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20-07-2019 | Research Article

A new approach to the rational expectations equilibrium: existence, optimality and incentive compatibility

Authors: Luciano I. de Castro, Marialaura Pesce, Nicholas C. Yannelis

Published in: Annals of Finance | Issue 1/2020

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Abstract

Rational expectations equilibrium seeks a proper treatment of behavior under private information by assuming that the information revealed by prices is taken into account by consumers in their decisions. Typically agents are supposed to maximize a conditional expectation of state-dependent utility function and to consume the same bundles in indistiguishable states [see Allen (Econometrica 49(5):1173–1199, 1981), Radner (Econometrica 47(3):655–678, 1979)]. A problem with this model is that a rational expectations equilibrium may not exist even under very restrictive assumptions, may not be efficient, may not be incentive compatible, and may not be implementable as a perfect Bayesian equilibrium (Glycopantis et al. in Econ Theory 26(4):765–791, 2005). We introduce a notion of rational expectations equilibrium with two main features: agents may consume different bundles in indistinguishable states and ambiguity is allowed in individuals’ preferences. We show that such an equilibrium exists universally and not only generically without freezing a particular preferences representation. Moreover, if we particularize the preferences to a specific form of the maxmin expected utility model introduced in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), then we are able to prove efficiency and incentive compatibility. These properties do not hold for the traditional (Bayesian) Rational Expectation Equilibrium.

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Sun et al. (2013) provides a counterexample for a large economy in which REE does not possess the desirable property of incentive compatibility for each agent.

Nothing changes in the analysis if we assume that individual 2 considers all convex combinations of \(s_{1}\) and \(s_{2}\) as possible.

Another way of describing the same problem is to think that the decision on quantities is measurable with respect to the information partition that the individual has after observing prices.

Note that she is indifferent taking in account the information that she has when making decisions. Obviously, she is not indifferent ex post.

It is known that if \(u_i(s,\cdot )\) is continuous and monotone, then it is also monotonically increasing, i.e., \(x\ge y\) implies \(u_i(s,x)\ge u_i(s,y)\).

For simplicity, we will often use the symbol \(x_i(s)\in {\mathbb {R}}_+^\ell \) to denote \(x(i,s)\in {\mathbb {R}}_+^\ell \). Similarly, \(x_i(\cdot )\) refers to the function \(x(i, \cdot ):S \rightarrow {\mathbb {R}}_+^\ell \). Finally, x(s) refers to the function \(x(\cdot , s) :I \rightarrow {\mathbb {R}}_+^\ell \).

In particular if \(\mu \in \mathcal {C}^F\), then \( \mu (s^\prime )=0\) for any \(s^\prime \notin F\) and \(\sum _{s^\prime \in F }\mu (s^\prime )=1\). For applications of those preferences see Ravanelli and Svindland (2019).

The information about the bundle chosen by the “Walrasian auctioneer” is available to the individual i only after all choices are made and, therefore, cannot affect her behavior.

By traditional in this paper we mean Bayesian.

Hence, this example does not contradict Lemma 8.2 in the “Appendix” and Proposition 4.7.

We are grateful to Z. Liu and L. Sun for having checked the computations in Example 4.1.

We can consider also the general MEU formulation (5) provided that for all agent i and state s, the set \({\mathcal {M}}_i^s\) contains only positive priors (see Sect. 8.5).

This property stated in Proposition 4.10 holds true even with the general MEU formulation (5).

Only to prove the statements of Theorems 5.4 and 5.9 under the first condition (i.e., \(\sigma (u_i, e_i)\subseteq {\mathcal {F}}_i\) for all \(i\in I\)), any set \({\mathcal {M}}_i^s\) must contain only positive priors (see Sect. 8.5).

This assumption is quite common in the literature of asymmetric information economies (see for example Angeloni and Martins-da Rocha 2009 and Correia-da Silva and Hervés-Beloso 2012) (see Remark 4.9 in Sect. 4.2).

The reader is also referred to Krasa and Yannelis (1994), Koutsougeras and Yannelis (1993) and Podczeck and Yannelis (2008) for an extensive discussion of the Bayesian incentive compatibility in asymmetric information economies.

The existence holds for the general function V, while for the efficiency results we may consider a more general MEU framework by adopting suitable modifications.

We thank Liu Zhiwei for having suggested this example to us.

Notice that the private information measurability assumption of utility functions is not too strong when we deal with coalitional incentive compatibility notions (see for example Koutsougeras and Yannelis 1993; Krasa and Yannelis 1994; Angeloni and Martins-da Rocha 2009 where the utility functions are assumed to be state-independent, and therefore \(\mathcal F_i\)-measurable).

We mean that \(H(s_1)=H(s_2)\) if \(\sigma (p)(s_1)=\sigma (p)(s_2).\)

Kreps’s example can also be used to show that an ex post efficient allocation may not be maxmin Pareto optimal (see Remark 5.11).

Notice that \((t_1(a), z_1(a))\gg 0\) because \(t_1(a) z_1(a)> \frac{3}{2}>0\).

Clearly, \((t_i(b), z_i(b))\gg (0,0)\) for each \(i=1,2\).

Notice that for all i, \(\sigma (p)\subseteq \mathcal {G}^p_i=\mathcal {F}_i\vee \sigma (p)\). Thus, for all i, \(p(\cdot )\) is \(\mathcal {G}^p_i\)-measurable. Therefore, condition (i) implies that \(p(a)=p(b)\).

Notice that for any \(s \in \mathcal G_j(a)\), \(\sum _{h=1}^\ell p^h( s )>0\), because \(p( s )\in {\mathbb {R}}_+^{\ell }{\setminus } \{0\}\) for any \(s \in S\).

Notice that \(\sum _{h=1}^\ell p^h( a )>0\), because agents’ utility functions are monotone and consequently \(p( s )\in {\mathbb {R}}_+^{\ell }{\setminus } \{0\}\) for any \(s \in S\).

Actually Proposition 4.3 requires strict quasi-concavity, while \(u_i\) is concave and satisfies a weaker condition according to which the inequality \(u(\alpha x+ (1-\alpha )y)>\min \{u(x), u(y)\}\) holds when \(u(x) \ne u(y)\).

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Title: A new approach to the rational expectations equilibrium: existence, optimality and incentive compatibility
Authors: Luciano I. de Castro
Marialaura Pesce
Nicholas C. Yannelis
Publication date: 20-07-2019
Publisher: Springer Berlin Heidelberg
Published in: Annals of Finance / Issue 1/2020
Print ISSN: 1614-2446
Electronic ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-019-00349-w

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