Limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium with or without short sales

Markets and Games: A Simple Equivalence among the Core, Equilibrium and Limited Arbitrage

“…2 This excludes the classic Arrow Debreu model in which the trading space is R''+, and preferences which are linear or have half lines in their indifferences. The results in Chichilnisky (1991Chichilnisky ( , 1992Chichilnisky ( , 1993aChichilnisky ( , 1993bChichilnisky ( , 1994aChichilnisky ( , 1994bChichilnisky ( , 1995aChichilnisky ( , 1995b include 118 N+ and all the preferences mentioned above . 3 See Chichilnisky (1991Chichilnisky ( , 1992Chichilnisky ( , 1993aChichilnisky ( , 1994bChichilnisky ( , 1995a.…”

“…Therefore net trades in directions which lead to unbounded utility gains are unaffordable by all traders from their initial endowments at price p. The economy of Figure 2 does not 7 For a proof see Chichilnisky (1995b) . 8 For a proof see Chichilnisky (1992Chichilnisky ( , 1995a. 9 Without the condition that preferences contain no half lines the arguments provided in this paper do not hold .…”

“…., H} . The trading opportunities which could yield unbounded utility increases for the ith trader are described by net trades in the global cone defined ash A;(Q;) = {y E RN :`dx E 118N, 3A > 0: ui(Qi + Ay) > ui(x)} : this is a concept originally introduced in Chichilnisky (1991) and(1992), and contains global information about the trader. I make three assumptions on preferences, b' is (1) non-satiation: A ;(52 ;) * 0; (2) uniformity : the smallest closed set containing Ai(Qi), denoted Ai(Qi), is the set of directions of net trades from S2 i along which ui does not decrease and is the same b' Ski E RN .…”

“…the nonemptiness of the core in the special cases considered in this paper. The relationship between the results of Heal (1984, 1993) and the Other literature is discussed in Chichilnisky (1992) and also below. 2 This excludes the classic Arrow Debreu model in which the trading space is R''+, and preferences which are linear or have half lines in their indifferences.…”

“…the 1984 results in Heal (1984, 1993) which introduced a noarbitrage condition C and proved it is sufficient for the existence of an Arrow Debreu equilibrium in markets with short sales (finite or infinite dimensional) . Subsequently, existence results which are special cases of the above were given in Werner (1987), Nielsen (1989), Page (1987), andChichilnisky (1992) . The latter provides necessary as well as sufficient conditions.…”