Arbitrage and global cones: Another counterexample*

Transcription

1 Soc Choice Welfare (999) 6: 7±46 Arbitrage and global cones: Another counterexample* Paulo K. Monteiro, Frank H. Page, Jr.,**, Myrna H. Wooders,*** CORE, 4 voie du Roman Pays, B-48 Louvain-la-Neuve, Belgium ( pklm@impa.br) Department of Finance, University of Alabama, Tuscaloosa, AL 5487, USA ( fpage@cba.ua.edu) Department of Economics, University of Toronto, 5 St. George Street, Toronto, Ontario, Canada M5S A ( mwooders@chass.utoronto.ca) Received: 8 August 997/Accepted: January 998 Abstract. Chichilnisky (997) claims that another variant of her condition limiting arbitrage is necessary and su½cient for existence of equilibrium and nonemptiness of the core in an economy with short sales allowing half lines in indi erence surfaces. Her proof, however, is based on a proposition purporting to relate her notion of ``global cone'' (see Chichilnisky (997) for references) to the Page-Wooders ``increasing cone.'' In this paper, we present a counterexample showing that parts (i) and (ii) of Chichilnisky's proposition are false. Thus, Chichilnisky's claimed result is without proof. Introduction In an exchange economy with unbounded short sales, Chichilnisky (997) states that another variant of her condition limiting arbitrage is necessary and su½cient for existence of equilibrium and nonemptiness of the core. An aspect of her model is that, in a limited way, half lines in indi erence surfaces are allowed. Chichilnisky's claimed result rests on Proposition 5 of the paper, * The authors are grateful to Kenneth Arrow, Peter Hammond, and an anonymous referee for helpful comments. ** F. H. Page gratefully acknowledges nancial support from the RGC program at the University of Alabama. *** M. H. Wooders is indebted to the Social Sciences and Humanities Research Council of Canada for support. Monteiro, Page, and Wooders (997) discuss previous versions of her condition and, in particular, the global cones used in her previous conditions (see Sect. below). For example, her model does not allow an agent to have half lines in indi erence surfaces in some directions and no half lines in others.

2 8 P. K. Monteiro et al. which purports to establish the relationship between global cones, introduced in her prior papers, and Page-Wooders increasing cones. In this paper, we present a counterexample showing that parts (i) and (ii) of Chichilnisky's proposition are false. In a related paper, Monteiro, Page, and Wooders (997) present a counterexample showing that Chichilnisky's previous notions of limited arbitrage (no arbitrage conditions based on global cones) are either not well de ned or fail to be su½cient for existence of equilibrium or nonemptiness of the core. Thus, her claimed existence and nonemptiness results remain without proof. When agents' indi erence surfaces contain no half lines, Chichilnisky's (997) condition of limited arbitrage is in fact equivalent to the conditions limiting arbitrage typically used in the literature.4 Moreover, in various exchange economy models, some more general than Chichilnisky's (for example, Werner (987)), a number of authors have shown that whenever agents' indi erence surfaces contain no half lines, no unbounded arbitrage is necessary and su½cient for existence of equilibrium.5 In addition, Page and Wooders (99, 994, 996a,b) have shown that no unbounded arbitrage is necessary and su½cient for compactness of the set of utility possibilities and nonemptiness of the core.6 Thus, what remains to be shown is whether the connections between unbounded arbitrage, equilibrium, and the core continue to hold outside of the no half lines case, that is, in the case where half lines in indi erence surfaces are allowed. This appears to have been one of the main objecties of Chichilnisky's papers on arbitrage. The counterexample below, together with the counterexample in Monteiro, Page, and Wooders (995), show that this objective has not yet been achieved.. The model Chichilnisky (997a) considers an exchange economy R L ; o j ; u j n jˆ where each agent j has consumption set R L, endowment o j A R L, and preferences Increasing cones were introduced in Page (98, 989) to obtain necessary and suf- cient conditions for the existence of equilibrium in an asset market model with short sales and named in Page and Wooders (99). Page and Wooders (994, 996a,b) extend the notion of increasing cone to accommodate thick indi erence curves. 4 See Hart (974), Milne (98), Kreps (98), Hammond (98), Grandmont (977, 98), Page (98, 984, 987, 989, 996), Werner (987), Nielsen (989), and Page and Wooders (99, 994, 996a,b). 5 See Grandmont (977, 98), Milne (98), Hammond (98), Page (98, 989, 996), Werner (987), and Page and Wooders (99, 994, 996a,b). 6 The equivalence of nonemptiness of the core and existence of an equilibrium is immediate from Debreu and Scarf (96). In unbounded economies, Page and Wooders (996a,b) make this connection directly for the core and the partnered core introduced in Reny, Winter, and Wooders (994) and Reny and Wooders (996). Page and Wooders (996a) show that no unbounded arbitrage is even necessary and su½cient for the existence of a Pareto optimal point. Aliprantis et al. (997) extend the equivalence of nonemptiness of the core and existence of equilibrium to in nite dimensional Riesz spaces.

3 Arbitrage and global cones 9 over R L speci ed via a utility function u j : R L! R. Chichilnisky makes the following special assumptions: [G-] for each agent j, u j is continuous, quasi-concave, and monotonically increasing with u j ˆ and sup fu j x : x A R L gˆy;7 [G-] for each agent j, either (i) (no half lines): all indi erence surfaces contain no half lines, or (ii) (closed gradient): the set of gradient directions along any indi erence surface is closed; [G-] (uniform nonsatiation): for each agent j, preferences are uniformly nonsatiated, that is, u j has a bounded rate of increase. In particular, for preferences represented by a twice continuously di erentiable utility function u j, there exists positive numbers K and e such that K > k`u j x k > e for all x A R L.. Global cones and increasing cones According to Chichilnisky (997), the global cone corresponding to the jth agent's utility function u j at consumption vector x is given by, A j x ˆ fy A R L :E x A R L b l x > such that u j x l x y >u j x g: Thus, a consumption vector y is an element of the global cone A j x if and only if given any consumption vector x, there exists a scalar l x > such that the jth agent strictly prefers x l x y to x. The increasing cone, introduced in Page (98) (also see Page (989, 996) and Page and Wooders (99)), is given by, I j x ˆfy A R L : u j x ly > u j x my if l > m V g: An example in Monteiro et al. (997) shows that the cones de ned by () and () are not equivalent. In Page and Wooders (994, 996a,b) the de nition of the increasing cone is extended to accommodate thick indi erence curves: ^I j x ˆfyAR L :EmV ; b l>m such that u j x ly >u j x my g: Chichilnisky (997) modi es her arbitrage condition by using the increasing cone ^I j x, but alternatively stated in her paper as: G j x ˆfyA R L : :b max l V u j x ly g: 4 The equivalence of the Page-Wooders increasing cone de ned by () and the increasing cone de ned by (4) is immediate.8 Note that under condition [G-], the cones de ned by (), () and (4) are all equivalent, and not equivalent to the cone de ned by () ± see Monteiro, Page, and Wooders (997). 7 According to Chichilnisky, u j ˆ and sup fu j x : x A R L gˆy can be assumed without loss of generality (see Kannai and Yusim 99). 8 We are indebted to Kenneth Arrow for private correspondence bringing to our attention this change in Chichilnisky (995) from her prior condition based on her global cone, A j o j, to a condition based on the Page-Wooders increasing cone, I j o j.

4 4 P. K. Monteiro et al.. Chichilnisky's proposition Chichilnisky's (997) Proposition 5, counterexampled below, can now be stated formally as follows: Proposition 5 (Chichilnisky 997): Let u j : R L! R be a utility function satisfying [G-], [G-], and [G-] (i.e., uniform nonsatiation). Then the following statements are true: (i) The interior (denoted int) of the cone G j o j is nonempty and equal to the interior of global cone A j o j. Moreover, the interior of G j o j contains those directions in which utility increases to in nity. Thus, int G j o j ˆint A j o j ˆfy A G j o j : lim l!y u j o j ly ˆyg 6ˆq: 5 (ii) The boundary of the cone G j o j, denoted by qg j o j, contains (a) those directions along which utility increases toward a bounded value that is never reached, and (b) those directions along which the utility eventually achieves a constant value. (iii) The interiors, boundaries, and closures of the global cones A j x are uniform across all vectors x A R L. The cones G j x are also uniform across all vectors x A R L. (iv) For general nonsatiated preferences (i.e., preferences that are not uniformly nonsatiated), the cones G j x may not be uniform across all vectors x A R L. Before stating Proposition 5, Chichilnisky writes: ``The following proposition establishes the structure of the global cones, and is used in proving the connection between limited arbitrage, equilibrium and the core:'' The counterexample presented below shows that parts (i) and (ii) of Chichilnisky's Proposition 5 are false. No proof of Proposition 5 nor of the claimed connection between limited arbitrage, equilibrium and the core is provided in Chichilnisky (997). For the proof of Proposition 5, Chichilnisky instead refers the reader to forthcoming papers and Chichilnisky (995) which states and purports to prove the same proposition. Note that Monteiro, Page, and Wooders (997) also provides a counterexample to footnote 6 of Chichilnisky (997) which claims the equivalence of G j o j and A j o j. Note also that we do not present a counterexample to the current version of Chichilnisky's claim that limited arbitrage is necessary and su½cient for existence. However, given the importance of Proposition 5 to the validity of this claim, our counterexample shows that the claim is still without proof.

5 Arbitrage and global cones 4 Fig. 4. The counterexample Overview We give an example of a utility function u : R! R which satis es assumptions [G-], [G-] (i) (no half lines), and [G-] (uniform nonsatiation). The utility function in our example has global cone, A o, depicted in Fig.. Here, global cone A o is given by the nonnegative orthant minus the origin. We show for the utility function in our example that for any x A R L u x ly is increasing in l V for y A A o ; lim l!y u x ly ˆy for all y A A o ; and lim l!y u x ly ˆ y for all y B A o : Thus, in our example the global cone A x and the increasing cone G x are equal for all x A R L, and, int A j o j 6ˆfy A G j o j : lim l!y u j o j ly ˆyg; negating part (i) of Chichilnisky's Proposition 5. Also, in our example lim u j o j ly ˆy for all nonzero y A qg x ; l!y negating part (ii) of Chichilnisky's Proposition 5. Details Consider the utility function u : R! R given by u x ; x ˆ x x = p ArcTan c dc x x = p ArcTan c dc: 6

6 4 P. K. Monteiro et al. Lemma (Basic Properties of Utility Function (6)): Utility function (6) is strictly concave, strictly increasing, and continuously di erentiable in nitely many times (i.e. C y ), with u ˆ. Lemma ( Evaluation of Limits): For utility function (6), the following limit statements are true: (a) lim l!y u x ly ˆy for any x in R and any nonzero vector y contained in the nonnegative orthant (i.e., any vector y ˆ y ; y 6ˆ with y V and y V ). (b) lim l!y u x ly ˆ y for any x in R and any y not contained in the nonnegative orthant. From Lemma we can conclude that the global cone corresponding to the utility function given in (6) is invariant with respect to x and is given by A x ˆfy A R j y ˆ y ; y 6ˆwithy V and y V g; 7 as in Fig.. It only remains to show that utility function (6) satis es [G-], uniform nonsatiation. Lemma ( Utility Function (6) Satis es Uniform Nonsatiation): For utility function (6), uniform nonsatiation is satis ed (i.e., [G-] holds). In particular, for all x A R L, < qu x ; x qx qu x ; x qx < : Appendix Proof of Lemma : The strict concavity of u is an immediate consequence of the strict concavity of the function v given by c v c ˆ p ArcTan c dc; and the fact that the mapping x ; x! x x ; x x specifying the limits of integration in () is linear injective. Moreover because v is continuously di erentiable in nitely many times, the utility function u is C y. The fact that u is strictly increasing follows by inspection of the partial derivatives qu x ; x ˆ qx p ArcTan x x qu x ; x ˆ qx p ArcTan x x p ArcTan x x ; and p ArcTan x x : Q.E.D.

7 Arbitrage and global cones 4 Proof of Lemma : First note that the antiderivative of ArcTan(c) is c Arctan c log c a constant: Therefore, ( u x ˆ x x x x = p c ArcTan c log ) c x x = ( x x = p c ArcTan c log ) c x x = : Let y A R and l > and consider u x ly l ˆ x x y y l x x y p l y p x x l u x ly. We have l y y ArcTan x x l y y ArcTan x x l y y log x pl x l y y log x x l y y : As l! y, the limit of the terms above involving log is zero. Moreover, z lim l!y ArcTan c lz ˆjzj p : Thus, lim l!y u x ly l ˆ y y y y y y : 8 Limit statement (b) follows immediately from (8). Also it follows from (8) that lim l!y u x ly ˆy for all x in R and all vectors y ˆ y ; y with y > and y > : Now let us evaluate the limit lim l!y u x ly for the more delicate case y ˆ ;. The case y ˆ ; is totally analogous. For the case y ˆ ;

8 44 P. K. Monteiro et al. we have u x ly ˆu x l; x ˆ x l x ( x x l = p c ArcTan c log ) c x x l = ( x l x = p c ArcTan c log ) c x l x = : Thus, u x l; x ˆ x x l p x x l p We need to calculate x x l ArcTan x x l ArcTan x x log x p x l log x x l : l l lim l!y l p ArcTan x x l x x l ArcTan l log x p x l log x x l : First note that the limit of the terms not containing log is nite. To see this we apply L'Hopital rule: lim l!y ArcTan x x = l= p =l ArcTan x l x = p by L'Hoà pital's rule 9p x ˆ lim x = l= p x l x = l!y =l ˆ p :

9 Arbitrage and global cones 45 Thus, we have l lim l!y p l ArcTan x x l x x larctan l log x p x l log x x l ˆ y; and thus we can conclude that lim l!y u x l; x ˆy for all x ; x A R. Q.E.D. Proof of Lemma : First recall that qu x ; x ˆ qx p ArcTan x x p ArcTan x x ; and qu x ; x ˆ qx p ArcTan x x p ArcTan x x : To see that utility function (6) satis es uniform nonsatiation [G-] it su½ces to consider the sum norm. Thus, for all x A R L we have qu x ; x qx qu x ; x qx ˆ p ArcTan x x p ArcTan x x p ArcTan x x p ArcTan x x ˆ 4 p ArcTan x x p ArcTan x x : Since for c A R, p < ArcTan c < p, we have < qu x ; x qx qu x ; x qx < : Q:E:D References Aliprantis CD, Border KC, Burkinshaw O (997) Economies with Many Commodities. Journal of Economic Theory 74: 6±5 Chichilnisky G (997) Market Arbitrage, Social Choice and the Core. Social Choice and Welfare 4: 6±98 Chichilnisky G (995) A Uni ed Perspective on Resource Allocation: Limited Arbitrage is Necessary and Su½cient for the Existence of a Competitive Equilibrium, the Core and Social Choice. Core Discussion Paper 957

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