0205" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is an interval of the second direction iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is negative&#46;</p></li><li class="elsevierStyleListItem" id="lsti0035"><span class="elsevierStyleLabel">&#40;3&#41;</span><p id="par0210" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is the null interval iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is null&#46;</p></li><li class="elsevierStyleListItem" id="lsti0040"><span class="elsevierStyleLabel">&#40;4&#41;</span><p id="par0215" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">P</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> iff either both <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are positive or null and &#963;&#40;x1x2&#41;&#8919;&#963;&#40;x3x4&#41;&#59; or both <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are negative and &#963;&#40;x1x2&#41;&#8918;&#963;&#40;x3x4&#41;&#59; or <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is of the first direction&#44; or null <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41;&#44; and is negative&#46;</p></li><li class="elsevierStyleListItem" id="lsti0045"><span class="elsevierStyleLabel">&#40;5&#41;</span><p id="par0220" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">E</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> iff both <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41; are of the same direction and &#963;&#40;x1x2&#41;&#8784;&#963;&#40;x3x4&#41;&#46;</p></li><li class="elsevierStyleListItem" id="lsti0050"><span class="elsevierStyleLabel">&#40;6&#41;</span><p id="par0225" class="elsevierStylePara elsevierViewall">&#963;&#40;x1x3&#41;&#8784;&#963;&#40;x1x2&#41;&#8728;&#963;&#40;x2x3&#41;&#46;</p></li><li class="elsevierStyleListItem" id="lsti0055"><span class="elsevierStyleLabel">&#40;7&#41;</span><p id="par0230" class="elsevierStylePara elsevierViewall">If AB&#8784;&#963;&#40;x1x2&#41; for some &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41;&#44; then&#44; for any interval CD <elsevierMultimedia ident="201703220242154573"></elsevierMultimedia> AB &#40;or CD <elsevierMultimedia ident="201703220242154574"></elsevierMultimedia> AB&#41;&#44; there exist x1&#8242;&#44;x2&#8242;&#8712;&#937; such that &#963;x1x2&#8242;&#8784;CD&#8784;&#963;x1&#8242;x2&#46;</p></li></ul></p><p id="par0235" class="elsevierStylePara elsevierViewall">It is easy to see that &#9001;&#923;&#44;&#8728;&#9002; is a group with the null interval as identity element&#46; The following result is immediate&#44; as it is based upon the existence of the numerical representation &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>&#41;&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia></span><span class="elsevierStyleEnunciation" id="enun0025"><span class="elsevierStyleLabel">Theorem 1</span><p id="par0240" class="elsevierStylePara elsevierViewall">If &#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002; is an algebraic difference structure then there exists a geometric representation &#963;&#58;&#937;2&#8594;&#923;&#8201;of&#8201;R&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0030"><span class="elsevierStyleLabel">Proof</span><p id="par0245" class="elsevierStylePara elsevierViewall">Consider any numerical representation <span class="elsevierStyleItalic">&#981;</span> of R&#46; If is <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> positive&#44; &#968;&#40;x1x2&#41;&#61;&#981;&#40;x2&#41;&#8722;&#981;&#40;x1&#41; is a positive real number and so there are points <span class="elsevierStyleItalic">A</span>&#44; <span class="elsevierStyleItalic">B</span> on the line such that A&#8826;B and <span class="elsevierStyleItalic">&#968;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is equal to the interval number &#40;cut&#41; &#91;AB&#58;1&#8594;&#93;&#46; Let <span class="elsevierStyleItalic">&#961;</span> be the application mapping &#91;a&#58;1&#8594;&#93; into <span class="elsevierStyleItalic">a</span>&#44; and define <span class="elsevierStyleItalic">&#963;</span> as follows &#40;cf&#46; <a class="elsevierStyleCrossRef" href="#bib0050">H&#246;lder 1997&#44; pp&#46; 351&#8211;352</a>&#41;&#58;<elsevierMultimedia ident="eq0015"></elsevierMultimedia></p><p id="par0250" class="elsevierStylePara elsevierViewall">Clearly&#44; by construction&#44; <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is of the first direction iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is positive&#59; of the second iff it is negative&#59; and null iff it is neither&#46; Since AB&#8784;BA&#44; notice that &#963;&#40;x1x2&#41;&#8784;&#963;&#40;x2x1&#41;&#46;</p><p id="par0255" class="elsevierStylePara elsevierViewall">Suppose that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are nonnegative and let <span class="elsevierStyleItalic">A</span>&#44; <span class="elsevierStyleItalic">B</span>&#44; <span class="elsevierStyleItalic">C</span>&#44; <span class="elsevierStyleItalic">D</span>&#44; be points such that &#968;&#40;x1x2&#41;&#61;&#91;AB&#58;1&#8594;&#93; and &#968;&#40;x1x2&#41;&#61;&#91;CD&#58;1&#8594;&#93;&#46; Then we have<elsevierMultimedia ident="eq0020"></elsevierMultimedia></p><p id="par0260" class="elsevierStylePara elsevierViewall">If both <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are negative&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> are positive and we have<elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0265" class="elsevierStylePara elsevierViewall">Given <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">P</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> when the intervals are of opposite signs&#44; the case when <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is negative and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> is nonnegative is excluded because in such a case we would have<elsevierMultimedia ident="eq0030"></elsevierMultimedia>and so <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">P</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#46; Hence&#44; the only case remaining is when <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is nonnegative and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> is negative&#46;</p><p id="par0270" class="elsevierStylePara elsevierViewall">It has to be shown that &#963;&#40;x1x3&#41;&#8784;&#963;&#40;x1x2&#41;&#8728;&#963;&#40;x2x3&#41;&#46; I refer the reader to <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#44; as I shall consider case by case&#46; Keep in mind that it is always true that<elsevierMultimedia ident="eq0035"></elsevierMultimedia>&#40;cf&#46; <a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder&#44; 1996</a>&#44; eqn&#46; 19&#44; p&#46; 243&#41;&#46; Also&#44; for every <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937;&#44;<elsevierMultimedia ident="eq0040"></elsevierMultimedia></p><p id="par0275" class="elsevierStylePara elsevierViewall">In all cases&#44; let &#968;&#40;x1&#44;x2&#41;&#61;&#91;AB&#58;1&#8594;&#93; and &#968;&#40;x2&#44;x3&#41;&#61;&#91;BC&#58;1&#8594;&#93;&#46; It will suffice to show that &#91;AC&#58;1&#8594;&#93;&#61;&#968;&#40;x1&#44;x3&#41;</p><p id="par0280" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Case 1</span>&#58; A&#8826;B&#8826;C&#46; We have<elsevierMultimedia ident="eq0045"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0285" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Case 2</span>&#58; A&#8826;B&#8826;C&#46; We have<elsevierMultimedia ident="eq0055"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0290" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Case 3</span>&#58; B&#8826;A&#8826;C&#46; We have<elsevierMultimedia ident="eq0065"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0070"></elsevierMultimedia></p><p id="par0295" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Case 4</span>&#58; B&#8826;C&#8826;A&#46; We have<elsevierMultimedia ident="eq0075"></elsevierMultimedia>or<elsevierMultimedia ident="eq0080"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0085"></elsevierMultimedia></p><p id="par0300" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Case 5</span>&#58; C&#8826;A&#8826;B&#46; We have<elsevierMultimedia ident="eq0090"></elsevierMultimedia>or<elsevierMultimedia ident="eq0095"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0100"></elsevierMultimedia></p><p id="par0305" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Case 6</span>&#58; C&#8826;B&#8826;A&#46; We have<elsevierMultimedia ident="eq0105"></elsevierMultimedia>and so<elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0310" class="elsevierStylePara elsevierViewall">Hence&#44; at any rate&#44; <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">AC</span> and so axiom &#40;6&#41; of <a class="elsevierStyleCrossRef" href="#enun0020">Definition 4</a> is shown to be satisfied&#46;</p><p id="par0315" class="elsevierStylePara elsevierViewall">Finally&#44; assume that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> and let &#91;AB&#58;1&#8594;&#93;&#61;&#968;&#40;x3x4&#41;&#44; so that <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">AB</span>&#46; By axiom 3 of <a class="elsevierStyleCrossRef" href="#enun0010">Definition 2</a>&#44; there exist x1&#8242; and x2&#8242; such that x1x2&#8242;Ex3x4Ex1&#8242;x2 Setting CD&#8784;&#963;x1x2&#8242;&#8784;&#963;x1&#8242;x2&#44; condition 7 of <a class="elsevierStyleCrossRef" href="#enun0020">Definition 4</a> is satisfied&#46;</p></span>&#9633;</p><p id="par0320" class="elsevierStylePara elsevierViewall">We have shown the existence of a geometric representation of an algebraic difference structure&#46; The point of having this representation is that it provides an adequate conceptual and linguistic apparatus to express the differentiability condition we are looking for&#46; Moreover&#44; it can be shown that the existence of a geometric representation for a difference structure D&#61;&#9001;&#937;2&#44;R&#9002; guarantees that D is an algebraic difference structure&#46; For we can express the properties defining the concept of an algebraic difference structure purely in terms of the geometric representation&#44; and show that the structure has these properties out of the axioms regulating <span class="elsevierStyleItalic">&#963;</span>&#46; The wages of doing this is that we can also express in terms of the geometric representation all the properties of an algebraic difference structure&#44; plus the required differentiability condition&#44; and establish in this way the existence of a <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span> numerical representation of <span class="elsevierStyleItalic">R</span>&#46;</p><p id="par0325" class="elsevierStylePara elsevierViewall">I will prove in what follows that the existence of a geometric representation of difference structure D implies that D is an algebraic difference structure&#46; The following four lemmas&#44; all of which share the assumption that such representation exists&#44; are devoted to this end&#46; I will introduce later the differentiability condition&#46; For the sake of brevity&#44; from now on&#44; that an interval is of the first direction will be expressed by saying that &#8220;it is I&#8221;&#59; and that &#8220;it is II&#8221; if it is of the second direction&#46; The null interval will be denoted as 0&#8594;<span class="elsevierStyleEnunciation" id="enun0035"><span class="elsevierStyleLabel">Lemma 1</span><p id="par0330" class="elsevierStylePara elsevierViewall">If <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> then <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0040"><span class="elsevierStyleLabel">Proof</span><p id="par0335" class="elsevierStylePara elsevierViewall">Suppose that both <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are in X&#43;&#8746;X0&#46; This means that both <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41; are I or null&#44; with <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; <elsevierMultimedia ident="201703220242154575"></elsevierMultimedia><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41; Hence&#44; <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; are II or null&#44; with <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41; <elsevierMultimedia ident="201703220242154576"></elsevierMultimedia><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41;&#46; It follows that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46;</p><p id="par0340" class="elsevierStylePara elsevierViewall">If both are II&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> implies that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><elsevierMultimedia ident="201703220242154577"></elsevierMultimedia><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> and that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> are I&#46; Hence&#44; again&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46;</p><p id="par0345" class="elsevierStylePara elsevierViewall">Notice that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> implies that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> cannot be I or null if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is II&#46; Hence&#44; the only remaining case is when <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is I or null&#44; and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> is II&#46; In this case&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> is II or null and <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span> is I&#46; It follows that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">P</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> and so&#44; finally&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46;</p></span>&#9633;<span class="elsevierStyleEnunciation" id="enun0045"><span class="elsevierStyleLabel">Lemma 2</span><p id="par0350" class="elsevierStylePara elsevierViewall">If x1x2Rx1&#8242;x2&#8242; and x2x3Rx2&#8242;x3&#8242; then x1x3Rx1&#8242;x3&#8242;&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0050"><span class="elsevierStyleLabel">Proof</span><p id="par0355" class="elsevierStylePara elsevierViewall">The proof of this lemma is easy but laborious&#44; since there are several cases to be considered&#46; Excluding the cases precluded by the hypothesis of the proposition&#44; there are still nine cases to consider&#46; They are given in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>&#46; The proof is interesting because it yields more insight into the meaning of the geometric representation&#46;</p><elsevierMultimedia ident="tbl0010"></elsevierMultimedia><p id="par0360" class="elsevierStylePara elsevierViewall">Case 1 is straightforward because all intervals are I or null and so x1x2Rx1&#8242;x2&#8242;&#44; and x2x3Rx2&#8242;x3&#8242; is tantamount to <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; <elsevierMultimedia ident="201703220242154578"></elsevierMultimedia>&#963;x1&#8242;x2&#8242; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; <elsevierMultimedia ident="201703220242154579"></elsevierMultimedia>&#963;x2&#8242;x3&#8242;&#46; We have<elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0365" class="elsevierStylePara elsevierViewall">Hence&#44; by <a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder&#39;s &#40;1996&#44; p&#46; 238&#41;</a> conclusion 2&#44;<elsevierMultimedia ident="eq0120"></elsevierMultimedia>or&#44; equivalently&#44;<elsevierMultimedia ident="eq0125"></elsevierMultimedia></p><p id="par0370" class="elsevierStylePara elsevierViewall">In case 4 there is nothing to prove because <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; is I or null and &#963;x1&#8242;x3&#8242; is II&#46;</p><p id="par0375" class="elsevierStylePara elsevierViewall">In cases 2 and 3&#44; &#963;x1&#8242;x3&#8242; can be I or null&#44; or II&#46; When it is II&#44; we are done&#44; because <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; is I or null&#46; When &#963;x1&#8242;x3&#8242; is or null&#44; we have<elsevierMultimedia ident="eq0130"></elsevierMultimedia>if &#963;x2&#8242;x3&#8242;0&#8594;&#44; or<elsevierMultimedia ident="eq0135"></elsevierMultimedia></p><p id="par0380" class="elsevierStylePara elsevierViewall">if &#963;x1&#8242;x2&#8242;0&#8594;</p><p id="par0385" class="elsevierStylePara elsevierViewall">In case 5 we have <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; <elsevierMultimedia ident="2017032202421545710"></elsevierMultimedia>&#963;x1&#8242;x2&#8242; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; <elsevierMultimedia ident="2017032202421545711"></elsevierMultimedia>&#963;x2&#8242;x3&#8242;&#46; We have five subcases&#44; setting <span class="elsevierStyleItalic">B</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">B</span>&#8242; &#40;see <a class="elsevierStyleCrossRef" href="#fig0010">Fig&#46; 2</a>&#41;&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><p id="par0390" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Subcase &#40;5&#46;1&#41;</span>&#46; A&#8826;A&#8242;&#8826;C&#8242;&#8826;C&#8826;B&#61;B&#8242;&#46; In this case we have that both <span class="elsevierStyleItalic">&#968;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; and &#963;x1&#8242;x3&#8242; are I or null with<elsevierMultimedia ident="eq0140"></elsevierMultimedia></p><p id="par0395" class="elsevierStylePara elsevierViewall">It follows that x1x3Rx1&#8242;x3&#8242;&#46;</p><p id="par0400" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Subcase &#40;5&#46;2&#41;</span>&#46; A&#8826;C&#8242;&#8826;A&#8242;&#8826;C&#8826;B&#61;B&#8242;&#46;</p><p id="par0405" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Subcase &#40;5&#46;3&#41;</span>&#46; A&#8826;C&#8242;&#8826;C&#8826;A&#8242;&#8826;B&#61;B&#8242;&#46;</p><p id="par0410" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Subcase &#40;5&#46;4&#41;</span>&#46; C&#8242;&#8826;A&#8826;C&#8826;A&#8242;&#8826;B&#61;B&#8242;&#46;</p><p id="par0415" class="elsevierStylePara elsevierViewall">In cases &#40;5&#46;2&#41;&#8211;&#40;5&#46;4&#41;&#44; AC&#8784;&#963;&#40;x1x3&#41; is I or null&#44; whereas A&#8242;C&#8242;&#8784;&#963;x1&#8242;x3&#8242; is II&#46; It is immediate that x1x3Rx1&#8242;x3&#8242; in these cases&#46;</p><p id="par0420" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Subcase &#40;5&#46;5&#41;</span>&#46; C&#8242;&#8826;C&#8826;A&#8826;A&#8242;&#8826;B&#61;B&#8242;&#46; In this final case&#44; both AC&#8784;&#963;&#40;x1x3&#41; and A&#8242;C&#8242;&#8784;&#963;x1&#8242;x3&#8242; are II but <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; <elsevierMultimedia ident="2017032202421545712"></elsevierMultimedia>&#963;x1&#8242;x3&#8242;&#44; and so x1x3Rx1&#8242;x3&#8242;&#46;</p><p id="par0425" class="elsevierStylePara elsevierViewall">Case 7 is entirely analogous to case 5&#46;</p><p id="par0430" class="elsevierStylePara elsevierViewall">In cases&#44; 6 and 8 <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; can be I&#44; null&#44; or II&#46; If it is I or null&#44; we are done&#46; If <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; is II&#44; in case 6 we have C&#8826;A&#8826;B and C&#8242;&#8826;B&#8242;&#8826;A&#8242; with<elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0435" class="elsevierStylePara elsevierViewall">Thus&#44; x1x3Rx1&#8242;x3&#8242;&#46; An analogous argument leads to the same conclusion in case 8&#46;</p><p id="par0440" class="elsevierStylePara elsevierViewall">Finally&#44; in case 9&#44; all segments are II and we have both <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; <elsevierMultimedia ident="2017032202421545713"></elsevierMultimedia>&#963;x1&#8242;x2&#8242; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; <elsevierMultimedia ident="2017032202421545714"></elsevierMultimedia>&#963;x2&#8242;x3&#8242; Hence&#44; &#963;x1&#8242;x3&#8242;<elsevierMultimedia ident="2017032202421545715"></elsevierMultimedia><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41; and so x1x3Rx1&#8242;x3&#8242;&#46;</p></span>&#9633;<span class="elsevierStyleEnunciation" id="enun0055"><span class="elsevierStyleLabel">Lemma 3</span><p id="par0445" class="elsevierStylePara elsevierViewall">if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#44; then there exist x1&#8242;&#44;x2&#8242;&#8712;&#937; such that x1x2&#8242;Ex3x4 and x3x4Ex1&#8242;x2&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0060"><span class="elsevierStyleLabel">Proof</span><p id="par0450" class="elsevierStylePara elsevierViewall">Assume that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">R</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#46; If <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">E</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#44; there is nothing to prove&#44; and so we may suppose that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">S</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#46;</p><p id="par0455" class="elsevierStylePara elsevierViewall">Let CD&#8784;&#963;&#40;x3x4&#41;&#46; By axiom 7 of <a class="elsevierStyleCrossRef" href="#enun0020">Definition 4</a>&#44; there exist menus x1&#8242;&#44;x2&#8242; such that &#963;x1&#8242;x2&#8784;CD&#8784;&#963;x1x2&#8242;&#46; It follows that x1&#8242;x2Ex3x4Ex1x2&#8242;&#46;</p></span>&#9633;<span class="elsevierStyleEnunciation" id="enun0065"><span class="elsevierStyleLabel">Lemma 4</span><p id="par0460" class="elsevierStylePara elsevierViewall">Every strictly bounded sequence is finite&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0070"><span class="elsevierStyleLabel">Proof</span><p id="par0465" class="elsevierStylePara elsevierViewall">Let &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span>&#41; be a bounded standard sequence and <span class="elsevierStyleBold">x</span>&#44;<span class="elsevierStyleBold">x</span>&#8242;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937; be such that <span class="elsevierStyleBold">xx</span>&#8242;<span class="elsevierStyleItalic">S</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span> for all <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span> in the sequence&#46; Let <span class="elsevierStyleItalic">AB</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span>&#43;1</span>&#41; and <span class="elsevierStyleItalic">CD</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">xx</span>&#8242;&#41;&#46; Then CDkAB for every k such that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span> is in the sequence&#46; But&#44; &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">k</span></span>&#41; if were not finite&#44; there would be one such positive integer k with kABCD &#40;cf&#46; <a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder&#44; 1996</a>&#44; p&#46; 239&#41;&#46;</p></span>&#9633;</p><p id="par0470" class="elsevierStylePara elsevierViewall">Using the previous lemmas and <a class="elsevierStyleCrossRef" href="#enun0025">Theorem 1</a>&#44; we can establish the following proposition&#46;<span class="elsevierStyleEnunciation" id="enun0075"><span class="elsevierStyleLabel">Theorem 2</span><p id="par0475" class="elsevierStylePara elsevierViewall">There exists a geometric representation of a difference structure D iff D is an algebraic-difference structure&#46;</p><p id="par0480" class="elsevierStylePara elsevierViewall">Hence&#44; the existence of a geometric representation of D is necessary and sufficient for D to be an algebraic-difference structure&#44; and indeed implies the existence of a numerical representation of D&#46; Yet&#44; as the reader shall presently see&#44; the geometric language has more expressive power than the algebraic one&#46;</p></span></p></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">The theory of preference</span><p id="par0485" class="elsevierStylePara elsevierViewall">A preference relation among consumption menus in &#937; can be defined out of the difference relation&#46; As we said&#44; if <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is I or null&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is weakly preferred by the agent to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#46; We may express this preference relation by means of symbol &#8827;&#732;&#46;<span class="elsevierStyleEnunciation" id="enun0080"><span class="elsevierStyleLabel">Definition 5</span><p id="par0490" class="elsevierStylePara elsevierViewall">For consumption menus <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> in &#937;&#44; say that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> is weakly preferred to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#44; and write x1&#8827;&#732;x2&#44; iff <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span> is I or null&#46;</p><p id="par0495" class="elsevierStylePara elsevierViewall">The kin notions of strict preference and indifference&#44; denoted by symbols &#8827; and &#8764;&#44; are defined as usual&#46; Clearly&#44; <span class="elsevierStyleItalic">&#981;</span> as characterized by the numerical representation of the difference relation is a utility function representing &#8827;&#732;&#44; for we have<elsevierMultimedia ident="eq0150"></elsevierMultimedia></p><p id="par0500" class="elsevierStylePara elsevierViewall">Actually&#44; all the properties that have been attributed to the preference relation in microeconomic textbooks can be defined in terms of relations among geometric intervals within the given Euclidean straight line&#44; just as we did with the properties of the difference relation&#46; This shows that the language of intervals&#44; being more powerful than the usual language used in economic theory&#44; is a suitable way of expressing the theories of difference and preference&#46; I hope the reader will find natural this way of expressing the properties of <span class="elsevierStyleItalic">R</span> &#40;some usual ones are given below&#41;&#44; particularly the one implying differentiability&#46; We keep assuming that &#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002; is an algebraic difference structure&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0085"><span class="elsevierStyleLabel">Definition 6</span><p id="par0505" class="elsevierStylePara elsevierViewall">&#8827;&#732; is <span class="elsevierStyleItalic">monotonic</span> iff&#44; for all <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937;&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> implies that <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41; is an interval of the first direction&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0090"><span class="elsevierStyleLabel">Definition 7</span><p id="par0510" class="elsevierStylePara elsevierViewall">&#8827;&#732; is continuous at <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937; iff&#44; for every interval a&#8712;&#923;&#44; as small as you wish&#44; there is a <span class="elsevierStyleItalic">&#948;</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0 such that <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">h</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#937; and &#963;&#40;x0&#40;x0&#43;h&#41;&#41;a whenever h&#60;&#948;&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0095"><span class="elsevierStyleLabel">Definition 8</span><p id="par0515" class="elsevierStylePara elsevierViewall">&#8827;&#732; is strictly convex iff&#44; for every <span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#91;0&#44;1&#93; and menus x1&#44;x2&#8712;&#937;&#44;x1&#8800;x2&#44;&#945;x1&#43;&#40;1&#8722;&#945;&#41;x2&#8712;&#937;&#44; and <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>&#40;1<span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#945;</span>&#41;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is of the first direction whenever <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is of the first direction&#46;</p></span></p><p id="par0520" class="elsevierStylePara elsevierViewall">The great advantage of the language of intervals over the languages typically used to formulate preference theories is that it provides resources by means of which we can also express natural&#44; intuitive differentiability conditions for the preference relation&#46; We turn now to these&#46;</p></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0065">The empirical meaning of differentiability</span><p id="par0525" class="elsevierStylePara elsevierViewall">Derivates are&#44; and cannot be&#44; but ratios between homogeneous magnitudes&#46; That is why it is necessary to represent satisfaction differences by means of intervals within the same geometric space in which distances among consumption vectors are represented&#46; Notice that there is a natural mapping <span class="elsevierStyleItalic">&#964;</span> from the segments within &#937; into &#923;&#44; namely <span class="elsevierStyleItalic">&#964;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; is &#40;the equivalence class of&#41; that interval in &#923; whose Euclidean norm is x1&#8722;x2&#46; Notice that the cut &#91;&#964;&#40;x1x2&#41;&#58;1&#8594;&#93; is precisely x1&#8722;x2&#46; Using H&#246;lder&#39;s interval numbers &#40;cuts&#41; or measure-numbers &#40;<a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder&#44; 1996</a>&#44; p&#46; 242&#41;&#44; as we did in the proof of <a class="elsevierStyleCrossRef" href="#enun0025">Theorem 1</a>&#44; it is possible to map the &#937;-segments into &#923;&#46; In particular all segments of length 1 are mapped by <span class="elsevierStyleItalic">&#964;</span> into segment 1&#8594;&#8784;NE&#8201;in&#8201;&#923;&#46;</p><p id="par0530" class="elsevierStylePara elsevierViewall">Recall that a real-valued function defined on an open subset <span class="elsevierStyleItalic">D</span> of &#8477;L is continuously differentiable at <span class="elsevierStyleBold">x</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">D</span> iff all its partial derivatives exist throughout a neighborhood of <span class="elsevierStyleBold">x</span> and are continuous at <span class="elsevierStyleBold">x</span>&#46; Hence&#44; our aim is to find conditions over the satisfaction differences &#40;or their proxies in &#923;&#41; implying the existence of a function <span class="elsevierStyleItalic">&#981;</span> fulfilling these requirements&#46;</p><p id="par0535" class="elsevierStylePara elsevierViewall">The <span class="elsevierStyleItalic">empirical meaning of the differentiability condition</span> is that the agent&#39;s tastes have a certain sort of stability&#46; That is to say&#44; the rate of change of the agent&#39;s satisfaction is almost constant within a small vicinity of any consumption menu <span class="elsevierStyleBold">x</span> and&#44; at any rate&#44; it varies continuously in any given direction&#46; This means that&#44; within the infinitesimal neighborhood of <span class="elsevierStyleBold">x</span> &#40;within the &#8220;halo&#8221; or &#8220;monad&#8221; of <span class="elsevierStyleBold">x</span>&#41; the rate of change of satisfaction of the agent is &#8220;almost&#8221; constant&#46; This implies&#44; in particular&#44; that in an arbitrarily given direction&#44; determined by the unit vector <span class="elsevierStyleBold">u</span>&#44; the ratio of the satisfaction difference between any two menus to their physical difference is &#8220;almost&#8221; constant&#46; How can this condition be expressed in a formal way&#63;</p><p id="par0540" class="elsevierStylePara elsevierViewall">Let <span class="elsevierStyleBold">x</span> be an arbitrary point in the interior of &#937; and <span class="elsevierStyleBold">u</span> a unit vector in a given fixed direction&#46; Following Newton&#39;s conception of the theory of proportions&#44; the ratio of one magnitude to another of the same kind is to be expressed as a real positive number&#44;<a class="elsevierStyleCrossRef" href="#fn0030"><span class="elsevierStyleSup">6</span></a> and so the required condition is that&#44; for any infinitesimal number <span class="elsevierStyleItalic">¿</span>&#44; the ratio of the satisfaction segment <span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">¿</span><span class="elsevierStyleBold">u</span>&#41;&#41; to the quantity segment <span class="elsevierStyleItalic">&#964;</span>&#40;<span class="elsevierStyleBold">x</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">¿</span><span class="elsevierStyleBold">u</span>&#41;&#41; be infinitely close to a certain positive real number which we can conveniently identify with the cut &#91;&#916;&#40;x&#41;&#58;1&#8594;&#93;&#46; Naturally&#44; we want to identify the cut &#91;&#916;&#40;x&#41;&#58;1&#8594;&#93; with a certain directional derivative&#46; Hence&#44; the condition we are looking for can be formulated&#44; in the language of intervals&#44; in the following way&#46;</p><p id="par0545" class="elsevierStylePara elsevierViewall">If <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> are menus in &#937;&#44; let us denote with &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; the set<elsevierMultimedia ident="eq0155"></elsevierMultimedia></p><p id="par0550" class="elsevierStylePara elsevierViewall">Notice that the points in &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; are interior points of &#937; whenever at least one of the two points <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> is an interior point &#40;see <a class="elsevierStyleCrossRef" href="#fig0015">Fig&#46; 3</a>&#41;&#46;<span class="elsevierStyleEnunciation" id="enun0100"><span class="elsevierStyleLabel">Definition 9</span><p id="par0555" class="elsevierStylePara elsevierViewall">&#8827;&#732; is <span class="elsevierStyleItalic">uniformly differentiable or smooth</span> on &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; if there is a function<elsevierMultimedia ident="eq0160"></elsevierMultimedia>such that&#44; for any menus <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41;&#44; any <span class="elsevierStyleBold">x</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">&#8712;</span><span class="elsevierStyleHsp" style=""></span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41; and <span class="elsevierStyleItalic">¿</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0&#44; there exist positive integers &#956;&#44;v&#44;&#956;&#8242;&#44;v&#8242;&#44;&#956;&#8243;&#44;v&#8243; and <span class="elsevierStyleItalic">&#948;</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0 such that&#44; whenever h&#60;&#948; with x&#43;h&#8712;&#40;x1&#44;x2&#41;&#44;&#8201;&#124;&#956;&#8243;&#47;v&#8243;&#8722;&#956;&#47;v&#124;&#60;&#949; and<elsevierMultimedia ident="eq0165"></elsevierMultimedia>if<elsevierMultimedia ident="eq0170"></elsevierMultimedia>or<elsevierMultimedia ident="eq0175"></elsevierMultimedia>if<elsevierMultimedia ident="eq0180"></elsevierMultimedia></p><p id="par0560" class="elsevierStylePara elsevierViewall">We say that &#8827;&#732; is <span class="elsevierStyleItalic">uniformly differentiable</span> iff it is uniformly differentiable on every set &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; with at least one of <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span> being an interior point of &#937;&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0105"><span class="elsevierStyleLabel">Theorem 3</span><p id="par0565" class="elsevierStylePara elsevierViewall">If &#8827;&#732; is uniformly differentiable&#44; then there exists a utility function representing &#8827;&#732; which is continuously differentiable in the interior of &#937;&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0110"><span class="elsevierStyleLabel">Proof</span><p id="par0570" class="elsevierStylePara elsevierViewall">We will show&#44; first&#44; that there is a uniformly differentiable function &#981;&#710; on any open interval&#46; Let &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; be any such interval&#44; <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span> any menus in &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; with <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#8800;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#44; <span class="elsevierStyleBold">x</span> any menu in &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">4</span>&#41; and <span class="elsevierStyleItalic">¿</span> any positive number&#44; as small as you wish&#46;</p></span></p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0575" class="elsevierStylePara elsevierViewall">It follows that there exist positive integers &#956;&#44;v&#44;&#956;&#8242;&#44;v&#8242;&#44;&#956;&#8243;&#44;v&#8243; and <span class="elsevierStyleItalic">&#948;</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0 such that&#44; whenever h&#60;&#948; with x&#124;&#43;h&#8712;&#40;x1&#44;x2&#41;&#44;&#8201;&#124;&#956;&#8243;&#47;v&#8243;&#8722;&#956;&#47;v&#124;&#60;&#949; and<elsevierMultimedia ident="eq0185"></elsevierMultimedia>if<elsevierMultimedia ident="eq0190"></elsevierMultimedia>or<elsevierMultimedia ident="eq0195"></elsevierMultimedia>if<elsevierMultimedia ident="eq0200"></elsevierMultimedia></p><p id="par0580" class="elsevierStylePara elsevierViewall">Assume that<elsevierMultimedia ident="eq0205"></elsevierMultimedia>let &#968;&#40;x1x2&#41;&#61;&#91;&#963;&#40;x1x2&#41;&#58;1&#8594;&#93;&#44; and choose <span class="elsevierStyleItalic">&#981;</span> with <span class="elsevierStyleItalic">&#981;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#981;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#968;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41;&#46; Since<elsevierMultimedia ident="eq0210"></elsevierMultimedia>for any segment a&#8712;&#923; &#40;cf&#46; <a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder&#44; 1996</a>&#44; p&#46; 244&#41;&#44;<elsevierMultimedia ident="eq0215"></elsevierMultimedia></p><p id="par0585" class="elsevierStylePara elsevierViewall">If we let &#981;&#710; be the function defined by condition<elsevierMultimedia ident="eq0220"></elsevierMultimedia>it follows that<elsevierMultimedia ident="eq0225"></elsevierMultimedia>with<elsevierMultimedia ident="eq0230"></elsevierMultimedia>Hence&#44;<elsevierMultimedia ident="eq0235"></elsevierMultimedia></p><p id="par0590" class="elsevierStylePara elsevierViewall">The assumption that<elsevierMultimedia ident="eq0240"></elsevierMultimedia>leads&#44; by an analogous argument&#44; to<elsevierMultimedia ident="eq0245"></elsevierMultimedia></p><p id="par0595" class="elsevierStylePara elsevierViewall">Thus&#44; at any rate&#44;<elsevierMultimedia ident="eq0250"></elsevierMultimedia></p><p id="par0600" class="elsevierStylePara elsevierViewall">This establishes that <span class="elsevierStyleItalic">&#981;</span> is uniformly differentiable on the interval &#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">2</span>&#41; &#40;with derivative &#981;&#710;&#40;x&#41; at point <span class="elsevierStyleBold">x</span>&#41; and&#44; therefore&#44; the derivative of any point within the interval is continuous&#46;</p><p id="par0605" class="elsevierStylePara elsevierViewall">Consider now&#44; for any interior point <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span> of &#937;&#44; in particular&#44; intervals of the form &#40;x0&#8722;&#945;el&#44;x0&#43;&#945;el&#41;&#8834;&#937; where <span class="elsevierStyleBold">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span> is the canonical vector in direction <span class="elsevierStyleItalic">l</span>&#46; The derivative of <span class="elsevierStyleItalic">&#981;</span> at <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span> is then nothing but the partial derivative of <span class="elsevierStyleItalic">&#981;</span> with respect to <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span> evaluated at <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span>&#58;<elsevierMultimedia ident="eq0255"></elsevierMultimedia></p><p id="par0610" class="elsevierStylePara elsevierViewall">Since this derivative is continuous for every <span class="elsevierStyleItalic">l</span> &#40;<span class="elsevierStyleItalic">l</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44;<span class="elsevierStyleHsp" style=""></span>&#8230;&#44;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">L</span>&#41;&#44; we may conclude that <span class="elsevierStyleItalic">&#981;</span> is continuously differentiable at <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span>&#46; As <span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">0</span> was arbitrarily chosen&#44; we may conclude that <span class="elsevierStyleItalic">&#981;</span> is continuously differentiable in the interior of &#937;&#46;</p><p id="par0685" class="elsevierStylePara elsevierViewall">&#9633;</p></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">New foundations of preference theory</span><p id="par0615" class="elsevierStylePara elsevierViewall">The previous argument shows that we can provide a new conceptual apparatus for preference theory by means of which all the usual properties of the preference relation&#44; as well as the differentiability condition&#44; can be expressed&#46; Is it possible to find another conceptual apparatus that allows the expression of some differentiability condition&#63; That is unlikely because differentiability requires the comparison of satisfaction distances with quantity distances within the same space&#46; At any rate&#44; it is incumbent upon those who believe that it is feasible to do so to produce such an apparatus&#46;</p><p id="par0620" class="elsevierStylePara elsevierViewall">By <a class="elsevierStyleCrossRef" href="#enun0075">Theorem 2</a>&#44; a difference structure D is an algebraic-difference structure iff D admits a geometric representation&#46; Hence&#44; the new conceptual apparatus for preference theory can be briefly summarized as follows&#46; I use the term &#8216;geometric-difference&#8217; to avoid confusion&#46;<span class="elsevierStyleEnunciation" id="enun0115"><span class="elsevierStyleLabel">Definition 10</span><p id="par0625" class="elsevierStylePara elsevierViewall">D is a geometric-difference structure iff there exist &#937;&#44; <span class="elsevierStyleItalic">R</span> and <span class="elsevierStyleItalic">&#963;</span> such that<ul class="elsevierStyleList" id="lis0015"><li class="elsevierStyleListItem" id="lsti0060"><span class="elsevierStyleLabel">&#40;1&#41;</span><p id="par0630" class="elsevierStylePara elsevierViewall">D&#61;&#9001;&#937;2&#44;R&#44;&#963;&#44;&#923;&#9002;&#59;</p></li><li class="elsevierStyleListItem" id="lsti0065"><span class="elsevierStyleLabel">&#40;2&#41;</span><p id="par0635" class="elsevierStylePara elsevierViewall">&#9001;&#937;<span class="elsevierStyleSup">2</span>&#44;<span class="elsevierStyleItalic">R</span>&#9002; is a weak order&#59;</p></li><li class="elsevierStyleListItem" id="lsti0070"><span class="elsevierStyleLabel">&#40;3&#41;</span><p id="par0640" class="elsevierStylePara elsevierViewall">&#963;&#58;&#937;2&#8594;&#923; is a geometric representation of <span class="elsevierStyleItalic">R</span>&#46;</p></li></ul></p><p id="par0645" class="elsevierStylePara elsevierViewall">As pointed out in section &#8216;The theory of preference&#8217;&#44; the preference relation &#8827;&#732; can be defined in terms of <span class="elsevierStyleItalic">&#963;</span> and all the properties usually attributed to it can be expressed using the language of geometric intervals &#40;see the examples there&#41;&#46; The novelty &#8212; as I have just shown &#8212; is that the smoothness condition can also be so expressed&#46; Thus&#44; using the notion introduced in <a class="elsevierStyleCrossRef" href="#enun0100">Definition 9</a> we can define the concept of a smooth preference structure&#46;</p></span><span class="elsevierStyleEnunciation" id="enun0120"><span class="elsevierStyleLabel">Definition 11</span><p id="par0650" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="2017032202421545716"></elsevierMultimedia> is a <span class="elsevierStyleItalic">smooth preference structure</span> iff there exist &#937; and &#8827;&#732; such that<ul class="elsevierStyleList" id="lis0020"><li class="elsevierStyleListItem" id="lsti0075"><span class="elsevierStyleLabel">&#40;1&#41;</span><p id="par0655" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="2017032202421545717"></elsevierMultimedia></p></li><li class="elsevierStyleListItem" id="lsti0080"><span class="elsevierStyleLabel">&#40;2&#41;</span><p id="par0660" class="elsevierStylePara elsevierViewall">&#8827;&#732; is a preference relation induced by a geometric-difference structure&#59;</p></li><li class="elsevierStyleListItem" id="lsti0085"><span class="elsevierStyleLabel">&#40;3&#41;</span><p id="par0665" class="elsevierStylePara elsevierViewall">&#8827;&#732; is smooth&#46;</p></li></ul></p><p id="par0670" class="elsevierStylePara elsevierViewall">By virtue of <a class="elsevierStyleCrossRef" href="#enun0105">Theorem 3</a>&#44; there is a continuously differentiable utility function &#981;&#58;&#937;&#8594;&#8477; representing &#8827;&#732;&#46; The relevance and usefulness of having such a function lies in that it allows the application of non-linear programming techniques in order to find the optimal points&#46;</p></span></p><p id="par0675" class="elsevierStylePara elsevierViewall">It would be desirable&#44; because the empirical condition would be even more intuitive&#44; to formulate entirely the smoothness condition in non-standard language&#44; by means of the notion of an infinitesimal segment&#44; as intimated in the informal discussion preceding the formal introduction of the condition&#46; That is entirely feasible because Euclid&#39;s Archimedian axiom is logically independent of the rest&#44;<a class="elsevierStyleCrossRef" href="#fn0035"><span class="elsevierStyleSup">7</span></a> but it would require a complete reformulation of H&#246;lder&#39;s theory&#44; as well as of the theory of algebraic-difference measurement&#46;</p></span></span>"
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          "titulo" => "Introduction"
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          "titulo" => "The conditions of Debreu and Mas-Colell"
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          "titulo" => "The algebraic theory of difference"
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          "titulo" => "The geometric theory of difference"
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          "titulo" => "The theory of preference"
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          "titulo" => "The empirical meaning of differentiability"
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          "titulo" => "New foundations of preference theory"
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            2 => "Rational choice"
            3 => "Foundations of consumer theory"
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        "titulo" => "Abstract"
        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">This is a paper on the foundations of individual rational choice&#44; specifically on the foundations of consumer theory&#46; Neoclassical consumer theory requires that the behavior of the consumer be explained by means of a preference relation&#44; and that all the required properties of the corresponding utility representation be derived from the properties of this relation&#46; Yet&#44; it is not clear what is the meaning of the property of the preference relation required in order to show that it is representable by means of a continuously differentiable <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span> utility function&#46; The aim of the present paper is to propose an explanation of such property and to prove the existence of a <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span> representation&#46;</p></span>"
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      "es" => array:2 [
        "titulo" => "Resumen"
        "resumen" => "<span id="abst0010" class="elsevierStyleSection elsevierViewall"><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Este es un art&#237;culo sobre los fundamentos de la teor&#237;a de la elecci&#243;n racional individual&#44; espec&#237;ficamente sobre los fundamentos de la teor&#237;a del consumidor&#46; La teor&#237;a neocl&#225;sica del consumidor requiere que el comportamiento del consumidor sea explicado mediante una relaci&#243;n de preferencia y que todas las propiedades requeridas de la correspondiente funci&#243;n de utilidad que la represente sean derivadas de las propiedades de esta relaci&#243;n&#46; No obstante&#44; no est&#225; claro cu&#225;l es el significado de la propiedad de la relaci&#243;n de preferencia requerida para mostrar que la misma es representable mediante una funci&#243;n continuamente diferenciable &#40;<span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">1</span>&#41;&#46; El prop&#243;sito del presente art&#237;culo es proponer una explicaci&#243;n de tal propiedad y demostrar la existencia de una representaci&#243;n&#46;</p></span>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0005">Peer Review under the responsibility of Universidad Nacional Aut&#243;noma de M&#233;xico&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0010">See&#44; for instance&#44; <a class="elsevierStyleCrossRef" href="#bib0055">Katzner &#40;1970&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0005">Barten and B&#246;hm &#40;1981&#41;</a>&#44; and <a class="elsevierStyleCrossRef" href="#bib0075">Mas-Colell&#44; Whinston and Green &#40;1995&#41;</a>&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0015">Because the dimension of &#937; is&#44; which is the same as that of &#8477;L&#46; This implies that the relative interior of &#937; with respect to the linear space &#8477;L is nonempty and so&#44; for sufficiently small <span class="elsevierStyleItalic">¿</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0&#44; the open ball <span class="elsevierStyleItalic">Be</span>&#40;<span class="elsevierStyleBold">x</span>&#41; centered in <span class="elsevierStyleBold">x</span> is contained in &#937;&#46; For a discussion of the notion of relative interior that relates it to economic theory&#44; see <a class="elsevierStyleCrossRef" href="#bib0060">Koopmans &#40;1951&#41;</a>&#44; especially p&#46; 45&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0020">Precisely in the sense of Definition 3 in <a class="elsevierStyleCrossRef" href="#bib0065">Krantz&#44; Luce&#44; Suppes and Tversky &#40;1971&#44; p&#46; 151&#41;</a>&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0025">See Theorem 2 in <a class="elsevierStyleCrossRef" href="#bib0065">Krantz et al&#46; &#40;1971&#44; p&#46; 151&#41;</a>&#59; see p&#46; 158 for a proof&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0030"><span class="elsevierStyleItalic">Von gleicher Richtung</span> in the original&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0035">Cf&#46; <a class="elsevierStyleCrossRef" href="#bib0045">H&#246;lder &#40;1996&#44; p&#46; 241&#44; 8&#41;</a>&#46;</p>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0040">Cf&#46; <a class="elsevierStyleCrossRefs" href="#bib0035">Hilbert &#40;1950&#44; pp&#46; 21&#8211;22&#41;&#59; Fleuriot &#40;2001&#44; chap&#46; 4&#41;</a>&#46;</p>"
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">CB</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">A&#8826;C&#8826;B&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">AC</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">3&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">BA</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">BC</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">B&#8826;A&#8826;C&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">AC</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">4&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">BA</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">BC</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">B&#8826;C&#8826;A&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">CA</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">5&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">AB</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">CB</span>&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">C&#8826;A&#8826;B&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">CA</span>&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">CB</span>&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top"><span class="elsevierStyleItalic">CA</span>&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head  " colspan="7" align="center" valign="top" scope="col" style="border-bottom: 2px solid black">Feasible cases in <a class="elsevierStyleCrossRef" href="#enun0055">Lemma 3</a></th></tr><tr title="table-row"><th class="td" title="table-head  " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Case&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</th><th class="td" title="table-head  " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">&#963;x1&#8242;x2&#8242;&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</th><th class="td" title="table-head  " align="left" valign="top" scope="col" style="border-bottom: 2px solid black"><span class="elsevierStyleItalic">&#963;</span>&#40;<span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleBold">x</span><span class="elsevierStyleInf">3</span>&#41;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">&#963;x1&#8242;x3&#8242;&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">3&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">5&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">6&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="top">7&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">I or null&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">II&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="left" valign="top">any&nbsp;\t\t\t\t\t\t\n
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Article information

ISSN: 01861042
Original language: English

DOI:10.1016/j.cya.2015.06.002

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2019 September	6	3	9
2019 August	7	2	9
2019 July	10	9	19
2019 June	35	11	46
2019 May	63	26	89
2019 April	27	3	30
2019 March	3	0	3
2019 February	5	7	12
2019 January	8	1	9
2018 December	2	8	10
2018 November	4	22	26
2018 October	5	4	9
2018 September	6	1	7
2018 August	7	8	15
2018 July	3	0	3
2018 June	5	0	5
2018 May	7	7	14
2018 April	2	0	2
2018 March	2	0	2
2018 February	6	0	6
2017 December	2	0	2
2017 November	8	0	8
2017 October	9	1	10
2017 September	6	0	6
2017 August	6	4	10
2017 July	6	2	8
2017 June	16	8	24
2017 May	16	1	17
2017 April	14	8	22
2017 March	14	13	27
2017 February	5	2	7
2017 January	1	2	3
2016 December	12	1	13
2016 November	18	3	21
2016 October	22	7	29
2016 September	13	1	14
2016 August	15	1	16
2016 July	19	4	23
2016 June	31	13	44
2016 May	33	21	54
2016 April	19	11	30
2016 March	35	18	53

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