Arbitrage and equilibrium in economies with short-selling and ambiguty

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1 Arbitrage and equilibrium in economies with short-selling and ambiguty Thai Ha-Huy, Cuong Le Van, Myrna Wooders April 20, 2015 Abstract We consider a model with an finite number of states of nature where short sells are allowed. We present a weaker notion of no-arbitrage price than the one of Werner, and prove that in the case of separable averse at risk utility functions, the existence of one common weak no-arbitrage price is equivalent to the existence of equilibrium. In the general case, this equivalence is true under one of three conditions: closed gradient, no half-line, maximal. Keywords: asset market equilibrium, individually rational attainable allocations, individually rational utility set, no-arbitrage prices, no-arbitrage condition. JEL Classification: C62, D50, D81,D84,G1. University of Evry-Val-D Essonne. thai.ha-huy@m4x.org IPAG Business School, CNRS, Paris School of Economics, VCREME. Cuong.Le- Van@univ-paris1.fr Vanderbilt University. myrna.wooders@gmail.com 1

2 1 Introduction Various conditions have been given for the existence of general equilibrium in an exchange economy. There are different approaches. We can classify them in three main categories. The first category based on conditions on net trade, for example Hart (1974) [?], Page (1987) [17], Nielsen (1989) [16], Page and Wooders (1996) [18], Allouch (1999) [?], Page, Wooders and Monteiro (2000) [19]. We define Individual arbitrage opportunity the set of directions which agent wants to trade with infinite quantities. Hart (1974) [?] is the first who consider the problem of existence of general equilibrium of assets market with short-sales. In the case the agents disagree too much, some agents can make an arbitrage, which is An opportunity with a mutually compatible set of net trades which are utility non-decreasing and, at most, costless to make. Taking the fact that this opportunity can be repeated indefinitely, equilibrium may do not exists. The Weak-no-market-arbitrage (WNMA) requires that that all mutually compatibles net trades which are non-decreasing be useless. Page (1987) proposed the no-unbounded-arbitrage (NUBA), a situation in which there is no group of agents can make mutually compatible, unbounded and utility increasing trades. In 2000, Page Wooders and Monteiro [19], use Inconsequential arbitrage condition to insure the existence for an equilibrium. The second category based on conditions on prices, for example Green (1973) [?], Grandmont (1972, 1977) [?] [?], Hammond (1983) [?] and Werner (1987) [21]. Those authors define Non-arbitrage-price as the dual strictly positive on the set of useful vectors. If the intersection of No-arbitrage-price cones of all agents are non empty (existence of No-arbitrage-price-system NAPS), then the exists general equilibrium. Other authors, like Brown and Werner (1995) [?], Dana, Le Van, Magnien (1999) [7] use condition on the compactness of attainable utility set to ensure the existence of equilibrium. In the case utility functions are quasi-concave, Allouch, Le Van and Page (2002) [?], Ha-Huy and Nguyen (2009) [?] by different approaches, prove the equivalence between the existence of No-arbitrage-price-system (NAPS) and NUBA or WNMA. Allouch, Le Van and Page [?] (2002) extend the definition of no arbitrage price system (NAPS) introduced by Werner [21] (1987) to the case where some agent in the economy has only useless vectors. They show that an extension of NAPS condition of Werner [21] (1987) is actually equivalent to the weak no market arbitrage (WNMA) condition introduced by Hart (1974). They mention that this result implies one given by Page and Wooders (1996) [18] who prove that no unbounded arbitrage (NUBA) condition, a special case of Page 2

3 (1987), is equivalent to NAPS when no agent has non-null useless vectors. The first part of this chapter is to show that when the statement NUBA implies NAPS (Page and Wooders, 1996) is true then we have WNMA implies NAPS (Allouch, Le Van and Page [?] 2002). But it is obvious that if the second statement holds then the first one also holds. The novelty of the result of this note is that the results are self-contained. While Allouch, Le Van and Page [?] (2002) prove WNMA implies NAPS by using a difficult Lemma in Rockafellar [20] (1970) and then deduce that NUBA implies NAPS when no agent has nonnull useless vectors, we show that these conditions are actually circular. In some mathematical senses, these conditions let us to think of the circular tours of Brouwer and Kakutani fixed-point theorems. Moreover, proofs are simple and elementary. In 2010, Dana and Le Van [8], in considering the relationships between the agents beliefs and risk where there is ambiguity, propose to use the set of derivatives of utility function like no-arbitrage-price set. By using this notation, they give a description of weak no-arbitrage price and useful vector. Furthermore, they give an equivalence between non-emptiness of intersection of interior of no-arbitrage-price cones and NUBA condition, or non-emptiness of intersection of relative interior of no-arbitrage-price cones and WNMA condition. Hence, if this intersection is no empty, the existence of general equilibrium is ensured. We can prove that the existence of equilibrium ensures the no emptiness of intersection of weak no-arbitrage prices cones (definition of Dana and Le Van [8]), but do not imply the same result of intersection of interiors. The NUBA and WNMA conditions are quite strong. In this chapter, we will give an example in which NUBA and WNMA are violated, whereas the equilibrium exists. We give in this chapter the conditions based on the closeness of the gradients set and the existence of maximal value on affine sub space. Under convenable conditions, the existence of equilibrium is equivalent with the existence of a common weak no-arbitrage price. In this paper, we reconsider the equilibrium theory of assets with shortselling when there is risk and ambiguity. Using the notion of weak no-arbitrage price proposed by Dana and Le Van [8], we prove the equivalence between existence of general equilibrium and non-emptiness of intersection of no-arbitrageprice cones. 1.1 Existence of a General equilibrium with separable expected utilities We consider now the economy E with m agents i = 1, 2,..., m, a standard model of Arrow-Debreu of complete contingent security markets. There are 3

4 two dates, 0 and 1. At date 0, there is uncertainty about which state s from a state space {1, 2,..., S} will occur at date 1. At date 0, agents who are uncertain about their future endowments trade contingent claims for date 1. Agent i is characterized by an endowment e i R S of contingent claims, a consumption set R S and a probability π i = (π1 i, πi 2,..., πi S ) which represents his belief about future, πs i > 0 s and s πi s = 1. For each contingent claim (x 1, x 2,..., x S ) R S, the utility of agent i is: U i (x) = S πsu i i (x s ) with u i : R R is a concave, strictly increasing differentiable function. Observe that U i (x) = (πsu i i (x s )) S. Denote by ai and b i the limits of derivation of u i at infinity: a i = lim x + ui (x) and b i = lim x ui (x). Proposition 1 If a i < u i (x) x, or b i > u i (x) x, then P i is open. Proof: See Dana and Le Van [8]. Proposition 2 Suppose that a i < u i (x) x, or b i > u i (x) x, then we have: m P i NUBA condition U is compact there exists equilibrium. Proof: Easy. Now we consider the case utility function u i is linear at far, i.e. there exists z i such that u i (x) = z i for x > z i and u i (x) = b i for x < z i. We present here an example in which the weak no-arbitrage prices cones are closed, their intersection is no empty, the model does not satisfy NUBA or WNMA conditions, but there exists equilibrium. Example 1 We consider an economy with two agents, the number of states is S = 2. The belief of agent 1 is represented by probability π 1 1 = π1 2 = 1 2. The one of agent 2 is π 2 1 = 1 3, π2 1 = 2 3. u 1 (x) = u 2 (x) = log(x) if x [1/3, 1/2] 2x 1 log 2 if x 1/2 3x 1 log 3 if x 1/3 log(x) if x [1/3, 4/9] 9 4 x 1 + log( 4 9 ) if x 4 9 3x 1 log 3 if x 1 3 4

5 We have u 1 (+ ) = 2, u 1 ( ) = 3 and u 2 (+ ) = 9 4, u2 ( ) = 3. Therefore, the cone of no-arbitrage prices of agent 1 is P 1 = {λ(ζ 1, ζ 2 )} λ>0 with 1 ζ 1 3 2, 1 ζ The one of agent 2 is P 2 = {λ(ζ 1, ζ 2 )} λ>0 with 3 4 ζ 1 1, 3 2 ζ 2 2. The common no-arbitrage prices is the intersection of two cones P 1 P 2 = {λ(1, 3 2 )} λ>0. Denote S 1, S 2 the interiors of P 1 and P 2. We have S 1 S 2 =. If we consider useful vector w 1 = (1, 2 3 ) of agent 1, the useful vector w2 = ( 1, 2 3 ) of agent 2, observe that w 1 + w 2 = 0. Whenever w 1, w 2 are not useful vectors of agent 1 and agent 2. Our economy does not satisfies NUBA or WNMA conditions. In the next of this subsection, we will prove the existence of equilibrium un such model. Example 2 We consider an economy with two agents, the number of states is S = 2. The belief of agent 1 is represented by probability π1 1 = π1 2 = 1 2. The one of agent 2 is π1 2 = 1 3, π2 1 = 2 3. u 1 (x) = u 2 (x) = 5 2x 1 if x [ 1, 1] 2x 1 2x if x 1 3x + 1 2x if x x 1 if x [ 1, 1] 9 4 x 1 2x if x 1 3x + 1 4x 1 2 if x 1 We have u 1, u 2 are continuous, differentiable, and for all x R, 2 < u 1 (x) < 9 3, 4 < u2 (x) < 3. Therefore, the cone of no-arbitrage prices of agent 1 is P 1 = {λ(ζ 1, ζ 2 )} λ>0 with 1 < ζ 1 < 3 2, 1 < ζ 2 < 3 2. The one of agent 2 is P 2 = {λ(ζ 1, ζ 2 )} λ>0 with 3 4 < ζ 1 < 1, 3 2 < ζ 2 < 2. The weak no-arbitrage prices cones are open, and coincide with the Werner s no-arbitrage cones, S 1 and S 2. Since P 1 P 2 =, equilibrium does not exists. The agent 1 has w 1 = (1, 2 3 ) like useful vector. And the agent 2 has w 2 = ( 1, 2 3 ) like useful vector. We can verify easily that lim λ + U 1 (λw 1 ) < + and lim λ + U 2 (λw 2 ) < +. The intersection of closures P 1 P 2 = {λ(1, 3 2 )} λ>0. The limited arbitrage condition hence is satisfied. But the equilibrium does not exists. Theorem 1 In the model with separable utility, we have: m P i U is compact there exists equilibrium. Proof: Suppose that i P i. We use the Proposition??, and one of the Theorems??,??,?? to prove the compactness of utility set U and existence of equilibrium. 5

6 For the applying of the Theorem??, we prove now that if (w 1,..., w m ) satisfies i wi = 0 and w i R i for all i, for all x i, there exists λ 0 such that U i (x i + λw i ) = U i (x i + λw i ), λ λ. Take p i P i, there exists λ i > 0, x i such that p s = λ i u i (x i s). Easily, we can prove that p w i = 0 i. This implies U i (x i ) w i = 0. Hence for all i, a i S + π i sw i s + b i S π i sw i s = 0. For all i, fix x i. If u i (z) > a i for all z or u i (z) < b i for all z, then P i is open (see Dana and Le Van [8]). In this case, there is no half line in indifferent surfaces, this implies w i = 0. And the claim is evidently true. Suppose that there exists z i such that u i (z) = a i for z > z i and u i (z) = b i for z < z i. Take λ 0 big enough such that for s S + (w i ), x i s + λw i s > z i, and for s S (w i ), x i s + λw i s < z i. For all λ λ we have U i (x i + λw i ) U i (x i + λw i ) (λ λ) S πsu i i (x i s + λws)w i s i = (λ λ) a i = 0. s S + (w i ) π i sw i s + b i s S (w i ) Since w i is a useful vector of i, U i (x i + λw i ) = U i (x i + λw i ), for all λ λ. πsw i s i If there exists general equilibrium, we use the Theorem?? to prove that i P i. 2 General equilibrium in the presence of risk and ambiguity Define = {π R S + such that S π s = 1}. In this section, we consider a standard Arrow-Debreu model of complete contingent security markets like the subsection 1.1. The only different is the utility functions. In this model, each agent has a preference represented by utility function under the form: U i (x) = min π i m π s u i (x s ) with u i : R R is a concave, strictly increasing, differentiable function, and i which is a convex, compact subset of interior of. Easily, we can prove that U i is concave. We define R i the set of useful vectors of U i. Denote a i = inf z R ui (z) = u i (+ ). 6

7 b i = sup u i (z) = u i ( ). z R Lemma 1 The vector w R S is useful vector of function U i if and only if for all π i, x R S we have S π s u i (x s )w s 0. Proof: See Dana and Le Van [8]. We can now define P i the cone generated by the set {(π s u i (x s )) S }: P i = {p R S λ > 0, π i, x R S such that p s = λπ s u i (x s ) for all s}. Lemma 2 P i is a convex cone. Proof: See Dana and Le Van [8]. Lemma 3 Fix x R S. Suppose that π i satisfying U i (x) = s π su i (x s ), then the vector p = (π 1 u i (x 1 ), π 2 u i (x 2 ),..., π S u i (x S )) U i (x). Proof: Indeed, take any y R S. We have U i (y) U i (x) S S π s u i (ys) i π s u i (x s ) S π s u i (x s )(y s x s ) = p (y x). Theorem 2 Suppose that each probabilities set i is a convex hull of the set of M i points i = convex{π i 0, πi 2,..., πi M i }. Then: m P i U is compact there exists general equilibrium. Proof: Suppose that there exists general equilibrium, by using the Lemma, we have i P i. Suppose now that i P i. Take p i P i. Without lost of generality, we can assume that for each i, there exists λ i > 0, x i R S such that p s = λ i π i 0,s ui (x s ), i, s. 7

8 Firstly, we prove that U is bounded. Indeed, for all (x 1,..., x m ) A: m λ i U i (x i ) m S λ i Hence for all i we have: π i 0,su i (x i s) λ i U i (e i ) λ i U i (x i ) p (e m x j ) + j=1 m S λ i π i 0,su i (x i s) m m λ i π0,su i i (x i s)(x i s x i s) = p (e m j=1 λ j m x i ). S π j 0,s uj (x j s) j i S λ i λ j U j (e j ). π i 0,su i (x i s) We will construct a sequence of economies E 0, E 1,..., E t,.... There is m agents in each economy, each agent is represented by utility function U i t (x) = inf i t m π s u i (x i s) with the probabilities set i t is convex hull of M i t + 1 points: the probability π i 0 that we denoted above and other M i t points. For each economy E t = {U i t, i t} m, denote A t = {(x 1, x 2,..., x m ) (R S ) m m x i = e and U i (e i ) Ut i (x i )}. U t = {(v 1, v 2,..., x m ) R m x A t such that U i (e i ) v i U i t (x i )}. We denote also R i t the set of useful vectors of U i t, and W t the set W t = {(w 1, w 2,..., w m ) R 1 t R 2 t R m t m w i = 0}. We denote by Pt i the set of weak-no-arbitrage-prices of agent i in the economy E t. And we define T (E t ) as the number of probabilities which together with π0 i determine the set of ambiguity of agent i: m T (E t ) = card(mt i ). Our initial economy is E 0 : i 0 = i, M0 i = M i, and so evidently, U0 i(xi ) = U i (x i ) for all i, x i R S. If for all (w 1,..., w m ) W 0, for all π i i we have: a i πsw i s i + b i πsw i s i = 0 s S + (w i ) 8 s S (w i )

9 then we stop. Suppose that there exists (w 1, w 2,..., w m ) W 0 such that: a i πsw i s i + b i πsw i s i > 0 s S + (w i ) s S (w i ) for an i, and a probability π i i 0. Without loss of generality, we can assume that this inequality is true for agent m, and a probability π M m 0. We define economy E 1 as an economy with m agents, agent i with 1 i m 1 is represented by i 1 = i 0, U 1 i(xi ) = U0 i(xi ), for all x i. For the agent m, we define 1 is convex hull of {π0 m, πm 1,..., πm M0 m 1}. The utility function is defined as: U m 1 (x m ) = inf π m 1 = min S π s u m (x m s ) 1 k M0 m 1 S πk,s m um (x m s ). We will prove that for this new economy, U 1 = U 0. Observe that for 1 i m 1, U i 0 (xi ) = U i 1 (xi ), and U m 0 (xm ) U m 1 (x m), so U 0 U 1. We also have an observation that i P i 1 0. Indeed, πi 0 i 1, then p P i 1 for all i. The U 1 is then bounded. Denote C > 0 this upper bound. Suppose that (v 1, v 2,..., v m ) U 1. There exists (x 1, x 2,..., x m ) such that i xi = e and U i (e i ) v i U i 1 (xi ) for all 1 i m. Take λ > 0 big enough such that x m s + λw m s > z m s S + (w m ) and x m s + λw m s < z m s S (w m ). Observe that for z > z m we have u m (z) = u m (z m ) + a m (z z m ), and for z < z m we have u m (z) = u m (z m ) + b m (z z m ). Then: = + S πm m 0 m,s u m (x m s + λws m ) s S + (w m ) s S (w m ) And we have: S πm m 0 m,s u m (x m s +λws m ) = Z+λ[a m with Z is a constant: S Z = πm m 0 m,s u m (x m s ) + + s S (w m ) S πm m 0 m,s u m (x m s ) π m M m 0,s [u m (z m ) + a m (x m s + λw m s z m )] π m M m 0,s [u m (z m ) + b m (x m s + λw m s z m )] s S + (w m ) s S + (w m ) π m M m 0,s w m s +b m π m M m 0,s [u m (z m ) + b m (x m s z m )]. 9 s S (w m ) π m M m 0,s [u m (z m ) + a m (x m s z m )] π m M m 0,s w m s ]

10 Since a m s S + (w m ) πm M0 m,swm s + b m s S (w m ) πm M0 m,swm s > 0, let λ tends to infinity, we have the right hand term tends to infinity. Then for λ > 0 big enough, m πm M0 m,sum (x m s + λws m ) > C the upper bound of U 1. Hence we have min S 1 k M0 m 1 π m k,s um (x m s + λw m s ) = min 1 k M m 0 S πk,s m um (x m s + λws m ) and then U0 m (x m + λw m ) = min 1 k M0 m 1 S πk,s m um (x m s + λws m ) = U1 m (x m + λw m ). So for λ > 0 big enough, we have for all 1 i m: U i (e i ) v i U1(x i i ) U1(x i i + λw i ) = U0(x i i + λw i ). Observe that i (xi + λw i ) = i ei. Hence U 1 U 0. Now we are ready to finish the proof. The economy E 1 has U 1 = U 0. Since π i 0 i 1 for all i, we have ip i 1. This implies U 1 is bounded. Observe that T (E 1 ) = T (E 0 ) 1. If in the economy E 1, there exists w W 1 such that i and π i i 1 satisfying a i s S + (w i ) π i sw i s + b i s S (w i ) π i sw i s > 0 then by the same argument, we can construct a new economy E 2, and so on. After each step, the number of probabilities whose determine the ambiguity set of agents diminish by 1: T (E t+1 ) = T (E t ) 1. Since there exists a finite probabilities determining i, so the procedure has to stop at step T. Hence we have T + 1 economy, E 0,..., E T. The utility sets are equal U 0 = U 1 = = U T. The economy E T satisfies the propriety: for all w W T, for all π i i T we have a i πsw i s i + b i πsw i s i = 0. s S + (w i ) s S (w i ) Now we will prove that E T satisfies Inconsequential arbitrage condition. Suppose that (w 1,..., w m ) W T, and {(x 1 n,..., x m n )} A, a positif sequence λ n 0 such that lim n λ n x i n = w i. Take ϵ > 0 arbitrarily, for n big enough, for s S + (w i ), we have x i n,s ϵw i s > z i, and for s S (w i ) we have x i n,s ϵw i s < z i. For every n, there exists π i n i T such that U i T (xi n) = s πi n,su i (x i n,s ϵw i s). 10

11 And we have for n big enough: U i (x i n ϵw i ) U i (x i n) S S πn,su i i (x i n,s ϵws) i πn,su i i (x i n,s) m πn,s( ϵw i s)u i i (x i n,s ϵws) i = ϵ[a i πn,sw i s i + b i = 0. s S + (w i ) s S (w i ) πn,sw i s] i The economy E T satisfies IC condition, then U T is compact, then U = U T is compact. The our initial economy has general equilibrium. References [1] Allouch, N., C. Le Van and F.H. Page (2002): The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, [2] Arrow, K. J., L. Hurwicz, and H. Uzawa (1958): Programming in Linear spaces. Stanford University Press, Stanford, California. [3] Brown, D. and J. Werner (1995): Existence theorems in the classical asset pricing model. Econometrica 59, [4] Brown, D. and J. Werner (1995): Arbitrage and existence of equilibrium in infinite asset markets. Review of Economic Studies, 62, [5] Cheng, H. (1991): Asset market equilibrium in infinite dimensional complete markets. Journal of Mathematical Economics, 20, [6] Chichilnisky, G. (1997): Topological invariant for competitive markets, Journal of Mathematical Economics, 28, [7] Dana, R. A and C. Le Van (1996): Asset Equilibria in L p Spaces: A Duality Approach. Journal of Mathematical Economics, 25, [8] Dana, R. A and C. Le Van (2010): Overlapping risk adjusted sets of priors and the existence of efficient allocations and equilibria with short-selling. Journal of Economic Theory, 145, [9] Dana, R.A, C. Le Van and F. Magnien (1997): General Equilibirum in asset markets with or without short-selling. Journal of Mathematical Analysis and Applications 206,

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