Advances in Equilibrium Theory: Proceedings of the by Donald G. Saari (auth.), Prof. C. D. Aliprantis, Prof. O.

By Donald G. Saari (auth.), Prof. C. D. Aliprantis, Prof. O. Burkinshaw, Prof. N. J. Rothman (eds.)

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Additional info for Advances in Equilibrium Theory: Proceedings of the Conference on General Equilibrium Theory Held at Indiana University-Purdue University at Indianapolis, USA, February 10–12, 1984

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OPTIMALITY PROPERTIES OF QUASIEQUILIBRIA Example 2 shows that a quasiequilibrium need not be a weak optimum. x with respect to pt: 0 EXAMPLE 2. See the single consumer economy represented in Figure 3 and note that w > x. 2 > ... p 1 x FIGURE 3 In order to obtain sufficient conditions for a quasiequilibrium to be a weak optimum or an optimum we first establish an important preliminary lemma. 30 is aonvu and >i is aontinuoUB. rat :Z:i E Xi and P E F ars such that "2 >i:Z:i imp USS p. 2 ;> p':Z:i" and P':Z:i > Inf{p'2: 2EXi}' Then "2 >i:Z:i impUss P'2 > P·:Z:i".

Given the non-side-payment game (A, V), a pair ~(vl*) (l*,u*) where a satisfying. •• ,l;u;) = is defined by vl*(S) = sup{ 6 l;ua:u E V(S)} aES The payoff vector u is said to be a l-transfer va~ue of the game (A, V) • lt is easily verified that the pair (A,V l *) satisfies the definition of a side-payment game, whence ~(vl*) (and consequently the not ions of valuation and l-transfer value) is weIl defined. The l-transfer value represents a generalization of the value notion for side-payment games to the setting of non-side-payment games: Conceptually, given the non-side-payment game would occur if side-payments were realizable.

Take Z>lX l trarily close to Xl. (z+v+x 3 + ... , p. (z + v) ~ p. (Xl + X2). Therefore, X Letting Xl z - we conclude p·v is a quasiequilibrium with respect to ~ arbi- P·x2. • Exarnple 8 shows that hypothesis (a] cannot be dispensed with (or weakened to [b]) in [XI]. EXAMPLE 8. by p = See Figure 9, where the optimum X can only be supported o. w ~--~r-----------------------~2 "'2: Complete Indifference 1 FIGURE 9 Allocation X in Example 1 shows that [XI] cannot be strengthened to replace optimum by weak optimum.

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