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Definable utility in o-minimal structures
Journal of Mathematical Economics 34 Ž2000. 159–172 www.elsevier.comrlocaterjmateco
Definable utility in o-minimal structures Marcel K. Richter a , Kam-Chau Wong
b,)
a
b
Department of Economics, UniÕersity of Minnesota, Minneapolis, MN 55455, USA Department of Economics, Chinese UniÕersity of Hong Kong, Shatin, Hongkong, China
Received 26 June 1997; received in revised form 30 July 1999; accepted 4 August 1999
Abstract We obtain definable utility representations for both continuous and upper semi-continuous definable preferences in the ‘‘tame topology’’ of o-minimal expansions of real closed ordered fields wPillay, A., Steinhorn, C., 1986. Definable sets in ordered structures. I. Transactions of American Mathematical Society 295, 565–592; Knight, K., Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures. II. Transactions of American Mathematical Society 295, 593–605; Van den Dries, L., 1998. Tame Topology and O-minimal Structures. Cambridge Univ. Press, Cambridge; etc.x. Such preferences have significant applications, for example, in establishing local determinacy of competitive equilibrium wBlume, L., Zame, W., 1992. The algebraic geometry of competitive equilibrium. In: Neuefeind, W., Riezman, R.G. ŽEds.., Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin.x, and in modeling bounded rationality wRichter, M.K., Wong, K.-C., 1996. Bounded rationalities and definable economies. Working Paper No. 295, University of Minnesota.x. Our proofs are based on geometric theorems for definable sets, and provide new alternatives to the classical tools of separability wDebreu, G., 1954. Representation of a preference ordering by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. ŽEds.., Decision Processes. Wiley, New York ŽReprinted in Mathematical Economics by Debreu, G., 1983, Cambridge Univ. Press, Cambridge.; Rader, T., 1963. The existence of a utility function to represent preferences. Review of Economic Studies 30, 229–232.x and metric distance wArrow, K., Hahn, F.H., 1971. General Competitive Analysis. Holden-Day, San Francisco.x. The results extend Theorem 1 of wBlume, L., Zame, W., 1992. The algebraic geometry of competitive
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equilibrium. In: Neuefeind, W., Riezman, R.G. ŽEds.., Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin.x in several directions. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: D11 Keywords: Definability; O-minimality; Utility representation
1. Introduction We prove analogues of the Debreu–Rader utility representation theorems for definable preferences in the new framework of o-minimal structures. Our theorems extend an o-minimality result of Blume and Zame Ž1992, Theorem 1.. Since the framework is new, we begin by explaining why definability and o-minimal systems are useful for economic modeling. The standard tools of classical mathematics do not always provide the best framework for developing economic analysis. If we are interested in describing the motivations and behavior of boundedly rational agents, for example, it is not natural to allow arbitrarily complex utility functions, preferences, production functions, and technologies. Although we can describe mathematically an agent who maximizes an arbitrary utility function on a budget set determined by an arbitrary endowment vector and an arbitrary price vector, the required complexity may exceed what a real-world agent could conceive of, or accomplish. Realism may require a simpler framework. One way to model simplicity is with agents who are not able to use arbitrary preferences, technologies, commodity bundles, or prices, but only ones that are definable — in some specified language. 1 Definable economic concepts are especially interesting in o-minimal structures. 2 Roughly, o-minimal systems are ‘‘simple’’ mathematical structures in which the only definable sets of basic elements are finite unions of intervals. That finiteness property leads to a ‘‘tame topology’’ that has very useful properties. O-minimal models have a further interesting property that makes them attractive tools for modeling economies. From each o-minimal model, we can obtain a corresponding model in which not only the preference and technology relations are definable, but so is every element of the whole underlying universe ŽRichter and Wong, 1996.. 1
Since there are many possible languages, there are many specific instances of bounded rationality. O-minimal Žorder-minimal. systems are a relatively new development in mathematics. The subject has grown into a wide-ranging generalization of semialgebraic and subanalytic topology. It can be viewed as an implementation of Grothendieck’s proposal for topologie moderee ´ ´ Žtame topology. ŽGrothendieck, 1984.. Cf. van den Dries Ž1984a; 1998., Knight et al. Ž1986., Pillay and Steinhorn Ž1986., etc. 2
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Another reason for seeking simplicity arises when we require the results of agent planning to be computable — either because computability is one way of representing bounded rationality of agents, or because economists use computers to approximate economic equilibria. In o-minimal structures, requiring definability of the underlying preference and technology relations results in a simplicity that can help in achieving computability. There is yet another way in which o-minimal structures can help us avoid unmotivated assumptions. Many economic models provide useful results only under special mathematical restrictions — Lipschitz or even continuous differentiability assumptions, for example. Rather than imposing such ad hoc mathematical assumptions, it may be more natural to require that the underlying preference or technology relations, and even the magnitudes over which they range, are definable. For example, Blume and Zame Ž1992, Theorems 3 and 4. obtained analogues of Debreu’s local uniqueness result ŽDebreu, 1970., replacing Debreu’s C 1 assumptions on demands by assuming that all preferences were definable relations. 3 Blume and Zame also proved an analogue of a special case of Debreu’s utility theorem, by obtaining continuous definable utility functions to represent definable preferences in o-minimal structures over real number systems ŽBlume and Zame, 1992, Theorem 1.. In this paper, we drop their special assumptions, 4 obtaining a full analogue of Debreu’s theorems. Moreover, we prove an analogue of Rader’s theorem: every upper semicontinuous definable preference in an o-minimal structure has an upper semicontinuous definable utility function. Economically, our theorems have this consequence: In simple Žo-minimal. models, under the usual continuity assumptions it does not matter whether we start with simple Ždefinable. preferences, or with simple Ždefinable. utility functions. Technically, our proofs provide new alternatives to the classical tools of separability ŽDebreu, 1954; Rader, 1963. and metric distance ŽArrow and Hahn, 1971.. Our tools are based on geometric theorems for definable sets. In particular, we use a definable set of representatives from the indifference classes, which can be decomposed into finitely many preference-monotone paths, and gives definable utility functions. We do not require second countability of the consumption set ŽRader, 1963., or even countable density ŽDebreu, 1954.. Nor do we require it to be convex ŽArrow and Hahn, 1971.. 5
3
‘‘Definable’’ is our terminology. From their point of view, the relation belongs to the o-minimal structure. 4 Blume and Zame assumed the consumption set was a closed and convex subset of R n , since their proof used Arrow and Hahn’s metric distance between bundles and preference sets. We are able to drop these assumptions since we use a new method. 5 Cf. Remark 1b below.
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2. Results We begin by reviewing some notation from Knight et al. Ž1986., Pillay and Steinhorn Ž1986., Blume and Zame Ž1992., and Richter and Wong Ž1996.. Standard notions in first order predicate logic can be found in Enderton Ž1972., Chang and Keisler Ž1990., et al. For a survey of o-minimal structures, we recommend van den Dries Ž1998.. Recall Žcf. Enderton, 1972; Chang and Keisler, 1990. that for any structure M s Ž M, . . . ., a set X : M n is parametrically definable if there is a first order formula f Ž x 1 , . . . , x n . g LŽ M . Žthe first order predicate language of M together with names for the elements of M . such that X is the set of all b g M n satisfying f ŽP. in M ; i.e., X s b g M n : M < s f w b x4 . For brevity, we drop mention of parameters, and refer simply to ‘‘definable’’. 6 An o-minimal structure ŽPillay and Steinhorn, 1986; cf. van den Dries, 1984a; Knight et al., 1986. is an ordered structure M s Ž M,- , . . . . in which every definable Žwith parameters. subset of M is a finite union of points in M and intervals Ž a,b . where a,b g M j y`,q `4 . 7 Our main results apply to those M that are expansions of real closed ordered fields, so M s Ž M,- ,q,P,0,1, . . . .. 8 As usual, we endow M with the interval topology, i.e., intervals Ž a,b . form a basis; and M n is given the product topology. Let X : M n be definable. A definable set Y is open in Ž closed in. X if Y s X l Y X for some open Ž closed . set Y X : M n. 9,10 We say X is definably connected Žcf. Knight et al., 1986; Pillay and Steinhorn, 1986. if there do not exist non-empty definable sets Y1 , Y2 both open in X such that X s Y1 j Y2 and Y1 l Y2 sœ 0. A function f : X M is upper semicontinuous if the upper contour set f y1 Žw s,q `.. is closed in X for all s g M. A function f : X M m is definable Žcf. Knight et al., 1986; Pillay and Steinhorn, 1986. if its graph Ž x, f Ž x ..: x g X 4 : M nq m is a definable set. A preference # on X is a reflexive, transitive, and total binary relation on X.11 A preference # on X is upper semicontinuous Ž lower semicontinuous. if the weakly-preferred set y g X: y # x 4 Žweakly-worse set y g X: x # y4. is closed in
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This should not be confused with the stricter notion of definability without parameters, that we use in Richter and Wong Ž1996.. 7 Thus, in many o-minimal structures Že.g., ŽR,-, . . . .. all infinite countable sets in M are undefinable. 8 For a definition and discussion of real closed ordered fields, see Chang and Keisler Ž1990, Section 1.4.. X 9 By o-minimality, it is equivalent that Y also be definable, as shown in Knight et al. Ž1986., Proposition 2.1. 10 Clearly, a definable Y is closed in X if its complement Xr Y is open in X. X X X X X 11 As usual, x % x means ! x # x, and x ; x means x # x and x # x.
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X for all x g X. And # is continuous if it is both upper semicontinuous and lower semicontinuous. A preference # on X is linear if x § y for all distinct x, y g X. When X is an interval, we say a preference # on X is constant Ž increasing . Ž decreasing . if for all x, y g X: x ) y implies x ; y Ž x % y . Ž x $ y .. A preference # on X is definable Žcf. Blume and Zame, 1992; Richter and Wong, 1996. if the set Ž x, y .: x # y4 : M 2 n is definable. 12 We say a preference # on X is represented by a function u: X M if x # xX uŽ x . G uŽ xX . for all x, xX g X. Our main result is the following definable analogue of the Debreu–Rader theorems ŽDebreu, 1954; Rader, 1963..
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Theorem. Let M s Ž M,- ,q,P,0,1, . . . . be an o-minimal expansion of a real closed ordered field. Let X : M n be a definably-connected definable set, and let # be a definable preference on X. (a) If # is upper semicontinuous, then # can be represented by an upper semicontinuous definable function u: X M. (b) If # is continuous, then # can be represented by a continuous definable function u: X M.
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Remark 1. Ža. A predecessor of our theorem is Blume and Zame Ž1992., Theorem 1, which shows that in an arbitrary o-minimal expansion ŽR,- ,q,P,0,1, . . . . of the real number field, a continuous definable preference # on a definable, closed, and convex set X : R n can be represented by a continuous definable utility function u: X R. Our theorem extends it by permitting o-minimal expansions of arbitrary real closed ordered fields. ŽThis extra level of generality is important for the economic applications to bounded rationality mentioned in Section 1 and discussed in Richter and Wong, 1996.. We relax all the technical assumptions on X except definable connectedness, and in Part Ža. we even allow upper semicontinuous preferences. Žb. Our theorem replaces the classical separability assumptions Žcf. Rader, 1963; Debreu, 1954. by o-minimal definability, 13 and obtains the additional definability conclusion on utility representations. Of course, without separability, an M-valued representation need not guarantee an R-valued representation. Žc. The assumption of upper semicontinuity is still indispensable, since the lexicographic preference is definable and the classical arguments Žcf. Debreu, 1959, pp. 72–73. still apply to show that no utility representation exists.
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12 The class of definable preferences is wide, containing most preferences economists use in applications: Cobb–Douglas, lexicographic, piecewise linear, semialgebraic, subanalytic, etc. For implications of, and intuition about, definability, see Blume and Zame Ž1992. and Richter and Wong Ž1996.. 13 Our real closed ordered fields need not be separable, or second countable.
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The theorem is an immediate corollary of the following two propositions. Proposition 1. Consider any M , X, and # where: M s Ž M ,- ,q,P,0,1, . . . . is an o y minimal expansion of a real closed ordered field,
Ž 1.
n
X : M is a definable set, and # is a definable preference on X .
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If # is upper semicontinuous, then # is represented by some definable utility function u: X M. Proposition 2. Let M , X, and # satisfy (1). Let X be definably connected and let # be represented by some definable function U:X M. Then: (a) If # is upper semicontinuous, then # is represented by some upper semicontinuous definable function u:X M. (b) If # is continuous, then # is represented by some continuous definable function u: X M.
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Remark 2. Notice that connectedness is not required in Proposition 1 for obtaining definable representations. 14 It is used only in Proposition 2 for obtaining continuity and semicontinuity for the representations. The rest of this paper is devoted to proving Propositions 1 and 2.
3. Proofs The mathematical background already given in Section 2 is sufficient for proving Proposition 2. For the proof of Proposition 1, we require several basic theorems Že.g., cell-decomposition. in o-minimal structures, which are collected in Theorems A.1–A.3 and Proposition A.1 in Appendix A. Proof of Proposition 2. ŽPart b. If the representation U is not continuous, then we transform it along the lines of ŽBlume and Zame, 1992, proof of Theorem 1, paragraph 3., as follows. By o-minimality, the definable set U Ž X . : M is a finite union of disjoint intervals Ii
Ž 2.
whose left and right boundary points a i and bi are in M j y`, q` 4 . The continuity of # implies that each pair Ž Ii , Iiq1 . corresponds to neither a jump nor a gap; 15 otherwise, by carrying over standard arguments Žcf. Richter, 1980, 14
Since our set X is definable, it is a finite union of connected sets Žcells.. See Section 3 and Theorem A.3 below. 15 As usual, Ž Ii , Iiq1 . is said to be a jump if bi g Ii and a iq1 g Iiq1 ; and Ž Ii , Iiq1 . is said to be a gap if bi f Ii and a iq1 f Iiq1. Cf. Richter Ž1980., pp. 294, 296.
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proof of Theorem 1. to our definable context it is easy to show that continuity of the definable # would separate X into two disjoint definable open sets, contradicting the definable connectedness of X. Each Ž Ii , Iiq1 . corresponds to neither a jump nor a gap, so either Ž bi g Ii and a iq1 f Iiq1 . or Ž bi f Ii and a iq1 g Iiq1 . holds. Therefore, we can shift the disjoint intervals Ii , pasting them together into a single interval I, thereby transforming U into a definable representation u from X onto an interval I. 16 As usual, 17 continuity of u follows easily, since uŽ X . s I and # is continuous. ŽPart a. If the representation U is not upper semicontinuous, we modify it along the above lines, as follows. First, let I1 , a1 , b 1 , . . . , Im , a m , bm be as given in Eq. Ž2.. By easy arguments Že.g., a definable analogue of Rader, 1963., we see that the upper semicontinuity of the definable # ensures that for every c g A, if the pre-image Uy1 Žw c,`.. is not closed, then there is some i with c g Ž bi ,a iq1 x and bi g Ii and a iq1 f Iiq1. Again by shifting, we can paste together all such consecutive intervals Ii and Iiq1; this gives a definable representation U for which Uy1 w c,`. is closed for all c g M, so U is upper semicontinuous. Q.E.D. It remains to prove Proposition 1. We will extensively apply techniques for decomposing a definable set into finitely many cells. The notion of a cell and its dimension are standard Žcf. van den Dries, 1984a; Knight et al., 1986; Pillay and Steinhorn, 1986., defined inductively as follows: If X s a4 , where a g M, then X is a cell, and dimŽ X . s 0. If X is a non-empty interval Ž a,b ., where a,b g M j q`,y `4 , then X is a cell, and dimŽ X . s 1, If Y : M n is a cell and dim Ž Y . s k, then: if f :Y M is definable and continuous, then the set X s Ž y, f Ž y .: y g Y 4 : M nq 1 is a cell, and dimŽ X . s k; if f 1 , f 2 : Y M are definable and continuous, and such that f 1Ž y . - f 2 Ž y . for all y g Y, then the set X s Ž y, s .: y g Y and f 1Ž y . - s - f 2 Ž y .4 is a cell, and dimŽ X . s k q 1. Nothing else is a cell of dimension k or k q 1 on M n.
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Remark 3. Ža. By induction on n, it is easy to show that if a cell X : M n has dimŽ X . s 1, then there exists an interval Ž a,b . and a definable, continuous, and one-to-one function f mapping Ž a,b . onto X. Žb. More generally, for all n, k with k F n, if a cell X : M n has dimŽ X . s k, then there exists a cell Y : M k with dim Ž Y . s k, and there exists a definable, continuous, and one-to-one function f mapping Y onto X. Žc. For every densely ordered o-minimal structure M s Ž M,- , . . . ., every cell in M n is definably connected Žcf. Knight et al., 1986, Proposition 2.4..
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16 E.g., we can define u: X M by uŽ y . sUŽ y . for y gUy1 Ž I0 ., and uŽ y . sUŽ y .yÝijs1Ž a j y bjy 1 . for y gUy1 Ž Ii . and is1, . . . ,m. 17 Cf. Debreu Ž1954., proof of Lemma I.
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We will also extensively use the notion of a preference-jump point, which we define as follows. For any X, any preference # on X, and any K : X, we say x is a Ž# , X, K .-jump point if x g K and x has a separating point y g X Ži.e., x % y and x is in the boundary of the weakly-worse set z g K : y # z 4.. Remark 4. Ža. Clearly, if K : Y : X, then a Ž# < Y ,Y, K .-jump point is also a Ž# , X, K .-jump point. Žb. Note that if x g K X : K, and if x is a Ž# , X, K .-jump point then x is not necessarily a Ž# , X, K X .-jump point; but x is a Ž# , X, K X .-jump point when both K and K X are one-dimensional cells. Žc. If there are no Ž# , X, X .-jump points, then # is lower semicontinuous. Žd. The set of jump points clearly is definable. Proof of Proposition 1. Let M , M and # satisfy Ž1.. First, by the Definable Skolem Function Theorem ŽTheorem A.2., there exists a definable set Y : X of representatives; i.e., Y : X and for all x g X, there exists exactly one y g Y with x ; y. Then # is linear on Y. Next, by the Cell Decomposition Theorem ŽTheorem A.3., we can decompose the definable set Y into disjoint cells C1 , PPP ,Cn . Since # < Y is definable, linear, and upper semicontinuous, Lemma 2 below ensures that dimŽ Ci . - 2 for all i. Then Lemma 1 below ensures that each Ci has at most only finitely many Ž# < Y ,Y,Ci .-jump points. By separating out these jump points, we can modify the sets Ci if necessary so that for each Ci : either or
Ž a. Ž b.
dim Ž Ci . s 0 Ž i.e., Ci is a singleton. Ž 3. dim Ž Ci . s 1 and Ci has no Ž # < Y ,Y ,Ci . y jump points.
Then we order the cells Ci as follows. For any distinct cells Ci , C j , we define Ci % C j if there exists x g Ci and y g C j such that x % y. Since # is linear on Y s jCi , all pairs of cells are related by % . We now show that Ci % C j
´ !C % C . j
Ž 4.
i
To prove Eq. Ž4., by linearity of # < Y , it suffices to show that it is impossible to have some x, z g Ci and y g C j with x % y % z. We will prove that impossibility by contradiction. Suppose such x, y, z exist. Then Ci is not a singleton, so Ci satisfies Ž3b.. By applying Remark 3a, we can obtain a continuous definable function g from the interval w0,1x into the one-dimensional cell Ci such that g Ž0. s z and g Ž1. s x. Notice that for all t g w0,1x we have g Ž t . / y / Ci , so g Ž t . § y Žby linearity of # < Y .. The set t g w0,1x: g Ž t . % y4 is definable, so it is a finite union of intervals, and therefore has an infimum t g w0,1x. By continuity of g and upper semicontinuity of # , we have g Ž t . # y, so g Ž t . % y and t ) 0. For all t g w0,1x with t - t, we have y % g Ž t .; but as t t, we have g Ž t . g Ž t . % y % g Ž t ., so g Ž t . is a Ž# < Y ,Y,Ci .-jump point, contradicting Ž3b..
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Now by Eq. Ž4., to obtain a definable utility representation U on the finite union Y of disjoint Ci , it suffices to obtain a definable utility representation for each # < C i . This is trivial for singleton Ci , so we consider Ž3b.. By Remark 3a, for each one-dimensional cell Ci , we can pick an interval Ž a i ,bi ., a continuous definable bijection f i :Ž a i ,bi . Ci . By Ž3b., Ci has no Ž# < C i , Ci , Ci .-jump points, so the upper semicontinuous preference # < C i is (continuous on Ci . Let the ( ( definable preference # on Ž a i ,bi . be defined by x # y f i Ž x . # f i Ž y .. Then # ( is continuous and linear; so by Lemma 3 # is either increasing or decreasing. Consequently each # < C i can be represented either by the definable function f iy1 Ž x . or the function yf iy1 Ž x .. So we can find a definable representation U:Y M of # < Y . Finally, it is easy to extend U to all of X; e.g., for each x g X define uŽ x . s UŽ y ., where y is the unique y g Y with x ; y. Q.E.D.
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Lemma 1. Let M s Ž M,- ,q,P,0,1, . . . ,. be an o-minimal expansion of a real closed ordered field. Suppose Y : M n is a definable set, and # is a linear, upper semicontinuous, and definable preference on Y. Then for eÕery cell C : Y with dim(C) - 2, there exist at most finitely many (# ,Y,C)-jump points. Proof. Suppose not. Then there is a one-dimensional cell C : Y containing infinitely many Ž# ,Y,C .-jump points. Since the set of Ž# ,Y,C .-jump points is definable Žsee Remark 4d., by the Cell Decomposition Theorem ŽTheorem A.3. it can be decomposed into finitely many cells. There exists an infinite, hence, one-dimensional cell D : C. By Remark 4b, all elements of D are also Ž# ,Y, D .jump points. Since D is one-dimensional, we can identify it with an interval Ž a, b .. Taking a subinterval if necessary, by the Monotone Preference Proposition ŽProposition A.1. we can assume # is either increasing or decreasing on the interval D s Ž a,b .. Since the argument is similar for either alternative, we assume # is increasing. By the Definable Skolem Function Theorem ŽTheorem A.2., we can pick a definable function f : D Y that assigns to each w g D a separating point f Ž w . g Y, so by increasingness of # < D we have: w % f Ž w . % Õ for all Õ with w ) Õ g D. Picking a subinterval if necessary, by the Cell Decomposition Theorem again we can assume f is continuous. Now, for any z g D, we have z % x for all x g D with x ) z; but as x z, we have: f Ž x . f Ž z . % z % f Ž x ., contradicting the upper semicontinuity of # . Q.E.D.
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Lemma 2. Let M s Ž M,- ,q,P,0,1 . . . . be an o-minimal expansion of a real closed ordered field. Let Y : M n be a definable set, and let # be a linear, upper semicontinuous, and definable preference on Y. Then each cell C : Y has dimŽ C . - 2. Proof. Suppose not. Then Y contains a two-dimensional cell CX : Y; so to derive a contradiction, we can assume Y itself is a two-dimensional cell. By Remark 3b, Y s f Ž X . for some two-dimensional cell X : M 2 and some continuous and
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definable bijection f : X Y. Then f and # together clearly induce an upper semicontinuous, linear, and definable preference on X : M 2 . Therefore, we can pretend that the two dimensional cell Y : M 2 ; and from that we will derive a contradiction. Let X 1 be the Ždefinable. set of Ž a,b . g Y such that there exists some open interval I where I 2 a, and I = b4 ; Y, and I = b4 has no Ž# ,Y, I = b4.-jump points. Similarly, let X 2 be the Ždefinable. set of Ž a,b . g Y such that there exists some open interval I X where I X 2 b, and a4 = I X ; Y, and 18 a4 = I X has no Ž # ,Y , a4 = I X . y jump points, and # < a4=I X is constant, Ž 5. increasing, or decreasing on I X . By the Cell Decomposition Theorem ŽTheorem A.3., there exists a finite decomposition P of cells that partitions X 1 and X 2 . We will show that: each two y dimensional Ž hence open . cell C˜ g P Ž 6. has no Ž # ,Y ,C˜ . y jump points. Before that, we derive a contradiction from Eq. Ž6.. First, since Y is two-dimensional, there exists a two-dimensional C˜ g P, and by Eq. Ž6. the upper semicontinuous # < C˜ is continuous. We can pick any x, y, z g C˜ with x % y % z, and pick any continuous definable function g:w0,1x C˜ avoiding y Ži.e., y f g Žw0,1x. such that g Ž0. s z and g Ž1. s x. As in the above proof of Eq. Ž4., we define t s inf t g w0,1x: g Ž t . % y4 , so t ) 0, and y % g Ž t . for all t g w0,1x with t - t; but as t t, we have: g Ž t . g Ž t . % y % g Ž t ., contradicting the continuity of # < C˜. To prove Eq. Ž6., we will follow the lines of Knight et al. Ž1986, pp. 602–603, proof of Proposition 5.3.. We consider any two-dimensional cell C˜ g P. Since C˜ : Y : M 2 has dimŽ C˜ . s 2, we can choose an open interval I˜: M, and continuous definable functions f 1 , f 2 : I˜ M with f 1 - f 2 and C˜ s Ž a,b .:a g I˜ and f 1Ž a. - b - f 2 Ž a.4 . We claim:
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Ži.
C˜ : X 1 ,
Ž ii .
˜ the interval I X s Ž f 1Ž a . , f 2 Ž a . . satisfies Ž5. . C˜ : X 2 , and for every a g I,
Ž 7. ˜ To prove Eq. Ž6. from Eq. Ž7., we consider any point Ž a,b . g C. To show that Ž a,b . is not a Ž# ,Y,C˜ .-jump point, we assume z g Y with Ž a,b . % z. To see that z cannot be a separating point for the point Ž a,b ., it suffices to find an open box B : C˜ with Ž a,b . g B and B % z. 19 First, by Ž7ii. we can pick an interval X 18 Of course, in Eq. Ž5. by # < a4= I X being constant Žincreasing, decreasing. on I we mean that the ˜ defined by b# ˜ bX Ž a,b . # Ž a,bX . is constant Žincreasing, decreasing. on I X . induced preference # 19 For any set K, we write K % z to mean x % z for all x g K.
m
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w b 1 ,b 2 x : Ž f 1Ž a., f 2 Ž a.. with b 1 - b - b 2 and a4 = w b 1 ,b 2 x % z. we can pick open intervals I1 , I2 2 a such that
˜ and I1 = b1 4 , I2 = b 2 4 % z. I1 = b 1 4 , I2 = b 2 4 ; C,
20
169
Then by Ž7i.,
Ž 8.
Now we can pick any open box B s J = J X : C˜ for open intervals J, J X such a g J : I1 l I2 and b g J X : w b 1 ,b 2 x. Then for any Ž aX ,bX . g B s J = J X , by Eq. Ž8. we have Ž aX ,b 1 ., Ž aX ,b 2 . % z, so by Ž7ii. we have Ž aX ,bX . % z Žsince b 1 - bX - b 2 .. ˜ Then there It remains to prove Eq. Ž7.. To see Ž7i., we consider any Ž a,b . g C. ˜ By Lemma 1, we can pick an open exists an open interval I with I = b4 ; C. interval Iˆ: I so that Iˆ= b4 has no Ž# ,Y, Iˆ.-jump points. Then for all aX g I,ˆ we have Ž aX ,b . g X 1. Recall that P partitions X 1 and C˜ g P, so we have C˜ : X 1 , i.e., Ž7i. holds. Similarly, to prove Ž7ii., it follows from Proposition A.1 and Lemma 1 that C˜ : X 2 . Now let a g I.˜ Again by Lemma 1 and Proposition A.1, there is an m such that there exist b 0 , PPP ,bm g M with f 1Ž a. s b 0 - b 1 - PPP - bm s f 2 Ž a. and all the intervals IiX s Ž bi ,biq1 . satisfy property Ž5. with I X s IiX . If the least of such m is 1, then Ž7ii. holds and we are done. Otherwise, on every open interval I 2 b 1 , the ordering # < aX 4=I X cannot be constant, increasing, or decreasing, so Ž5. fails at the point Ž a,b 1 ., contradicting that Ž a,b 1 . g C˜ : X 2 . This proves Ž7ii., completing the proof of Lemma 2. Q.E.D.
Lemma 3. Let M s Ž M,- , . . . . be a densely ordered o-minimal structure. Let I s Ž a,b . be an interÕal in M, and let # be a definable preference on Ž a,b .. If # is linear and continuous, then # is either increasing or decreasing. Proof. Suppose # is neither increasing nor decreasing. Then by linearity, it follows that there exist elements w - u F Õ - z g I such that either w % u and z % Õ, or u % w and Õ % z. Since the argument is similar in either alternative, we assume u % w and Õ % z. By linearity of # , either w % z or z % w. Since the argument is similar in either alternative, we assume w % z. Then we can pick an x g Ž w, z . with x # u and x # Õ Že.g., pick the appropriate x g u,Õ4., so x % w % z. By continuity, the nonempty definable sets U s x g w x, z x: x # w4 and W s x g w x, z x:w # x 4 are closed; since the interval w x, z x is definably connected, there exist some xX g U l W, so xX / w exists Žsince xX g w x, z x 2 u w . and xX ; w, contradicting the linearity of # . Q.E.D.
20
Ch K oosing such an interval is possible because otherwise, for every b1 , b 2 with b1 - b- b 2 , X X there would be a b g w b1 ,b 2 x such that Ž a,b . $ z; but this would imply that Ž a,b . was in the boundary X X X X X of the set Ž a,b . g a4= I : z % Ž a,b .4, hence, Ž a,b . g a4= I would be a Ž #,Y, a4= I .-jump point, X X where I s Ž f 1Ž a., f 2 Ž a..; so I would not satisfy Ž5. Žwith as a., contradicting Ž7ii..
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Acknowledgements We wish to thank Professors L. van den Dries, G. Fuhrken, K. Prikry, W. Richter, and C. Steinhorn for helpful conversations, and to thank a referee for helpful suggestions. We also thank the Department of Economics, University of Minnesota for the hospitality and support provided to K.-C. during the winter and spring of 1996.
Appendix A. Basic tools for o-minimal structures Here we discuss four tools for o-minimal structures M s Ž M,- , . . . .. Theorem A.1 and Proposition A.1 apply to all o-minimal structures, Theorem A.2 applies to o-minimal M where the order - is dense, 21 and Theorem A.2 applies to o-minimal expansions of real closed ordered fields. Every definable unary function is piecewise constant or strictly monotone: Theorem A.1 (Monotone Function Restriction (Pillay and Steinhorn, 1986)). Let M s Ž M,- , . . . . be an o-minimal structure. Then for any interÕal Ž a,b . of M, and any definable function f : X M, there are finitely many points a s a 0 - a1 PPP - a n s b such that on each interÕal Ž a i ,a iq1 ., the function f is either constant Ži.e., f Ž x . s f Ž xX . for all x,x X g Ž a i ,a iq1 .. or an isomorphism (i.e., f < Ž a i , a iq 1 . is either an order-preserÕing or order-reÕersing mapping from Ž a i ,a iq1 . onto some interÕal Ž c,d . in M). 22
™
Theorem A.1 extends to definable preferences: Proposition A.1 (Monotone Preference Restriction). Let M s Ž M,- , . . . . be an o-minimal structure. Then for any interÕal Ž a,b . of M, and any definable preference # on Ž a,b ., there are finitely many points a s a0 - a1 - PPP - a n s b such that on each interÕal (a i ,a iq1 ., the preference # is constant, increasing, or decreasing. There exist definable Skolem functions for definable sets: Theorem A.2 (Definable Skolem Functions (van den Dries, 1984b)). Let M s Ž M,- ,q,P,0,1, . . . . be an o-minimal expansion of a real closed ordered 21
I.e. for any x, z g M with x - z, there is a y g M with x - y - z. Of course the order of an ordered field is dense. 22 Notice that when f < Ž a i, a iq 1 . is an isomorphism, then it and its Ždefinable. inverse is also continuous.
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field. Then for any definable set X : M nq m , there exists a definable Skolem function f : M n M m Ži.e., for all x g M n , if there is a y g M m with Ž x,y . g X, then Ž x,f Ž x .. g X ..
™
The next theorem decomposes the domain of any definable function into finitely many cells C1, PPP ,Cn , on each of which the function is continuous. n Moreover, given any finite family of definable sets Yj , the decomposition Ci 4is1 can be chosen so that it partitions each Yj . Theorem A.3 (Cell Decomposition (Knight et al., 1986)). Let M s Ž M,- , . . . . be an o-minimal structure, where - is a dense order. Then for any definable set X : M n , and any definable function f : X M m , there exists a finite family of disjoint cells C1 , PPP ,Cn : M n such that X s Ui ns 1 Ci , and f < C i is continuous for all i. MoreoÕer, giÕen any finite family Yj 4 of definable sets, the decomposition Ci 4 can be chosen so that it partitions each Yj (i.e., Ci : Yj or Ci l Yj sœ 0 for all Ci ..
™
Theorem A.1 is proved in Pillay and Steinhorn Ž1986, Theorem 4.2.; and Proposition A.1 follows from easy modifications of those arguments. 23 Theorem A.2 is proved in van den Dries Ž1984b.; see also Marker Ž1996, Proposition 2.14.. Theorem A.3 is a restatement of facts Ž3.5. and Ž3.6. in Knight et al. Ž1986, p. 598..
References Arrow, K., Hahn, F.H., 1971. General Competitive Analysis. Holden-Day, San Francisco. Blume, L., Zame, W., 1992. The algebraic geometry of competitive equilibrium. In: Neuefeind, W., Riezman, R.G. ŽEds.., Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin. Chang, C.C., Keisler, H.J., 1990. Model Theory, 3rd edn. North-Holland, Amsterdam. Debreu, G., 1954. Representation of a preference ordering by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. ŽEds.., Decision Processes. Wiley, New York ŽReprinted in Mathematical Economics by Debreu, G., 1983, Cambridge Univ. Press, Cambridge.. Debreu, G., 1959. Theory of Value, 1st edn. Wiley, New York. Debreu, G., 1970. Economies with a finite set of equilibria. Econometrica 38, 387–392. van den Dries, L., 1984a. Remarks on Tarski’s problem concerning ŽR, q, P, exp.. In: Lolli, Longo, Marcja ŽEds.., Logic Colloquium 82. North-Holland, Amsterdam, pp. 97–121.
23
Theorem A.1 gives the monotone restriction property asserted in Proposition A.1 for the special case where a definable preference # is generated by some defined preference f, i.e., x # y f Ž x .G f Ž y .. To extend it to the general case of any definable preference, we can use easy modifications of the Pillay–Steinhorn proof of Theorem A.1, e.g., replacing inequality ‘‘ f Ž x ) f Ž y .’’ by ‘‘ x % y,’’ replacing ‘‘Ž' t .w f Ž s . - t x’’ Žsee the fourth display in Pillay and Steinhorn, 1986, p. 581. by ‘‘Ž' t .w s% t x’’.
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van den Dries, L., 1984b. Algebraic theories with definable skolem functions. Journal of Symbolic Logic 49, 625–629. van den Dries, L., 1998. Tame Topology and O-minimal Structures. Cambridge Univ. Press, Cambridge. Enderton, H.B., 1972. A Mathematical Introduction to Logic. Academic Press, New York. Grothendieck, A., 1984. Esquisse d’un Programme, unpublished. ŽEnglish translation: Schneps, L., Lochak, P. ŽEds.., 1997. Geometric Galois Actions, Vol. 1. Cambridge Univ. Press, Cambridge, pp. 243–283.. Knight, K., Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures: II. Transactions of American Mathematical Society 295, 593–605. Marker, D., 1996. Introduction to the model theory of fields. In: Marker, D., Messmer, M., Pillay, A. ŽEds.., Model Theory of Fields. Springer, Berlin. Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures: I. Transactions of American Mathematical Society 295, 565–592. Rader, T., 1963. The existence of a utility function to represent preferences. Review of Economic Studies 30, 229–232. Richter, M.K., 1980. Continuous and semi-continuous utility. International Economic Review 21, 293–299. Richter, M.K., Wong, K.-C., 1996. Bounded rationalities and definable economies. Working Paper No. 295, University of Minnesota.
Definable utility in o-minimal structures Marcel K. Richter a , Kam-Chau Wong
b,)
a
b
Department of Economics, UniÕersity of Minnesota, Minneapolis, MN 55455, USA Department of Economics, Chinese UniÕersity of Hong Kong, Shatin, Hongkong, China
Received 26 June 1997; received in revised form 30 July 1999; accepted 4 August 1999
Abstract We obtain definable utility representations for both continuous and upper semi-continuous definable preferences in the ‘‘tame topology’’ of o-minimal expansions of real closed ordered fields wPillay, A., Steinhorn, C., 1986. Definable sets in ordered structures. I. Transactions of American Mathematical Society 295, 565–592; Knight, K., Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures. II. Transactions of American Mathematical Society 295, 593–605; Van den Dries, L., 1998. Tame Topology and O-minimal Structures. Cambridge Univ. Press, Cambridge; etc.x. Such preferences have significant applications, for example, in establishing local determinacy of competitive equilibrium wBlume, L., Zame, W., 1992. The algebraic geometry of competitive equilibrium. In: Neuefeind, W., Riezman, R.G. ŽEds.., Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin.x, and in modeling bounded rationality wRichter, M.K., Wong, K.-C., 1996. Bounded rationalities and definable economies. Working Paper No. 295, University of Minnesota.x. Our proofs are based on geometric theorems for definable sets, and provide new alternatives to the classical tools of separability wDebreu, G., 1954. Representation of a preference ordering by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. ŽEds.., Decision Processes. Wiley, New York ŽReprinted in Mathematical Economics by Debreu, G., 1983, Cambridge Univ. Press, Cambridge.; Rader, T., 1963. The existence of a utility function to represent preferences. Review of Economic Studies 30, 229–232.x and metric distance wArrow, K., Hahn, F.H., 1971. General Competitive Analysis. Holden-Day, San Francisco.x. The results extend Theorem 1 of wBlume, L., Zame, W., 1992. The algebraic geometry of competitive
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0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 3 5 - X
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equilibrium. In: Neuefeind, W., Riezman, R.G. ŽEds.., Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin.x in several directions. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: D11 Keywords: Definability; O-minimality; Utility representation
1. Introduction We prove analogues of the Debreu–Rader utility representation theorems for definable preferences in the new framework of o-minimal structures. Our theorems extend an o-minimality result of Blume and Zame Ž1992, Theorem 1.. Since the framework is new, we begin by explaining why definability and o-minimal systems are useful for economic modeling. The standard tools of classical mathematics do not always provide the best framework for developing economic analysis. If we are interested in describing the motivations and behavior of boundedly rational agents, for example, it is not natural to allow arbitrarily complex utility functions, preferences, production functions, and technologies. Although we can describe mathematically an agent who maximizes an arbitrary utility function on a budget set determined by an arbitrary endowment vector and an arbitrary price vector, the required complexity may exceed what a real-world agent could conceive of, or accomplish. Realism may require a simpler framework. One way to model simplicity is with agents who are not able to use arbitrary preferences, technologies, commodity bundles, or prices, but only ones that are definable — in some specified language. 1 Definable economic concepts are especially interesting in o-minimal structures. 2 Roughly, o-minimal systems are ‘‘simple’’ mathematical structures in which the only definable sets of basic elements are finite unions of intervals. That finiteness property leads to a ‘‘tame topology’’ that has very useful properties. O-minimal models have a further interesting property that makes them attractive tools for modeling economies. From each o-minimal model, we can obtain a corresponding model in which not only the preference and technology relations are definable, but so is every element of the whole underlying universe ŽRichter and Wong, 1996.. 1
Since there are many possible languages, there are many specific instances of bounded rationality. O-minimal Žorder-minimal. systems are a relatively new development in mathematics. The subject has grown into a wide-ranging generalization of semialgebraic and subanalytic topology. It can be viewed as an implementation of Grothendieck’s proposal for topologie moderee ´ ´ Žtame topology. ŽGrothendieck, 1984.. Cf. van den Dries Ž1984a; 1998., Knight et al. Ž1986., Pillay and Steinhorn Ž1986., etc. 2
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Another reason for seeking simplicity arises when we require the results of agent planning to be computable — either because computability is one way of representing bounded rationality of agents, or because economists use computers to approximate economic equilibria. In o-minimal structures, requiring definability of the underlying preference and technology relations results in a simplicity that can help in achieving computability. There is yet another way in which o-minimal structures can help us avoid unmotivated assumptions. Many economic models provide useful results only under special mathematical restrictions — Lipschitz or even continuous differentiability assumptions, for example. Rather than imposing such ad hoc mathematical assumptions, it may be more natural to require that the underlying preference or technology relations, and even the magnitudes over which they range, are definable. For example, Blume and Zame Ž1992, Theorems 3 and 4. obtained analogues of Debreu’s local uniqueness result ŽDebreu, 1970., replacing Debreu’s C 1 assumptions on demands by assuming that all preferences were definable relations. 3 Blume and Zame also proved an analogue of a special case of Debreu’s utility theorem, by obtaining continuous definable utility functions to represent definable preferences in o-minimal structures over real number systems ŽBlume and Zame, 1992, Theorem 1.. In this paper, we drop their special assumptions, 4 obtaining a full analogue of Debreu’s theorems. Moreover, we prove an analogue of Rader’s theorem: every upper semicontinuous definable preference in an o-minimal structure has an upper semicontinuous definable utility function. Economically, our theorems have this consequence: In simple Žo-minimal. models, under the usual continuity assumptions it does not matter whether we start with simple Ždefinable. preferences, or with simple Ždefinable. utility functions. Technically, our proofs provide new alternatives to the classical tools of separability ŽDebreu, 1954; Rader, 1963. and metric distance ŽArrow and Hahn, 1971.. Our tools are based on geometric theorems for definable sets. In particular, we use a definable set of representatives from the indifference classes, which can be decomposed into finitely many preference-monotone paths, and gives definable utility functions. We do not require second countability of the consumption set ŽRader, 1963., or even countable density ŽDebreu, 1954.. Nor do we require it to be convex ŽArrow and Hahn, 1971.. 5
3
‘‘Definable’’ is our terminology. From their point of view, the relation belongs to the o-minimal structure. 4 Blume and Zame assumed the consumption set was a closed and convex subset of R n , since their proof used Arrow and Hahn’s metric distance between bundles and preference sets. We are able to drop these assumptions since we use a new method. 5 Cf. Remark 1b below.
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2. Results We begin by reviewing some notation from Knight et al. Ž1986., Pillay and Steinhorn Ž1986., Blume and Zame Ž1992., and Richter and Wong Ž1996.. Standard notions in first order predicate logic can be found in Enderton Ž1972., Chang and Keisler Ž1990., et al. For a survey of o-minimal structures, we recommend van den Dries Ž1998.. Recall Žcf. Enderton, 1972; Chang and Keisler, 1990. that for any structure M s Ž M, . . . ., a set X : M n is parametrically definable if there is a first order formula f Ž x 1 , . . . , x n . g LŽ M . Žthe first order predicate language of M together with names for the elements of M . such that X is the set of all b g M n satisfying f ŽP. in M ; i.e., X s b g M n : M < s f w b x4 . For brevity, we drop mention of parameters, and refer simply to ‘‘definable’’. 6 An o-minimal structure ŽPillay and Steinhorn, 1986; cf. van den Dries, 1984a; Knight et al., 1986. is an ordered structure M s Ž M,- , . . . . in which every definable Žwith parameters. subset of M is a finite union of points in M and intervals Ž a,b . where a,b g M j y`,q `4 . 7 Our main results apply to those M that are expansions of real closed ordered fields, so M s Ž M,- ,q,P,0,1, . . . .. 8 As usual, we endow M with the interval topology, i.e., intervals Ž a,b . form a basis; and M n is given the product topology. Let X : M n be definable. A definable set Y is open in Ž closed in. X if Y s X l Y X for some open Ž closed . set Y X : M n. 9,10 We say X is definably connected Žcf. Knight et al., 1986; Pillay and Steinhorn, 1986. if there do not exist non-empty definable sets Y1 , Y2 both open in X such that X s Y1 j Y2 and Y1 l Y2 sœ 0. A function f : X M is upper semicontinuous if the upper contour set f y1 Žw s,q `.. is closed in X for all s g M. A function f : X M m is definable Žcf. Knight et al., 1986; Pillay and Steinhorn, 1986. if its graph Ž x, f Ž x ..: x g X 4 : M nq m is a definable set. A preference # on X is a reflexive, transitive, and total binary relation on X.11 A preference # on X is upper semicontinuous Ž lower semicontinuous. if the weakly-preferred set y g X: y # x 4 Žweakly-worse set y g X: x # y4. is closed in
™ ™
6
This should not be confused with the stricter notion of definability without parameters, that we use in Richter and Wong Ž1996.. 7 Thus, in many o-minimal structures Že.g., ŽR,-, . . . .. all infinite countable sets in M are undefinable. 8 For a definition and discussion of real closed ordered fields, see Chang and Keisler Ž1990, Section 1.4.. X 9 By o-minimality, it is equivalent that Y also be definable, as shown in Knight et al. Ž1986., Proposition 2.1. 10 Clearly, a definable Y is closed in X if its complement Xr Y is open in X. X X X X X 11 As usual, x % x means ! x # x, and x ; x means x # x and x # x.
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X for all x g X. And # is continuous if it is both upper semicontinuous and lower semicontinuous. A preference # on X is linear if x § y for all distinct x, y g X. When X is an interval, we say a preference # on X is constant Ž increasing . Ž decreasing . if for all x, y g X: x ) y implies x ; y Ž x % y . Ž x $ y .. A preference # on X is definable Žcf. Blume and Zame, 1992; Richter and Wong, 1996. if the set Ž x, y .: x # y4 : M 2 n is definable. 12 We say a preference # on X is represented by a function u: X M if x # xX uŽ x . G uŽ xX . for all x, xX g X. Our main result is the following definable analogue of the Debreu–Rader theorems ŽDebreu, 1954; Rader, 1963..
™
m
Theorem. Let M s Ž M,- ,q,P,0,1, . . . . be an o-minimal expansion of a real closed ordered field. Let X : M n be a definably-connected definable set, and let # be a definable preference on X. (a) If # is upper semicontinuous, then # can be represented by an upper semicontinuous definable function u: X M. (b) If # is continuous, then # can be represented by a continuous definable function u: X M.
™
™
Remark 1. Ža. A predecessor of our theorem is Blume and Zame Ž1992., Theorem 1, which shows that in an arbitrary o-minimal expansion ŽR,- ,q,P,0,1, . . . . of the real number field, a continuous definable preference # on a definable, closed, and convex set X : R n can be represented by a continuous definable utility function u: X R. Our theorem extends it by permitting o-minimal expansions of arbitrary real closed ordered fields. ŽThis extra level of generality is important for the economic applications to bounded rationality mentioned in Section 1 and discussed in Richter and Wong, 1996.. We relax all the technical assumptions on X except definable connectedness, and in Part Ža. we even allow upper semicontinuous preferences. Žb. Our theorem replaces the classical separability assumptions Žcf. Rader, 1963; Debreu, 1954. by o-minimal definability, 13 and obtains the additional definability conclusion on utility representations. Of course, without separability, an M-valued representation need not guarantee an R-valued representation. Žc. The assumption of upper semicontinuity is still indispensable, since the lexicographic preference is definable and the classical arguments Žcf. Debreu, 1959, pp. 72–73. still apply to show that no utility representation exists.
™
12 The class of definable preferences is wide, containing most preferences economists use in applications: Cobb–Douglas, lexicographic, piecewise linear, semialgebraic, subanalytic, etc. For implications of, and intuition about, definability, see Blume and Zame Ž1992. and Richter and Wong Ž1996.. 13 Our real closed ordered fields need not be separable, or second countable.
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The theorem is an immediate corollary of the following two propositions. Proposition 1. Consider any M , X, and # where: M s Ž M ,- ,q,P,0,1, . . . . is an o y minimal expansion of a real closed ordered field,
Ž 1.
n
X : M is a definable set, and # is a definable preference on X .
™
If # is upper semicontinuous, then # is represented by some definable utility function u: X M. Proposition 2. Let M , X, and # satisfy (1). Let X be definably connected and let # be represented by some definable function U:X M. Then: (a) If # is upper semicontinuous, then # is represented by some upper semicontinuous definable function u:X M. (b) If # is continuous, then # is represented by some continuous definable function u: X M.
™
™
™
Remark 2. Notice that connectedness is not required in Proposition 1 for obtaining definable representations. 14 It is used only in Proposition 2 for obtaining continuity and semicontinuity for the representations. The rest of this paper is devoted to proving Propositions 1 and 2.
3. Proofs The mathematical background already given in Section 2 is sufficient for proving Proposition 2. For the proof of Proposition 1, we require several basic theorems Že.g., cell-decomposition. in o-minimal structures, which are collected in Theorems A.1–A.3 and Proposition A.1 in Appendix A. Proof of Proposition 2. ŽPart b. If the representation U is not continuous, then we transform it along the lines of ŽBlume and Zame, 1992, proof of Theorem 1, paragraph 3., as follows. By o-minimality, the definable set U Ž X . : M is a finite union of disjoint intervals Ii
Ž 2.
whose left and right boundary points a i and bi are in M j y`, q` 4 . The continuity of # implies that each pair Ž Ii , Iiq1 . corresponds to neither a jump nor a gap; 15 otherwise, by carrying over standard arguments Žcf. Richter, 1980, 14
Since our set X is definable, it is a finite union of connected sets Žcells.. See Section 3 and Theorem A.3 below. 15 As usual, Ž Ii , Iiq1 . is said to be a jump if bi g Ii and a iq1 g Iiq1 ; and Ž Ii , Iiq1 . is said to be a gap if bi f Ii and a iq1 f Iiq1. Cf. Richter Ž1980., pp. 294, 296.
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proof of Theorem 1. to our definable context it is easy to show that continuity of the definable # would separate X into two disjoint definable open sets, contradicting the definable connectedness of X. Each Ž Ii , Iiq1 . corresponds to neither a jump nor a gap, so either Ž bi g Ii and a iq1 f Iiq1 . or Ž bi f Ii and a iq1 g Iiq1 . holds. Therefore, we can shift the disjoint intervals Ii , pasting them together into a single interval I, thereby transforming U into a definable representation u from X onto an interval I. 16 As usual, 17 continuity of u follows easily, since uŽ X . s I and # is continuous. ŽPart a. If the representation U is not upper semicontinuous, we modify it along the above lines, as follows. First, let I1 , a1 , b 1 , . . . , Im , a m , bm be as given in Eq. Ž2.. By easy arguments Že.g., a definable analogue of Rader, 1963., we see that the upper semicontinuity of the definable # ensures that for every c g A, if the pre-image Uy1 Žw c,`.. is not closed, then there is some i with c g Ž bi ,a iq1 x and bi g Ii and a iq1 f Iiq1. Again by shifting, we can paste together all such consecutive intervals Ii and Iiq1; this gives a definable representation U for which Uy1 w c,`. is closed for all c g M, so U is upper semicontinuous. Q.E.D. It remains to prove Proposition 1. We will extensively apply techniques for decomposing a definable set into finitely many cells. The notion of a cell and its dimension are standard Žcf. van den Dries, 1984a; Knight et al., 1986; Pillay and Steinhorn, 1986., defined inductively as follows: If X s a4 , where a g M, then X is a cell, and dimŽ X . s 0. If X is a non-empty interval Ž a,b ., where a,b g M j q`,y `4 , then X is a cell, and dimŽ X . s 1, If Y : M n is a cell and dim Ž Y . s k, then: if f :Y M is definable and continuous, then the set X s Ž y, f Ž y .: y g Y 4 : M nq 1 is a cell, and dimŽ X . s k; if f 1 , f 2 : Y M are definable and continuous, and such that f 1Ž y . - f 2 Ž y . for all y g Y, then the set X s Ž y, s .: y g Y and f 1Ž y . - s - f 2 Ž y .4 is a cell, and dimŽ X . s k q 1. Nothing else is a cell of dimension k or k q 1 on M n.
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Remark 3. Ža. By induction on n, it is easy to show that if a cell X : M n has dimŽ X . s 1, then there exists an interval Ž a,b . and a definable, continuous, and one-to-one function f mapping Ž a,b . onto X. Žb. More generally, for all n, k with k F n, if a cell X : M n has dimŽ X . s k, then there exists a cell Y : M k with dim Ž Y . s k, and there exists a definable, continuous, and one-to-one function f mapping Y onto X. Žc. For every densely ordered o-minimal structure M s Ž M,- , . . . ., every cell in M n is definably connected Žcf. Knight et al., 1986, Proposition 2.4..
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16 E.g., we can define u: X M by uŽ y . sUŽ y . for y gUy1 Ž I0 ., and uŽ y . sUŽ y .yÝijs1Ž a j y bjy 1 . for y gUy1 Ž Ii . and is1, . . . ,m. 17 Cf. Debreu Ž1954., proof of Lemma I.
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We will also extensively use the notion of a preference-jump point, which we define as follows. For any X, any preference # on X, and any K : X, we say x is a Ž# , X, K .-jump point if x g K and x has a separating point y g X Ži.e., x % y and x is in the boundary of the weakly-worse set z g K : y # z 4.. Remark 4. Ža. Clearly, if K : Y : X, then a Ž# < Y ,Y, K .-jump point is also a Ž# , X, K .-jump point. Žb. Note that if x g K X : K, and if x is a Ž# , X, K .-jump point then x is not necessarily a Ž# , X, K X .-jump point; but x is a Ž# , X, K X .-jump point when both K and K X are one-dimensional cells. Žc. If there are no Ž# , X, X .-jump points, then # is lower semicontinuous. Žd. The set of jump points clearly is definable. Proof of Proposition 1. Let M , M and # satisfy Ž1.. First, by the Definable Skolem Function Theorem ŽTheorem A.2., there exists a definable set Y : X of representatives; i.e., Y : X and for all x g X, there exists exactly one y g Y with x ; y. Then # is linear on Y. Next, by the Cell Decomposition Theorem ŽTheorem A.3., we can decompose the definable set Y into disjoint cells C1 , PPP ,Cn . Since # < Y is definable, linear, and upper semicontinuous, Lemma 2 below ensures that dimŽ Ci . - 2 for all i. Then Lemma 1 below ensures that each Ci has at most only finitely many Ž# < Y ,Y,Ci .-jump points. By separating out these jump points, we can modify the sets Ci if necessary so that for each Ci : either or
Ž a. Ž b.
dim Ž Ci . s 0 Ž i.e., Ci is a singleton. Ž 3. dim Ž Ci . s 1 and Ci has no Ž # < Y ,Y ,Ci . y jump points.
Then we order the cells Ci as follows. For any distinct cells Ci , C j , we define Ci % C j if there exists x g Ci and y g C j such that x % y. Since # is linear on Y s jCi , all pairs of cells are related by % . We now show that Ci % C j
´ !C % C . j
Ž 4.
i
To prove Eq. Ž4., by linearity of # < Y , it suffices to show that it is impossible to have some x, z g Ci and y g C j with x % y % z. We will prove that impossibility by contradiction. Suppose such x, y, z exist. Then Ci is not a singleton, so Ci satisfies Ž3b.. By applying Remark 3a, we can obtain a continuous definable function g from the interval w0,1x into the one-dimensional cell Ci such that g Ž0. s z and g Ž1. s x. Notice that for all t g w0,1x we have g Ž t . / y / Ci , so g Ž t . § y Žby linearity of # < Y .. The set t g w0,1x: g Ž t . % y4 is definable, so it is a finite union of intervals, and therefore has an infimum t g w0,1x. By continuity of g and upper semicontinuity of # , we have g Ž t . # y, so g Ž t . % y and t ) 0. For all t g w0,1x with t - t, we have y % g Ž t .; but as t t, we have g Ž t . g Ž t . % y % g Ž t ., so g Ž t . is a Ž# < Y ,Y,Ci .-jump point, contradicting Ž3b..
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Now by Eq. Ž4., to obtain a definable utility representation U on the finite union Y of disjoint Ci , it suffices to obtain a definable utility representation for each # < C i . This is trivial for singleton Ci , so we consider Ž3b.. By Remark 3a, for each one-dimensional cell Ci , we can pick an interval Ž a i ,bi ., a continuous definable bijection f i :Ž a i ,bi . Ci . By Ž3b., Ci has no Ž# < C i , Ci , Ci .-jump points, so the upper semicontinuous preference # < C i is (continuous on Ci . Let the ( ( definable preference # on Ž a i ,bi . be defined by x # y f i Ž x . # f i Ž y .. Then # ( is continuous and linear; so by Lemma 3 # is either increasing or decreasing. Consequently each # < C i can be represented either by the definable function f iy1 Ž x . or the function yf iy1 Ž x .. So we can find a definable representation U:Y M of # < Y . Finally, it is easy to extend U to all of X; e.g., for each x g X define uŽ x . s UŽ y ., where y is the unique y g Y with x ; y. Q.E.D.
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Lemma 1. Let M s Ž M,- ,q,P,0,1, . . . ,. be an o-minimal expansion of a real closed ordered field. Suppose Y : M n is a definable set, and # is a linear, upper semicontinuous, and definable preference on Y. Then for eÕery cell C : Y with dim(C) - 2, there exist at most finitely many (# ,Y,C)-jump points. Proof. Suppose not. Then there is a one-dimensional cell C : Y containing infinitely many Ž# ,Y,C .-jump points. Since the set of Ž# ,Y,C .-jump points is definable Žsee Remark 4d., by the Cell Decomposition Theorem ŽTheorem A.3. it can be decomposed into finitely many cells. There exists an infinite, hence, one-dimensional cell D : C. By Remark 4b, all elements of D are also Ž# ,Y, D .jump points. Since D is one-dimensional, we can identify it with an interval Ž a, b .. Taking a subinterval if necessary, by the Monotone Preference Proposition ŽProposition A.1. we can assume # is either increasing or decreasing on the interval D s Ž a,b .. Since the argument is similar for either alternative, we assume # is increasing. By the Definable Skolem Function Theorem ŽTheorem A.2., we can pick a definable function f : D Y that assigns to each w g D a separating point f Ž w . g Y, so by increasingness of # < D we have: w % f Ž w . % Õ for all Õ with w ) Õ g D. Picking a subinterval if necessary, by the Cell Decomposition Theorem again we can assume f is continuous. Now, for any z g D, we have z % x for all x g D with x ) z; but as x z, we have: f Ž x . f Ž z . % z % f Ž x ., contradicting the upper semicontinuity of # . Q.E.D.
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o
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Lemma 2. Let M s Ž M,- ,q,P,0,1 . . . . be an o-minimal expansion of a real closed ordered field. Let Y : M n be a definable set, and let # be a linear, upper semicontinuous, and definable preference on Y. Then each cell C : Y has dimŽ C . - 2. Proof. Suppose not. Then Y contains a two-dimensional cell CX : Y; so to derive a contradiction, we can assume Y itself is a two-dimensional cell. By Remark 3b, Y s f Ž X . for some two-dimensional cell X : M 2 and some continuous and
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definable bijection f : X Y. Then f and # together clearly induce an upper semicontinuous, linear, and definable preference on X : M 2 . Therefore, we can pretend that the two dimensional cell Y : M 2 ; and from that we will derive a contradiction. Let X 1 be the Ždefinable. set of Ž a,b . g Y such that there exists some open interval I where I 2 a, and I = b4 ; Y, and I = b4 has no Ž# ,Y, I = b4.-jump points. Similarly, let X 2 be the Ždefinable. set of Ž a,b . g Y such that there exists some open interval I X where I X 2 b, and a4 = I X ; Y, and 18 a4 = I X has no Ž # ,Y , a4 = I X . y jump points, and # < a4=I X is constant, Ž 5. increasing, or decreasing on I X . By the Cell Decomposition Theorem ŽTheorem A.3., there exists a finite decomposition P of cells that partitions X 1 and X 2 . We will show that: each two y dimensional Ž hence open . cell C˜ g P Ž 6. has no Ž # ,Y ,C˜ . y jump points. Before that, we derive a contradiction from Eq. Ž6.. First, since Y is two-dimensional, there exists a two-dimensional C˜ g P, and by Eq. Ž6. the upper semicontinuous # < C˜ is continuous. We can pick any x, y, z g C˜ with x % y % z, and pick any continuous definable function g:w0,1x C˜ avoiding y Ži.e., y f g Žw0,1x. such that g Ž0. s z and g Ž1. s x. As in the above proof of Eq. Ž4., we define t s inf t g w0,1x: g Ž t . % y4 , so t ) 0, and y % g Ž t . for all t g w0,1x with t - t; but as t t, we have: g Ž t . g Ž t . % y % g Ž t ., contradicting the continuity of # < C˜. To prove Eq. Ž6., we will follow the lines of Knight et al. Ž1986, pp. 602–603, proof of Proposition 5.3.. We consider any two-dimensional cell C˜ g P. Since C˜ : Y : M 2 has dimŽ C˜ . s 2, we can choose an open interval I˜: M, and continuous definable functions f 1 , f 2 : I˜ M with f 1 - f 2 and C˜ s Ž a,b .:a g I˜ and f 1Ž a. - b - f 2 Ž a.4 . We claim:
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Ži.
C˜ : X 1 ,
Ž ii .
˜ the interval I X s Ž f 1Ž a . , f 2 Ž a . . satisfies Ž5. . C˜ : X 2 , and for every a g I,
Ž 7. ˜ To prove Eq. Ž6. from Eq. Ž7., we consider any point Ž a,b . g C. To show that Ž a,b . is not a Ž# ,Y,C˜ .-jump point, we assume z g Y with Ž a,b . % z. To see that z cannot be a separating point for the point Ž a,b ., it suffices to find an open box B : C˜ with Ž a,b . g B and B % z. 19 First, by Ž7ii. we can pick an interval X 18 Of course, in Eq. Ž5. by # < a4= I X being constant Žincreasing, decreasing. on I we mean that the ˜ defined by b# ˜ bX Ž a,b . # Ž a,bX . is constant Žincreasing, decreasing. on I X . induced preference # 19 For any set K, we write K % z to mean x % z for all x g K.
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w b 1 ,b 2 x : Ž f 1Ž a., f 2 Ž a.. with b 1 - b - b 2 and a4 = w b 1 ,b 2 x % z. we can pick open intervals I1 , I2 2 a such that
˜ and I1 = b1 4 , I2 = b 2 4 % z. I1 = b 1 4 , I2 = b 2 4 ; C,
20
169
Then by Ž7i.,
Ž 8.
Now we can pick any open box B s J = J X : C˜ for open intervals J, J X such a g J : I1 l I2 and b g J X : w b 1 ,b 2 x. Then for any Ž aX ,bX . g B s J = J X , by Eq. Ž8. we have Ž aX ,b 1 ., Ž aX ,b 2 . % z, so by Ž7ii. we have Ž aX ,bX . % z Žsince b 1 - bX - b 2 .. ˜ Then there It remains to prove Eq. Ž7.. To see Ž7i., we consider any Ž a,b . g C. ˜ By Lemma 1, we can pick an open exists an open interval I with I = b4 ; C. interval Iˆ: I so that Iˆ= b4 has no Ž# ,Y, Iˆ.-jump points. Then for all aX g I,ˆ we have Ž aX ,b . g X 1. Recall that P partitions X 1 and C˜ g P, so we have C˜ : X 1 , i.e., Ž7i. holds. Similarly, to prove Ž7ii., it follows from Proposition A.1 and Lemma 1 that C˜ : X 2 . Now let a g I.˜ Again by Lemma 1 and Proposition A.1, there is an m such that there exist b 0 , PPP ,bm g M with f 1Ž a. s b 0 - b 1 - PPP - bm s f 2 Ž a. and all the intervals IiX s Ž bi ,biq1 . satisfy property Ž5. with I X s IiX . If the least of such m is 1, then Ž7ii. holds and we are done. Otherwise, on every open interval I 2 b 1 , the ordering # < aX 4=I X cannot be constant, increasing, or decreasing, so Ž5. fails at the point Ž a,b 1 ., contradicting that Ž a,b 1 . g C˜ : X 2 . This proves Ž7ii., completing the proof of Lemma 2. Q.E.D.
Lemma 3. Let M s Ž M,- , . . . . be a densely ordered o-minimal structure. Let I s Ž a,b . be an interÕal in M, and let # be a definable preference on Ž a,b .. If # is linear and continuous, then # is either increasing or decreasing. Proof. Suppose # is neither increasing nor decreasing. Then by linearity, it follows that there exist elements w - u F Õ - z g I such that either w % u and z % Õ, or u % w and Õ % z. Since the argument is similar in either alternative, we assume u % w and Õ % z. By linearity of # , either w % z or z % w. Since the argument is similar in either alternative, we assume w % z. Then we can pick an x g Ž w, z . with x # u and x # Õ Že.g., pick the appropriate x g u,Õ4., so x % w % z. By continuity, the nonempty definable sets U s x g w x, z x: x # w4 and W s x g w x, z x:w # x 4 are closed; since the interval w x, z x is definably connected, there exist some xX g U l W, so xX / w exists Žsince xX g w x, z x 2 u w . and xX ; w, contradicting the linearity of # . Q.E.D.
20
Ch K oosing such an interval is possible because otherwise, for every b1 , b 2 with b1 - b- b 2 , X X there would be a b g w b1 ,b 2 x such that Ž a,b . $ z; but this would imply that Ž a,b . was in the boundary X X X X X of the set Ž a,b . g a4= I : z % Ž a,b .4, hence, Ž a,b . g a4= I would be a Ž #,Y, a4= I .-jump point, X X where I s Ž f 1Ž a., f 2 Ž a..; so I would not satisfy Ž5. Žwith as a., contradicting Ž7ii..
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Acknowledgements We wish to thank Professors L. van den Dries, G. Fuhrken, K. Prikry, W. Richter, and C. Steinhorn for helpful conversations, and to thank a referee for helpful suggestions. We also thank the Department of Economics, University of Minnesota for the hospitality and support provided to K.-C. during the winter and spring of 1996.
Appendix A. Basic tools for o-minimal structures Here we discuss four tools for o-minimal structures M s Ž M,- , . . . .. Theorem A.1 and Proposition A.1 apply to all o-minimal structures, Theorem A.2 applies to o-minimal M where the order - is dense, 21 and Theorem A.2 applies to o-minimal expansions of real closed ordered fields. Every definable unary function is piecewise constant or strictly monotone: Theorem A.1 (Monotone Function Restriction (Pillay and Steinhorn, 1986)). Let M s Ž M,- , . . . . be an o-minimal structure. Then for any interÕal Ž a,b . of M, and any definable function f : X M, there are finitely many points a s a 0 - a1 PPP - a n s b such that on each interÕal Ž a i ,a iq1 ., the function f is either constant Ži.e., f Ž x . s f Ž xX . for all x,x X g Ž a i ,a iq1 .. or an isomorphism (i.e., f < Ž a i , a iq 1 . is either an order-preserÕing or order-reÕersing mapping from Ž a i ,a iq1 . onto some interÕal Ž c,d . in M). 22
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Theorem A.1 extends to definable preferences: Proposition A.1 (Monotone Preference Restriction). Let M s Ž M,- , . . . . be an o-minimal structure. Then for any interÕal Ž a,b . of M, and any definable preference # on Ž a,b ., there are finitely many points a s a0 - a1 - PPP - a n s b such that on each interÕal (a i ,a iq1 ., the preference # is constant, increasing, or decreasing. There exist definable Skolem functions for definable sets: Theorem A.2 (Definable Skolem Functions (van den Dries, 1984b)). Let M s Ž M,- ,q,P,0,1, . . . . be an o-minimal expansion of a real closed ordered 21
I.e. for any x, z g M with x - z, there is a y g M with x - y - z. Of course the order of an ordered field is dense. 22 Notice that when f < Ž a i, a iq 1 . is an isomorphism, then it and its Ždefinable. inverse is also continuous.
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field. Then for any definable set X : M nq m , there exists a definable Skolem function f : M n M m Ži.e., for all x g M n , if there is a y g M m with Ž x,y . g X, then Ž x,f Ž x .. g X ..
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The next theorem decomposes the domain of any definable function into finitely many cells C1, PPP ,Cn , on each of which the function is continuous. n Moreover, given any finite family of definable sets Yj , the decomposition Ci 4is1 can be chosen so that it partitions each Yj . Theorem A.3 (Cell Decomposition (Knight et al., 1986)). Let M s Ž M,- , . . . . be an o-minimal structure, where - is a dense order. Then for any definable set X : M n , and any definable function f : X M m , there exists a finite family of disjoint cells C1 , PPP ,Cn : M n such that X s Ui ns 1 Ci , and f < C i is continuous for all i. MoreoÕer, giÕen any finite family Yj 4 of definable sets, the decomposition Ci 4 can be chosen so that it partitions each Yj (i.e., Ci : Yj or Ci l Yj sœ 0 for all Ci ..
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Theorem A.1 is proved in Pillay and Steinhorn Ž1986, Theorem 4.2.; and Proposition A.1 follows from easy modifications of those arguments. 23 Theorem A.2 is proved in van den Dries Ž1984b.; see also Marker Ž1996, Proposition 2.14.. Theorem A.3 is a restatement of facts Ž3.5. and Ž3.6. in Knight et al. Ž1986, p. 598..
References Arrow, K., Hahn, F.H., 1971. General Competitive Analysis. Holden-Day, San Francisco. Blume, L., Zame, W., 1992. The algebraic geometry of competitive equilibrium. In: Neuefeind, W., Riezman, R.G. ŽEds.., Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin. Chang, C.C., Keisler, H.J., 1990. Model Theory, 3rd edn. North-Holland, Amsterdam. Debreu, G., 1954. Representation of a preference ordering by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. ŽEds.., Decision Processes. Wiley, New York ŽReprinted in Mathematical Economics by Debreu, G., 1983, Cambridge Univ. Press, Cambridge.. Debreu, G., 1959. Theory of Value, 1st edn. Wiley, New York. Debreu, G., 1970. Economies with a finite set of equilibria. Econometrica 38, 387–392. van den Dries, L., 1984a. Remarks on Tarski’s problem concerning ŽR, q, P, exp.. In: Lolli, Longo, Marcja ŽEds.., Logic Colloquium 82. North-Holland, Amsterdam, pp. 97–121.
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Theorem A.1 gives the monotone restriction property asserted in Proposition A.1 for the special case where a definable preference # is generated by some defined preference f, i.e., x # y f Ž x .G f Ž y .. To extend it to the general case of any definable preference, we can use easy modifications of the Pillay–Steinhorn proof of Theorem A.1, e.g., replacing inequality ‘‘ f Ž x ) f Ž y .’’ by ‘‘ x % y,’’ replacing ‘‘Ž' t .w f Ž s . - t x’’ Žsee the fourth display in Pillay and Steinhorn, 1986, p. 581. by ‘‘Ž' t .w s% t x’’.
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van den Dries, L., 1984b. Algebraic theories with definable skolem functions. Journal of Symbolic Logic 49, 625–629. van den Dries, L., 1998. Tame Topology and O-minimal Structures. Cambridge Univ. Press, Cambridge. Enderton, H.B., 1972. A Mathematical Introduction to Logic. Academic Press, New York. Grothendieck, A., 1984. Esquisse d’un Programme, unpublished. ŽEnglish translation: Schneps, L., Lochak, P. ŽEds.., 1997. Geometric Galois Actions, Vol. 1. Cambridge Univ. Press, Cambridge, pp. 243–283.. Knight, K., Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures: II. Transactions of American Mathematical Society 295, 593–605. Marker, D., 1996. Introduction to the model theory of fields. In: Marker, D., Messmer, M., Pillay, A. ŽEds.., Model Theory of Fields. Springer, Berlin. Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures: I. Transactions of American Mathematical Society 295, 565–592. Rader, T., 1963. The existence of a utility function to represent preferences. Review of Economic Studies 30, 229–232. Richter, M.K., 1980. Continuous and semi-continuous utility. International Economic Review 21, 293–299. Richter, M.K., Wong, K.-C., 1996. Bounded rationalities and definable economies. Working Paper No. 295, University of Minnesota.