Abstract
A sequential variation of the Arrow–Debreu abstract economy is developed to closely capture the timing of moves of the Walrasian general equilibrium model. Instead of inducing a pseudo- game, the extensive form game of our sequential variation is well defined. It is shown that when information is symmetric, Walrasian equilibrium allocations are equivalent to subgame-perfect equilibrium allocations. When information is asymmetric, rational expectations equilibrium allocations are shown to be equivalent to perfect Bayesian equilibrium allocations. These results are useful for understanding and characterizing Walrasian and rational expectations equilibrium allocations.
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Notes
There have been many studies of the simultaneous abstract economy allowing for general preferences, commodity spaces, and information structures. See, for example, Schmeidler (1973) on extensions allowing for measure spaces of consumers, Shafer and Sonnenschein (1975) for general preferences, Yannelis and Prabhakar (1983) and Tian (1992) for general commodity spaces, and Yannelis (2009) for asymmetric information. For applications of the abstract economy in the finance literature, the reader is referred to Rubinstein (1975).
The abstract economy is regarded as a pseudo-game in Ichiishi (1983, p. 60).
In the case an agent has no endowment, the court transfers to him the one unit penalized from the one who lied.
Glycopantis et al. (2005) further investigate, among other issues related to rational expectations, if REEs can be implemented as PBEs by allowing agents to strategically declare their private information. They show by example that if the implementation of a REE depends on certain compatibility of the agents’ information declarations, then not every REE can be achievable as a PBE of the game under their rules (see Section 4 in Glycopantis et al. 2005). The reason that we can have the equivalence between REE and PBE is because consumers in the abstract economy has no other strategic choice, such as disclosing private information to the auctioneer, besides making affordable choices of consumption bundles updating beliefs with regard to states of nature taking as given the announced prices of the auctioneer who is extra to the economy.
We use Greek letters in the definition of PBE in order to clearly distinguish from the definition REE.
We explain the ideas in our paper by using and developing the example in Mas-Colell et al. (1995, Example 19.H.2, p. 720).
Endowment \(\omega \) is assumed to be sufficient enough to support interior equilibrium allocations.
The price function helps consumer 2 update his belief, so the interim utility function turns to be the same as consumer 1’s.
Consumer i has to use his private signal when spot price vector p is not chosen by the auctioneer in state s, because no information can be extracted from observing the spot price vector for consumer i in this case.
See Fudenberg and Tirole (1991) for discussions.
For \(s,s'\in \mathcal S\) with \(p\not =\bar{\rho }(s)\), and \(\sigma _i(s) = \sigma _i(s')\), (16) implies that \(\Psi _i(\sigma _i(s), p,\bar{\rho }(s)) = \Psi _i(\sigma _i(s'), p,\bar{\rho }(s'))\). It follows from (17) that \(\bar{\mu }_i(\cdot | \sigma _i(s), p, \bar{\rho }(s))=\bar{\mu }_i(\cdot | \sigma _i(s'), p, \bar{\rho }(s'))\). Consequently, problem (19) at (s, p) and \((s', p)\) is the same. It follows that \(\bar{\chi }_i(\cdot )\) is well defined.
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Acknowledgements
We gratefully acknowledge helpful comments and suggestions from Zhiwei Liu, Xinxi Song, and participants at 2017 Nanjing International Conference on Game Theory and the Fourth Microeconomics Workshop, 2018 SAET Conference, and 2018 ISEM Workshop on Economics and Finance. We are also very grateful to Dionysius Glycopantis, the editor, and a referee for their constructive comments and suggestions that have greatly improved the paper. Financial support from the National Natural Science Foundation of China (Grant No. 71472110) is gratefully acknowledged.
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Qin, CZ., Yang, X. On the equivalence of rational expectations equilibrium with perfect Bayesian equilibrium. Econ Theory 69, 1127–1146 (2020). https://doi.org/10.1007/s00199-019-01192-w
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DOI: https://doi.org/10.1007/s00199-019-01192-w
Keywords
- Abstract economy
- Perfect Bayesian equilibrium
- Rational expectations equilibrium
- Subgame-perfect equilibrium
- Walrasian equilibrium