A Constructive Proof of the Existence of a Utility in Revealed Preference Theory

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1 A Constructie Proof of the Existence of a Utility in Reealed Preference Theory Eberhard, A.C. 1, D. Ralph 2 and J P. Crouzeix 3 1 School of Mathematical & Geospatial Sciences, RMIT Uniersity, Melbourne, VIC., Australia. 3 Judge Business School, Uniersity of Cambridge, UK. 2 LIMOS, Uniersite Blaise Pascal, de Clermont, France 12/01/2012 Abstract Within the context of the standard model of rationality within economic modelling we show the existence of a utility function that rationalises a demand correspondence, hence completely characterizes the associated preference structure, by taing a dense demand sample. This resoles the problem of reealed preferences under some ery mild assumptions on the demand correspondence which are closely related to a number of established axioms in preference theory. The proof establishes the existence of a limit of a sequences of indirect utilities that rationalise finite data sets, where the sample size increases to infinity. This limiting utility proides a rationalisation of the demand relation. Up to a rescaling this limiting indirect utility is unique and continuous on a set of full measure. 1 Introduction 1.1 Posing the question A demand relation X(p) describes what can be, in principle, obsered about the preferences of an agent such as an indiidual consumer: There are n commodities and a positie budget, that we normalise to 1, that constitutes the consumer s wealth. Then gien a ector of n nonnegatie prices p (with p 0) X(p) returns a commodity bundle x that the consumer is obsered to buy with their budget. In other words, out of all commodity bundles that are within budget, none is more preferred by the consumer than x X(p). When x X(p) and p, x y 0 then y is within budget but not chosen. We then say that x is a (directly) reealed preference to y and denote this by x R y. As not all obserations are directly reealed, a preference relation may be deduced ia transitiity of the directly reealed preferences, i.e., x R y and y R z would imply that x is preferred to z. This is only possible if obsered behaior is at least consistent under this transitie closure and this has become axiomatic in the theory, nown as the Generalized Axiom of Reealed Preferences (GARP). As in other studies of reealed preference theory we will posit this as a standing assumption. We do not now a priori whether there exists a utility function representing such a preference relation, an existence problem nown as the problem of reealed preferences. Typically in reealed preference theory [[Houthaer (1950)], [Samuelson (1947)], [Varian (2006)], [Varian (1982)]], additional structure of the demand relation is postulated to derie a preference relation. The debate about which axioms are essential or most justifiable is ongoing but these are critical to the building of utility functions. In the main such theories hae been relational in nature, often giing rise to geometric constructions. The role of the axioms in reealed preference theory in relation to (topological) conergence issues has not before been fully appreciated and the study of these issues will constitute the main contribution of this paper. Our specific goal is to understand when demand relations, and therefore preferences, can be described by a utility function and, in particular, when such a utility can be reconstructed from any appropriate sample of price-commodity pairs {(p i, x i )} i=1 where x i X(p i ) for each i. Not haing such a utility to study in the first place, we construct it by approximation: for a finite sample 1

2 {(p i, x i )} m i=1 of m price-commodity bundles satisfying the demand relation, we may construct a finite rationalisation, namely, a utility u m that is consistent with this data 1. Such rationalisations are well nown thans to Afriat s classical construction [Afriat (1967)]; alternatie rationalisations include the recent wor of [Crouzeix et al. (2011)]. What may be surprising is that although Afriat s finite rationalisations are concae, there may not exist a concae rationalisation of the entire demand relation (see example 7 in section 2.1). Thus an understanding of the asymptotics of finite rationalisations, i.e., as the sample size m grows to, is a challenge that needs techniques beyond conex analysis. A final introductory point is that our approach to the asymptotics of sampling preference relations is ia indirect utilities, which are dual to direct utilities in that they describe the maximum utility that could be gained within a unit budget. The notation for this follows Our assumptions and main result We see a utility that rationalises X( ). That is a quasiconcae function u on R n + that prescribes to each x R n + a number which represents the degree of preference, namely x R n + is preferred to y R n + if u(x) u(y). Gien a (normalized) unit budget, we define the indirect utility (p) by (p) := max{u(x) p, x 1}. (1) x R n + This assigns the greatest utility that can be attained, within a budget of 1, at the price p. Under mild conditions we can recoer u from ia the duality identity u(x) = inf {(p) p, x 1}, (2) p R n + see section 2.3. Similarly when such a utility exists we can recoer the demand relation ia X(p) = argmax{u(x) : p, x 1} := {x C u(x) (p)}. (3) x R n + Indeed we can recoer X( ) from the indirect utility by the connecting identity (see Proposition 18 in section 2.3) X(p) = N (p) {x x, p = 1}, (4) where N (p) is the normal cone to the (lower) leel set of gien by S (p) := {p (p ) (p)}. Conexity of these leel sets also holds that is, is quasiconex which allows application of conex analysis (e.g., the usual normal cone construction aboe). Gien a set S R n, co S denotes its conex hull, and cone S := λ 0 λs is its cone. We will construct an indirect utility that satisfies (4) as a ind of limit of a sequence of indirect utilities m each of which rationalises the data set {(p i, x i )} m i=1, that is x i N m (p i ) with x i, p i = 1 for i = 1,..., m. More generally, we say m is an indirect finite rationalisation or simply a finite rationalisation when it is clear that it is an indirect utility, if it is quasiconcae and the preious inclusion holds for i = 1,..., m. To date neither the existence of topological limit points of finite rationalisations { m }, as the sample size m goes to infinity, nor uniqueness of limit points has been studied. (Liewise, the asymptotics of the associated sequence of direct utilities has not been analysed.) These issues form the core questions around which this paper is directed. Critically we must mae certain assumption on the nature of X from which we are sampling. We state our assumptions here and, after giing some notation, summarize our main results. Then we will briefly reiew these assumptions by placing them in the literature on reealed preference theory. Recall that the graph of X( ) is the set graph(x) := {(p, x) x X(p)} so that a countable sample from this demand relation is gien by the inclusion {(p i, x i )} i=1 graph(x). Demand relation assumptions 1 Each u m rationalises {(p i, x i )} m i=1 in that x i argmax x R n + {u m (x) p i, x 1} for i = 1,..., m. 2

3 A1 The demand relation X( ) satisfies the generalised axiom of reealed preference, GARP (see section 1.2.1). A2 The images X(p) are nonempty and conex. A3 The graph of X( ) is closed (i.e., if x i X(p i ) and lim i (p i, x i ) = (x, p), then x X(p)). Sampling assumption A4 1. D, which denotes closure of {p i } i=1, is a conex set in Rn + that has nonempty interior, int D, and does not contain zero. 2. C, which denotes the closed, conex hull co{x i } 1, is a set in R n + that is bounded and does not contain zero. Our first main result says that a dense sample of a demand relation X( ) is enough to reconstruct the entire relation; more than that, we can gie a direct utility function u and indirect utility function representing X( ). We mae a few introductory remars to lin the assumptions and conclusions of Theorem 1. Axiom A1 implies that the leel sets S u (x) and S (p) are conex: quasiconexity of and quasi-concaity of u. In concert with A2 and A3, we also anticipate a lower semicontinuity property of, namely, the leel sets S (p) and strict leel sets S (p), defined by S (p) := {p (p ) < (p)}, hae the same closures when (p) > inf p (p ). This closure property is essentially a consequence of being able to recoer a preference relation by taing limits of dense sample data. Quasi-conex functions with this closure property are called g-pseudo-conex [Crouzeix et al. (2008)]. Interiority in Axiom A4 implies is solid, i.e., S (p) has interior when (p) > inf p (p ). It turns out that u is also solid, in short u is solid g-pseudo-concae. Finally we note that the class of utilities u, or indirect utilities, that represent a demand relation has at least one degree of freedom: for any strictly increasing function : R R we hae that u or represents, indirectly or directly, the same preferences (we will expand on this in section 1.1.2). We therefore focus on normalised utilities: is said to be normalised in some direction e R n, or e-normalised for short, if (te) = t. Theorem 1 Under Assumptions A1-A4: 1. There exists an indirect utility function : D R + that rationalises X( ), i.e., (4) holds, and that is solid and g-pseudo-conex and continuous almost eerywhere in the interior of D. 2. For e int D, being a fixed direction, we may further assume that is e-normalised and lower semicontinuous, in which case is the unique function with these properties that indirectly rationalises X(.) on int D. 3. In the general case (corresponding to part 1), the function u : C R + gien by (2) is a direct utility that rationalises X( ), i.e., satisfies (3) for p int D, and that is solid g-pseudo-concae and continuous almost eerywhere in int C. Parts 1 and 2 of this result are gien by Theorem 35 and Corollary 37 in Section 5. Part 3 is a consequence of Proposition 15 and the fact that any function u satisfying the duality (2) must be a monotonically nondecreasing functions with respect to the cone corresponding to the positie orthant. Thus the continuity is a consequence of the results of [Crouzeix et al. (1987)]. We stress the role of the relationship (3) in the subsequent proof of this result. Let Assumptions 1 4 hold and {f m } be any sequence of indirect utilities where each f m rationalises {(x i, p i )} m i=1 and is normalised in a direction e int D. Consider an epi-conergent subsequence of {f m }, which exists by Theorem 34, and whose (lower semicontinuous) epi-limit is denoted. Combining Theorem 34 35, any such satisfies N (p) = cone X(p) on int D which is equialent to (3). That is, normal cones to the leel sets of asymptotic indirect utilities are independent of the particular epi-limit. Indeed these normal cones are also independent of the demand relation sample {(x i, p i )} i=1 so long as the closure of {p i} is D. It is easy to establish (see Lemma 36) 3

4 that lower semicontinuous quasiconex functions are completely determined by the normal cones to their leel sets, and their alues in a particular direction (i.e., on the ray et for t > 0). It follows since is e-normalised that is it the unique lower semicontinuous e-normalised indirect utility that represents X. We now lin the properties of u, aboe to notions from the economics literature. Consider a solid g-pseudo-concae utility u. Instead of solidity of u we can stipulate that it is nonsatiated, which is a standard notion from preference theory: for all x R n +, δ > 0 there exists x B δ (x) with x x, x x and u(x) < u(x ). Liewise, is nonsatiated. Indeed if we tae a g-pseudo conex function, where is nonsatiated, the associated demand relation generated ia (4) possess the properties A1 A4. For instance the necessity of GARP is gien by [Eberhard et al. (2007)]. GARP is also nown to be characterised by the existence of a conex utility that rationalise any finite data set [Fostel et al. (2004)]. Conexity and closure of the images X(p), which follow from (4), or (3) if X is generated ia a quasi-conex utility u, are also standard assumptions in reealed preference theory and hae been justified in their own right [John (1998)]. A3, graph closure, is a ind of continuity that follows from g-pseudo conexity as discussed aboe. Axioms A2 and A3 are closely related to a maximality property for pseudo-monotone operators which are also a consequences of (4) when X is generated ia a solid, g-pseudo-conex, indirect utility. Regarding A4, prices and commodity bundles are clearly positie n-tuples this essentially asserts that one should be able to probe the demand with prices within an open neighbourhood and obtain non-triial preferences. This turns out to be closely related to the nonsatiation assumption. As D R n ++ is bounded one can argue that the boundedness of co{x i } 1 follows from the budget constraint and the positiity of commodity bundles Relating conergence to the wor [Crouzeix et al. (2011)] Here we begin to describe the framewor that leads to Theorem 1. The asymptotic analysis that is required is closely related to recent wor of [Crouzeix et al. (2011)] and which starts with a utility function u that represents the demand relation X( ) and is appropriately normalised 2 in a direction d int C. [Crouzeix et al. (2011)] constructs functions u m and u + m on R n + that rationalise {(p i, x i )} m i=1, are normalised in the direction d, and that sandwich the direct utility u as u m(x) u(x) u + m(x), for u R n +. Moreoer, u m and u + m sandwich any other normalised finite rationalisation and are monotonic in m: u m(x) u m+1 (x) u+ m+1 (x) u m(x) for x C. Since the construction of u m and u + m is entirely symmetric in x and p, we may apply it to samples ordered as (x i, p i ) instead of (p i, x i ) using a normalisation direction e int D. Assuming there is a normalised indirect utility that gies the demand relation ia (4), this dual construction produces normalised indirect rationalisations m and m + that sandwich. Thus by defining m := m + and m + := m(p), we sandwich and any indirect utility that rationalises {(p i, x i )} m i=1 and is e-normalised. Philosophically we depart from [Crouzeix et al. (2011)] in that we do not yet now whether our demand relation X( ) can be represented by an indirect utility or its direct counterpart u. Neertheless m and + m sandwich any indirect utility that is a normalised finite rationalisation of {(p i, x i )} m i=1. For example we can always use Afriat s construction to generate an indirect rationalisation m and then can normalise it (as explained in section to follow) to produce an indirect, normalised rationalisation f m. Then m and + m will sandwich f m. We summarise the properties of m and + m for later use. As explained aboe, this result is due to [Crouzeix et al. (2011)] although we hae transposed the original ersion from direct to indirect utilities. Lemma 2 Suppose the finite sample of data {(p i, x i )} m i=1 satisfies GARP. Then there exist real alued quasiconex functions m, m + on R n + that 1. indirectly rationalise {(p i, x i )} m i=1, 2. are e-normalised, 2 A utility u that is normalised in a direction d satisfies u(td) = t for t > 0. Normalising an indirect utility requires t on the right hand side. 4

5 3. are lower and upper bounds, respectiely, on any indirect utility f m that rationalises {(p i, x i )} m i=1 and is e-normalised: m(p) f m (p) + m(p), for p R n +, and 4. are monotonic in m: m(p) m+1 (p) + m+1 (p) m(p) for p R n +. See section for details of the normalisation procedure which conerts a conex (e.g., Afriat) rationalisation m into a normalised rationalisation f m that is quasiconex. When there is no conex indirect utility function associated with X( ), as in example 7 in section 2.1, we will still proe that {f m } conerges to gie existence of a quasi-conex (indeed, solid g-pseudoconex) indirect utility function. The normalisation process is critical in the proof because it will aoid theoretical and numerical difficulties that arise, as the sample size increases, when approximating a nonconex function by a conex function. As the sample size m increases, the lower minorant increases and the upper majorant decreases. If the sampling is dense then one arries (ia a monotonic limit) at an interal [ (p), + (p)] which must bracet a true indirect utility alue. Howeer [Crouzeix et al. (2011)] does not consider asymptotics of this sandwiching process. For instance the question of whether or when [m(p), m(p)] + shrins to a singleton is not addressed. Moreoer any sequence of rationalising, normalised mappings {f m } are unliely to be monotonic and therefore conergence of any ind is left open. For any sequence of normalized quasiconex indirect utilities {f m } m=1, where f m rationalises {(p i, x i ) m i=1, we may claim the following (assembled from Corollaries in Section 5): Theorem 3 Under Assumptions A1-A4: Let m and m + be the e-normalised indirect finite rationalisations as described in Lemma 2, and = lim m m and + = lim m m + denote their respectie pointwise (monotone) limits. Let {f m } be any sequence of mappings on R n + such that each f m that indirectly rationalises {(p i, x i )} m i=1 and is e-normalised. Then 1. There is a (unique) lower semicontinuous function : R n + R that is the lower closure of any quasiconex function satisfying (p) [ (p), + (p)] for p R n is the (unique) epi limit of {f m } m=1. 3. rationalises the full data set {(p i, x i )} i=1. 4. is independent of the particular sample {(p i, x i )} i=1, proided the closure of {x i} i=1 is D, and of the sequence of normalised indirect finite rationalisations { m } i=1 that are fit to these data. 5. rationalises the demand relation X. 6. For almost all p D, + is continuous at p and, as a result, (p) = (p) = + (p). See Remar 41 for some comments on applying this conergence framewor to direct rather than indirect utilities. We focus on the latter because they are more naturally related to sampling the demand relation, i.e., to probe consumption one would change the price slightly and obsere the change in consumption within the same budget. More importantly this is more closely related to elasticities [Kocosa et al. (2009)]. 1.2 The Anatomy of the Proof The purpose of this section is to proide an extended abstract of the proof of the main results. Much of the difficulty encountered in proing the results stems from the need to study epi-limits of g-pseudoconex functions which are not conex functions. Thus we wish to proide a conceptual roadmap of the crucial techniques that are deeloped in order to furnish a proof before entering into details. It is hoped that this will help the reader to understand the role of a number of technical constructions. 5

6 1.2.1 Constructing Approximate Indirect Utilities Gien {(x i, p i )} Graph X the transitie closure of R gies a partial order (of reealed preferences) that is we denotes x y when there exists x = x 0, x 1,..., x m = y with x i+1 R x i for all i. Gien x i X(p i ) and x j X(p j ), we say x j R x i if p j, x j < p j, x i while x i X(p j ). Similarly we denote x y when x y and there exists i with p i+1, x i+1 x i > 0 for (x i, p i ) Graph X. The generalised axiom of reealed preference (GARP) says that there cannot exist a cycle {(x i, p i )} m i=0 with x 0 = x n+1 and p i+1, x i+1 x i 0 unless p i+1, x i+1 x i = 0. This axiom is clearly necessary for a consistent transitie closure partial order in that p i+1, x i+1 x i > 0 for some i would imply the contradiction x n+1 x 0. Indeed it is nown that GARP is necessary and sufficient for the existence of a preference order R such that x R y wheneer x y, and x R y wheneer x y (see [Kanna (2004)]). It is also necessary and sufficient for the fitting of the Afriat indirect utility m to rationalise a finite data set {(x i, p i )} m i=1 [Afriat (1967)], [Fostel et al. (2004)]. Thus either ia Afriat s construction or that of [Crouzeix et al. (2011)] we can obtain a set of conex, polyhedral, leel cures of an indirect utility m : R n ++ R for each data set {(x i, p i )} m i=1 such x i N m (p i ) for i = 1,..., m. To mae a comparison which accounts for nonuniqueness of indirect utilities that represent a demand relation, we also normalize the indirect utility. For each m define the strictly decreasing, function ia the fitted indirect Afriat utility m (p) which has been translated so that m (0) = 0 e.g. m (x) 0. Begin by taing some e int D, hence e > 0, taing m sufficiently large so that e int co {p i } m i=1 and forming m (t) := m (et) where t > 0. As long as we sample (x i, p i ) with x i 0 then t m (t) is strictly decreasing. We then normalise our Afriat utilities to produce another sequence of equialent utilities f m (p) := 1 m ( m (p)) (5) which is the composition of a conex function on R n ++ and a concae continuous increasing mapping on R. Now e lies on the leel cure {p f m (p) = 1} for each m and also τ f m (τe) = τ is finite. Wheneer we hae f m (te) = t we say that the indirect utility f m has been normalized in the direction e, or simply normalized when e is already fixed. Note that when we fit the Afriat indirect utility we hae dom m = dom f m = R n +. Consequently by the chain rule of generalized gradients [Rocafellar et al. (1998)] and the fact that there exists µ m i > 0 such that µ m i x i f m (p i ) for i = 1,..., m (as f m rationalises {(p i, x i )} m i=1 as per (4)) we hae f m (p) = co { γµ m i x i γ 1 m ( m (p)) and for some µ m i > 0, µ m i x i m (p) }. (6) As m strictly decreasing continuous then m 1 is strictly increasing and so γ > 0. Denoting the leel set of f m at p by S fm ( p) = {p f m (p) f( p)}, then (6) implies the normal cone to S fm ( p) at p, denoted by N fm ( p) may be expresses as: N fm ( p) = cone m ( p). (7) This is a triial extension of the well nown relationship between normal cones of leel sets and subdifferentials of conex functions Epi/Quasi-conergence of a Subsequences In order to show that a utility exists that rationalises the data set we hae to establish that there is a well defined limit (in some sense) of the approximating sequence {f m } m=1. This strongly relies on the properties of epi-conergence and the properties of the underlying demand relation that we are interpolating. Epi-conergence can be directly associated with set-conergence of the leel sets of the family of functions. As epi-conergent sequences always produce lower semicontinuous functions we introduce a related but weaer form conergence called quasi-conergence that does not a priori introduce semicontinuities but still demands an orderly stratification of limiting leel sets (see section 2.2). Epi-conergence is then used as tool to study this weaer conergence. Our conergence analysis relies on a coordinate transformation from x R n to (y, t) R n 1 R in order to represent each nonconex function f m, that neertheless has conex leel sets S fm (p), by a family of conex functions {g m (, λ)} λ Λ on R n 1. Here Λ is a set of scalars and each λ Λ signifies a leel of f m, e.g., λ = f(p) corresponds to S fm (p). See section 3 for details. 6

7 Haing defined the sequence {g m (, λ) It will then be possible to describe normal cones to the leel sets of f m ia the subdifferential of g m (, λ), namely N fm (y, t) = cone g m (, λ)(p). Now Hausdorff set-conergence of a sequence of subdifferentials is implies by the epi-conergence of the corresponding sequence of conex functions [Attouch et al. (1991)]. Therefore this coordinate transformation from f m to obtain the family of functions g m (, λ) allows us to pursue conergence of the sequence of normal cones {N fm (y, t)} m=1 ia epi-conergence of a parameterised sequence of conex functions {g m (, λ)} λ Λ implied by the powerful epi-conergence theory for {f m } m=1. This results in a proposed limit, or perhaps limits, and an associated function f that is an indirect utility for X. A technical but critical aspect of conergence comes from the fact that although the analysis described aboe inoles conergence of cones ia conergence of epigraphs, our interest is in conergence of bounded sets that generate cones and liewise of subdifferentials. This can be inferred from (4), where the demand relation is gien as the base of a the normal cone N (p) where is an indirect utility that we will show exists as an epi-limit. We therefore explore, in section 2.4, the relationship between set-conergence of families of cones and set-conergence of their bases in a somewhat abstract setting. We e mentioned section 3 for the transformation from f m to g m and we further note that conergence of {f m } and {g m } is gien in section 4. We gie the milestones of this conergence analysis next. A number of steps must be taen before conergence of the sequence {f m } can be established. First we use the compactness property of epi-conergence to extract an epi-conergent subsequence of the family functions {f m (, )} m=1 and sets Λ m for which the associated leel cure functions p g m (p, λ) is conex for λ Λ m. We then note that there exists a sequence λ m λ Λ = lim Λ m such that {g m (, λ m )} =1 actually epi-conerges to a conex family of functions g (, λ) for all λ Λ. One may then associate a set functions {g m (, λ m )} =1 to a sequence of lower semicontinuous, solid, g-pseudoconex functions {f m } =1 which also epi-conerge (as its leel sets set-conerge) to an f while retaining the normalization property f (te) = t. The study of the correspondence between {g(, λ)} λ Λ and f allow one the establish the arious properties of {g(, λ)} λ Λ that force the limiting function f to be solid, g-pseudo-conex and nonsatiated. The associated conergence of subdifferentials along with the rationalisation property x i N fm (p i ) = g m (, λ i ) (p i ) (for λ i = f m (p i )) allows us to obsere that the limiting function f s subdifferential interpolates a countably dense set of points x i X(p i ). Once the problem of recoery of X from such a dense selection has been studied we may deduce that X(p) cone {x x g(, λ)(p) for λ = f(p)} {x x, p = 1} (8) = N f (p) {x x, p = 1}. (9) This later point turns out to be critically important to this study. Many multi-alued operators, such a monotone operators, possess this recoerability property where one only needs to form the smallest similar object that interpolates a dense selection in order to recoer the original operator. This is not surprising once one is able to connect X to maximal quasi-monotone operators, such as those studies in [Aussel et al. (2011)], that retain many of the properties of deriatie lie objects. The properties A1-A4 are sufficient to show this is true for both cone X(p) and N f (p). In fact N f must be the minimal operator of this type and cone X being also from this same family forces equality in (8). The desired direct utility is then gien by u(x) = inf {(p) p, x = 1} and can also be shown to be solid, g-pseudo-concae and nonsatiated. The proof of inclusion (8) is under taen in section 4 while the properties of maximal pseudo-monotone operators (such as N f and cone X) is undertaen in section 2.4 (see definitions there) Conergence\Uniqueness of any Rationalizing Approximation In section 5 we mae the obseration that the reconstructie properties of X means that the equality (9) holds irrespectie of the sampling made (as long as it is a dense selection) nor the means ia which one constructs a normalized sequence of approximate indirect utilities. This along with the normalization condition f (te) = t forces uniqueness of the epi-limit f that can be constructed ia this program. There a many ramifications of this obseration such as epiconergence of the normalized Afriat approximate utilities to the (common) lower closure of any other 7

8 indirect, normalised utility that rationalises X and the obseration that (p) = + (p) almost eerywhere with respect to p at points of continuity of +. 2 Preliminaries 2.1 Sets and functions We begin with notation for sets related to functions, such as epigraphs and leel sets, and arious classes of functions that are described through these sets. Denote the closure of a set S R n by S. The set of natural numbers is denoted N. Denote by R the set of real numbers and R = R { } {+ }. 3 R n denotes n-dimensional Euclidean space, R n + and R n ++ denote the set of n-dimensional ectors with nonnegatie and strictly positie components, respectiely. Gien f : R n R denote by epi f := {(x, α) α f (x)} and epi s f := {(x, α) α > f (x)}. Next, gien λ R, let us define the leel and strict leel sets of f by S λ (f) := {x f(x) λ}, Sλ (f) := {x f(x) < λ}, S f (x) := {x f(x ) f(x)}, Sf (x) := {x f(x ) f(x)}. Then, S λ (f) = µ>λ S µ (f) = µ>λ S µ (f), f(x) = inf { λ (x, λ) epi f } = inf { λ x S λ (f) } = inf { λ x S λ (f) }. Then f is said to be conex if epi f is conex and quasiconex if S λ (f) is conex for each λ R. The function f is said to be lower semicontinuous at a if for any λ < f(a) there is a neighborhood V of a such that f(x) > λ for all x V. The function f is said to be lower semicontinuous if lower semicontinuous at any a R n. It is nown that f is lower semicontinuous if and only if epi f is closed and, also, if and only if S λ (f) is closed for any λ R. Gien f : R n R, let us consider the function g be defined by g(x) = inf { λ (x, λ) epi f }. It is easily seen that epi g = epi f. Hence g is lower semicontinuous and is the supremum of all lower semicontinuous functions bounded from aboe by f. It is called the lower closure, or lower semicontinuous hull and is denoted by f. Next, let the function h be defined by h(x) = inf{ µ x S µ (f) }. Then, S λ (h) = µ>λ S µ(f). Then, h is lower semicontinuous and is the supremum of all lower semicontinuous functions bounded from aboe by f and therefore coincides with f. We summarize these results below. Proposition 4 Gien f : R n R, f(x) = inf{λ (x, λ) epi f } = inf{ λ x S λ (f) }, epi f = epi f, S λ ( f) = µ>λ S µ (f) = µ>λ S µ (f). It follows that f is always lower semicontinuous. As immediate consequences one has: Corollary 5 If f is conex (quasiconex) so is f. Corollary 6 f is lower semicontinuous at a if and only if f(a) = f(a). Proof. By construction f(a) f(a). Assume that there is λ such that f(a) > λ > f(a). Then a S λ (f). Any neighbourhood of a contains points p with f(p) λ. Therefore f is not lower semicontinuous at a. Conersely, assume that f(a) = f(a). Since f f and f is lower semicontinuous, for any λ < f(a) there exists a neighbourhood V of a such that f(p) f(p) > λ. Then f is lower semicontinuous at a. 3 We do not directly deal with functions that tae the alues ±. We allow for this possibility when taing limits of finite rationalisations, but then show limiting functions tae finite alues. 8

9 Since f f and f is lower semicontinuous one has S λ ( f) = S λ ( f) S λ (f) and therefore S λ ( f) S λ (f). The equality does not hold in general as seen in the following example: f(x 1, x 2 ) = x 2 if x 1 = 0, x 2 < 0 and f(x 1, x 2 ) = 0 otherwise. Then f is quasiconex and S 0 ( f) = R 2 {0} (, 0] = S 0 (f). We shall see in Appendix 3 that the equality holds under some condition. Define the normal cones to the leel and strict leel sets of f by N f (x) := {x x x, x 0 for all x S f (x)} if x dom f, (10) { Ñ f (x) := x x x, x 0 for all x S } f (x) if x S f (x), (11) (when x / S f (x) we define Ñf (x) = {0}). We say f is g-pseudo-conex function when it quasiconex and S λ (f) = S λ (f) for λ > inf f := inf{f(x) x R n }. We say f is solid when int S λ (f) for λ > inf f. Similarly we say g is pseudoconcae (resp. g-pseudo-concae) if f = g is pseudoconex (resp. g-pseudo-conex). For pseudoconcae function one must study the upper leel sets S λ ( g) and S λ ( g). This is a broader class than conex functions as the next example shows. Example 7 where u 1 (x) = 4 x 1 x 2 and u (x) := min {u 1 (x), u 2 (x)}, x 0 u 2 (x) = { x1 +3x 2 1+x 2 if x 1 + x 2 2 x 1 + x 2 if x 1 + x 2 2. This function u is pseudoconcae and increasing (also solid g-pseudo-concae) but no strictly increasing function exists such that x (u (x)) is concae. 2.2 Conergence of sequences of sets and functions It is well nown that epi-conergence characterises conergence of leel sets (see [Rocafellar et al. (1998), Proposition 7.7]) so we turn our attention to this type of conergence in our construction of a limiting utility. We gie some standard definitions, see [Rocafellar et al. (1998)] for example. Definition 8 1. For a family of sets {C N} in R n, denote the upper Kuratowsi-Painleé limit by lim sup C and the lower Kuratowsi-Painleé limit by lim inf C. Clearly lim inf C lim sup C. When these coincide we say that {C N} set-conerges, or simply conerges, to C (in the Kuratowsi-Painleé sense) and write C = lim C. Let {g, g R n R, N} be a family of proper extended-real-alued functions. Then the lower epi-limit e-li g is the function haing as its epi-graph the outer limit of the sequence of sets epi g : epi(e- li g ) := lim sup(epi g ). The upper epi-limit e-ls g is the function haing as its epigraph the inner limit of sets epi f : epi(e- ls g ) := lim inf(epi g ). When these two functions are equal, they are jointly called the epi-limit of {g } and denoted e-lim g. In this case the sequence g is said to epi-conerge (to its epi-limit as ). In particular it is possible to express the epi limit infimum (supremum) as follows e- ls g (p) = min lim inf {p p} g (p ) e- li g (x) = min lim sup g (p ), {p p} where the min signifies that there exist sequences that attain this infimum. Any epi-limit is a lower semicontinuous function. This may be iewed as undesirable because lower semicontinuity is not a fundamental notion when studying quasiconex functions. To this end we mae the following definition of a limiting function which does not imply lower semicontinuity in the limit, but, lie epi-conergence, demands an orderly inclusion of associated leel sets in the limit. 9

10 Definition 9 Gien a family of extended-real alued functions {g } N we say this family quasiconergent to g as (or in short q-conerge) if and only if for all λ we hae 1. For all λ λ and all µ > λ we hae 2. For some µ µ and all λ < µ we hae lim sup S λ (g ) S µ (g). lim inf S µ (g ) S λ (g) ; Denote by Q ( {g } N ) the equialence class of functions to which {g } N quasi-conerges. This is about the weaest (non-hausdorff) conergence one could demand and still obtain an orderly conergence of leel sets. This concept appears strictly weaer than epi-conergence (as we shall shortly show). Example 10 Consider the piecewise linear function 1 for 1 g x (x) := x for 1 < x < 1 1 for x 1 Then {g } N clearly quasi-conerges and Q ( {g } N ) contains all functions with g (x) = 1 if x < 0, g (x) = +1 if x > 0 and g (0) [ 1, 1]. This phenomenon, namely lac of uniqueness of prospectie limits is not academic in that it arises in utility approximation. The following demonstrates that epi-conergence is a stronger conergence than q-conergence. Proposition 11 ([Rocafellar et al. (1998)] Proposition 7.7 ) For functions {f, f R n R, N} we hae f = e- lim f if and only if both the following hold. 1. For all λ λ we hae 2. For some λ λ we hae lim sup S λ (f ) S λ f. lim inf S λ (f ) S λ f. Remar 12 The two inclusions imply the existence of a sequence λ λ (the one gien in 2) for which lim S λ f = S λ f. Also the theory of epi-conergence ensure that monotonic sequences epi-conerge. sequences also are q-conergent sequences. Thus monotonic As shown in [Martinez-Legaz (1991)] the leel sets of a quasiconex functions that enjoys a symmetric duality are most naturally assumed to be eenly conex. That is, the (conex) leel sets are intersections of open half spaces. How can one reconcile limiting processes with quasiconexity? Theorem 13 If we hae a family of extended-real alued, functions {g } N that quasi-conerges to g as then {g } N actually epi-conerges to g (and hence also quasi-conerges to g as well). Proof. See Appendix 1. for proof. Thus a quasiconex function constructed ia this ind of limiting process can only be unique up to lower closure. In particular, for any gien family of quasiconex functions the lower closure of any quasi-limit or, equialently, the epi-limit is a representaties of the larger class of all quasi-limits. 10

11 2.3 A Suitable Class of Utilities In order to determine a necessary set of assumption to impose on X( ) we will discuss the class of solid pseudo-conex\concae functions, see section 2.1. This class attracts our attention because it is, first, reasonably broad in the context of utility theory and, second, admits a self consistent and self contained theory including complete duality between direct and indirect utilities as described in [Eberhard et al. (2007)]. When presented with a utility function u( ) a consumer is deemed to consume a commodity bundle x X(p) at a price p and so gain u (x) utility. The indirect utility is the maximum utility that can be gained at price p within a unit budget, denoted (p), see (4). The indirect utility is quasiconex under mild assumptions and we hae that u can be defined from ia the dual formula (2) (first shown by [Diewert (1974)] but for the most general result see [Martinez-Legaz (1991)]). Indeed for (2) to hold we may assume is non-increasing (i.e. for p 1 p 2 we hae (p 1 ) (p 2 )), eenly quasi conex (i.e., its leel sets may be obtained ia the intersection of open half spaces), and satisfy (p) lim α 1 (αp) for all p R n +\R n ++. In this eent we deduce that u eenly quasi-concae, non-decreasing and satisfies u(x) lim α 1 u (αx) for all x R n +\R n ++. As open and closed conex sets are eenly conex, both upper and lower semicontinuous quasi conex are eenly quasi conex. Clearly the properties aboe placed on hold when is lower semicontinuous on R n + and decreasing on R n +\R n ++ (which is the case for the indirect utility we construct). The graph of the demand function is the set Graph X := {(p, x) u(x) = (p)} (12) When this symmetric duality (2) holds we hae the corresponding optimal solution set P (x) is defined by {p u(x) = (p)}. On comparison with (12) we see that the graph of Graph P corresponds to Graph X 1. That is, p P (x) if and only if x X(p). Remar 14 When the duality formula (2) holds then we must hae u is nondecreasing and is nonincreasing. Indeed when x 1 x 2 we hae { p R n + p, x 1 1 } { p R n + p, x 2 1 } and so u (x 1 ) = inf {(p) p, x 1 1} inf {(p) p, x 2 1} = u (x 2 ). We say u is nonsatiated if within any neighbourhood V of any gien point x 2 there exists x 1 V with u (x 1 ) > u (x 2 ). Proposition 15 Suppose the indirect utility : R n + R is a proper, solid and g-pseudo-conex function that admits the duality formula (2). Then the utility u : R n + R is a proper, solid and u is g-pseudo-conex. In particular we must hae u nonsatiated. Proof. See the Appendix 2 for a proof. We could hae framed the last results with the roles of u and (x and p) interchanged to obtain the following. Proposition 16 Suppose the direct utility u : R n + R is a proper, solid and u is pseudo-conex. Then the indirect utility : R n + R is a proper, solid and pseudo-conex. In particular we must hae nonsatiated. Remar 17 The last two propositions ensure that : R n + R that is a proper, solid and g-pseudoconex function that admits the duality formula (2) is nonsatiated. Proposition 18 Denote the indirect utility by and suppose that it is a proper, solid g- pseudo-conex function that admits the duality formula (2). Then for p dom we hae where the normal cone to the leel set S (p) at p is gien by [ N (p)] {x x, p = 1} = X(p) (13) N (p) := {y p p, y 0 for all p S (p)}. (14) 11

12 Proof. We hae from Proposition 15 that u is nonsatiated. Thus the optimal alue x of (1) satisfies x, p = 1 and if u(x) = (p) we hae from (1) that this is equialent to saying p, x 1 = (p ) (p). As u is nonsatiated we can also claim that when x R n +, x, p = 1 and p p, x < 0 implies (p ) > (p). This is because p, x < 1 and so x is strictly in budget. Thus it is possible to improe the utility obtained from x at price p (due to nonsatiation). That is there must exist x with p, x < 1 and u (x ) > u(x) and so (p ) > u(x) = (p). Thus we may write X(p) = { x R n + x, p = 1 and p p, x 0 implies (p ) (p) } Alternatiely, using the contrapositie there establishing the identity (13). = { x R n + x, p = 1 and p p, x < 0 implies (p ) > (p) } x X(p) iff x, p = 1 and p S (p) = p p, x < 0 iff x, p = 1 and p S (p) = p p, x 0 (15) 2.4 Minimal cccc and Maximal Pseudo-Monotonicity As we hae seen in the last section, normal cone operators are central to the description of the demand relation, e.g., Proposition 18. Indeed GARP is equialent to the cyclical pseudo-monotonicity of N (see [Eberhard et al. (2007)]). Clearly Ñ(x) as defined in (11) has closed conex conical images. When is solid, g-pseudo-conex then we also hae p S (p) = S (p) and hence N (p) = Ñ(p). When, in addition, is lower semicontinuous then N is upper semicontinuous (see Theorem 20 below). In the proof of our main results we will need to deal with such operators (in the abstract) prior to establishing they actually coincide with a normal cone operator to a leel set of a solid, g-pseudoconex indirect utility. A general study of a related class of abstract operators has recently been undertaen in [Aussel et al. (2011)]. In [Eberhard et al. (2007)] it is shown that a maximal cyclically pseudo monotone relation (see below for definitions) or a maximal pseudo monotone relation Γ : R n R n has conex, conic images. If x Γ (x) also hae a closed graph then these images must also be closed. To allow the origin to be included we must define pseudo-monotonicity carefully. Definition 19 [Eberhard et al. (2007)] Let Γ : R n R n be a set alued mapping. 1. Γ is pseudo-monotone (PM) on D R n if for all p, q D, the existence of x Γ(p)\ {0} with x, p q > 0 implies y, p q > 0 for all y Γ(q) \ {0}. 2. Γ is cyclically pseudo-monotone (CPM) if for any natural number m and (p i, x i ) with x i Γ(p i ) \ {0} for i = 0,..., m 1, we hae x i, p i+1 p i 0 implies x m, p 0 p m 0, x m Γ (p m ). The relation Γ that we will use later is the conic extension of the demand relation X( ). Recall cone X := λ 0 λx. The following theorem justifies our interest in such operators. Theorem 20 ([Eberhard et al. (2007)]) Suppose : D R is lower semicontinuous, solid and g-pseudo conex on V D. Then p N (p ) is both maximally pseudo monotone on int V and also (maximally) cyclically pseudo monotone with a closed graph. The closed graph property is not immediate for all pseudo-monotone operators but holds for the ones we construct here under the assumptions A1-A4. Lemma 21 Let X : R n R n be a set alued mapping and D be a compact set in R n. If {(p, x) : p D, x X(p)} is compact and contains no pairs of the form (p, 0) then the set mapping Γ : D R n defined by Γ(p) = cone X(p) has a closed graph. 12

13 Proof. Let (x n, p n ) Graph Γ\ {0} with x n D and (x n, p n ) (x, p). If x = 0 then x Γ(p). Now suppose x 0. Then there exists γ n > 0 and x n such that γ n x n X(p n ). As {(p, x) : p D, x X(p)} is compact there exists a conergent subsequence of {γ n } denoted by γ n. The γ n γ 0 and lim (p n, γ n x n ) = (p, x) with x X(p) and p D. By assumption x 0 and so x = γx implies γ 0 and so x 1 γ x cone X(p). Definition 22 Let Γ : R n R n be a set alued mapping with conic images. 1. The effectie domain of Γ, Edom Γ, is the set of p R n such that Γ(p) contains a nonzero ector. 2. We say Γ is orientable in the direction e R n on a set D EdomΓ, or orientable on D for short, if x, e < 0 for all x Γ(p)\{0} and p D. 3. We say Γ is C upper semicontinuous on a set D Edom Γ if for each p D and for each open, cone K such that Γ(p) K {0}, there is an open neighbourhood V (relatie to D) of p such that Γ(p ) K {0} for all p V. 4. We say Γ is ccc if its images (or set alues) are closed, conex and conic. We say Γ is cccc if it is ccc and, in addition, has a closed graph. We say Γ is cccc if it is ccc and, in addition, C upper semicontinuous. 5. We say Γ is maximal (cyclically) pseudo-monotone on a set D Edom Γ if it is (cyclically) pseudo-monotone and for any (cyclically) pseudo-monotone operator G : D R n for which D Edom G and Γ(p) cone G(p) for all p D then cone Γ(p) = cone G(p) for all p D. Lemma 23 Suppose D is a nonempty compact set in R n and Γ : D R n is cccc and orientable in the direction e. Denote p by (y, t) under a rotation of coordinates setting e/ e as the nth canonical basis ector and D # as the set of points corresponding to D in the new coordinate system. Then { z } Γ (y, t) := τ (z, τ) Γ(p) (16) is upper semicontinuous and uniformly bounded in diameter on the domain dom Γ D # (in fact, under the coordinate transformation, Edom Γ becomes dom Γ ). In particular Edom Γ is a relatiely closed subset of D and Γ is cccc. Proof. See Appendix 1 for proof. Now it is clear that when a relation Γ is simultaneously maximally pseudo monotone and cyclically pseudo monotone then its must also be maximally cyclically pseudo monotone. In [Eberhard et al. (2007)] it is shown that the normal cones to the leel sets of solid, g-pseudo-conex functions is indeed simultaneously maximally pseudo monotone and cyclically pseudo monotone, with a closed graph. We show that such multi-functions are closely related to the minimality of the following class of mappings. Theorem 24 Suppose Γ : R n R n is (cyclically) pseudo-monotone and cccc. If = D int(edom Γ) then Γ is simultaneously maximally (cyclically) pseudo monotone on D and the minimal cccc relation whose effectie domain contains D. Proof. See Appendix 3 for proof. A cusco is an upper semicontinuous conex, compact alued relation which hae been widely studied, see [Borwein et al. (1997)]. Corollary 25 Suppose Γ : R n R n is pseudo-monotone, cccc and orientable in the direction e. Suppose further that D int(edom Γ) is nonempty. Let D # and Γ be as in Lemma 23. Then Γ is minimal in the class of cusco s whose domains contain D #. 13

14 Proof. Lemma 23 says that Γ is cccc, hence Theorem 24 says it is minimal in the class of cccc relations whose effectie domains contain D #. Lemma 23 also says that Γ is a cusco whose domain contains D #. Let ϕ be a cusco on D # with = ϕ(y, t) Γ (y, t) for each (y, t) D #. Then the conical lifting of ϕ gien by Φ(y, t) := {(tz, t) : z ϕ(y, t)} is cccc such that Φ(y, t) Γ(y, t) for each (y, t) D #, and Edom Φ = D #. Thus Φ is pseudo-monotone being a subset of such an operator. Hence by Theorem 24 Φ is maximal pseudo-monotone and thus coincides with Γ on D. This implies in turn that ϕ coincides with Γ on D # giing the promised minimality of Γ. Theorem 26 Suppose Γ : R n R n is pseudo-monotone, cccc and orientable in the direction e. Suppose further that = D int(edom Γ). Let D # and Γ be as in Lemma 23. For any countably dense subset {p i } of D, its coordinate transformation to {(y i, t i )} is such that by taing any (z i, τ i ) Γ(y i, t i ) we recoer Γ ia { } Γ (y, t) = lim co zi : (y i, t i ) B δ (y, t) for all (y, t) D #. 0<δ 0 τ i Proof. Corollary 25 shows that Γ is a minimal usco on D. Hence using [Borwein et al. (1997), Proposition 1.4] and the minimality of Γ the result follows. 3 Leel Cures of g-pseudo-conex Functions Recall that a solid, g-pseudo conex function f has conex leel sets whose closures coincide with the closures of its strict leel sets, and that hae interior when the leel is aboe the minimum alue of f. This section describes a coordinate transformation from such a function f to a family of conex leel cure functions g(, λ) on R n 1, indexed by the leels λ R, and bac again. This a ind of duality between g-pseudo-conex functions and (families) of lower dimensional conex functions. We start with an approximate indirect utility f m, a rationalisation of the demand sample {(p i, x i )} m i=1 that is a normalised so that f m (te) = t for fixed e int D. This rationalisation is quasiconex but not necessarily conex. Section 3.1 conerts f m to the family conex functions g m (, λ) using the approach of [Borde et al. (1990)]. We also relate normal cones of leel sets of f m to the subdifferentials of g m (, λ). Recall that the normalisations f m are generally only g-pseudo-conex een though they are deried from conex rationalisations m. Section 3.2 gies conditions under which a family of conex functions g(, λ) can be used to derie a solid, g-pseudo-conex function f by defining its leel sets. This explains how to recoer f m from the family g m (, λ). More importantly, later, in Section 4 when we tae epilimits as m of the latter to gie a family of conex functions g(, λ), we will need this duality to derie a g-pseudo-conex function f that will be our indirect utility for the demand relation X. 3.1 A Coordinate Transformation In the following analysis we assume we hae at hand a sequence of approximate indirect utilities {f m } with f m (tp) t, that rationalise each of the finite data sets that these functions are built on (we do not care what construction is used). The constructions used in [Afriat (1967)] and [Crouzeix et al. (2011)] produce approximating utilities that are ultimately defined on any bounded region D inside R n ++. Following [Borde et al. (1990)] we will see that it is useful to mae the following change of basis of the local coordinate system around e int D R n ++, so that p (y, t) where y R n 1 and t is the scalar that gies projection of p on {te/ e t R}. Throughout this and later sections, we will abuse notation by writing f(p) as f(y, t) where (y, t) is the point p D in the new coordinates. Now a neighbourhood of e may be taen to hae the form D # = Y T in the new coordinate system, where Y and T are closed conex neighbourhoods in R n 1 and R respectiely and the resultant function we will denote by t f m (y, t) is decreasing. 14

15 p 1 Leel Cures of f t Direction of decreasing f 7 y p 2 Set λ = f m (e) and for simplicity of notation, translate e to the origin in Y. Let λ 0 = inf{f m (y, t) (y, t) Y T }. For λ > λ 0 define g m (y, λ) = inf{t f m (y, t) λ}, λ (λ 0, + ). (17) Although we won t analyse conergence of {f m }, or actually of {g m }, until section 4, we note a result to decouple the domain and range of f m from the index m. Lemma 27 Posit the axioms A1 to A4 of section 1.1. Suppose e int D. We will mae that change of basis of the local coordinate system as described aboe: let e # denote e in the new coordinates, and liewise D # and C # denote the regions D and C, respectiely, in the new basis. Suppose in addition that we hae at hand a sequence of approximate indirect utilities {f m } with f m (te) t that rationalises the data {(x i, p i )} m i=1. Then for m sufficiently large there exist compact regions Y Rn 1, T R such that Y T D #, e # int Y int T and T f m (Y T ) Λ m := {λ λ f m (y, t), (y, t) Y T } (18) In addition for all (y, t) Y T we hae N f m (y, t) = cone y g m (y, λ) for λ = f m (y, t) T. Proof. First note that as {p i } i=1 is dense in int D, D m := int co{(p i )} m i=1 and D m D m+1. Thus for sufficiently large m we hae Y T D m. # By construction dom f D m and so we may use the same sets Y and T for m. Clearly for all λ Λ m (Y, T ) we hae S f (y, t) and as f m (te) t and e/ e corresponds to the coordinate t and choice for T satisfies Λ m (Y, T ) { λ λ = f m (y, t), (y, t) int D m # } { λ λ = f m (0, t), (0, t) int D # m} T. As f m is a quasiconex function the remaining property follow from the analysis of [Borde et al. (1990)]. 3.2 Duality between conex leel sets and lower dimensional conex functions A family of g-pseudo-conex functions can always be fitted to samples that satisfy GARP, in fact we can rationalise finitely many data with a conex indirect utility function ia Afriat s construction. Symmetrically we hae obsered that when we do hae our demand relation generated by a solid, lower semicontinuous, g-pseudo-conex indirect utility then any such finite sample must satisfy GARP due to the fact that such samples are from a cyclically pseudo-monotone normal cone operator (see Theorem 20). As we will generate our utility from leel cures an important tool that we must deelop in 15