Equilibria and incentives in private information economies☆
Introduction
It is well-known that in a finite-agent private information economy, it is in general not possible to write contracts that are incentive compatible, individually rational and Pareto efficient. On the other hand, one may hope that such an inconsistency problem disappears in a large market where the informational influence of a single agent could be negligible. In particular, McLean and Postlewaite, 2002, McLean and Postlewaite, 2003, McLean and Postlewaite, 2005 used a notion of “informational smallness” and showed the existence of incentive compatible, approximate Walrasian equilibria and an approximate core equivalence result via countable independent replicas of a fixed private information economy with finitely many agents.1 Sun and Yannelis, 2007a, Sun and Yannelis, 2007b, Sun and Yannelis, 2008 obtained some exact results in private information economies with a continuum of agents, where the private information is negligible in the sense that the independent private signals of individual agents can influence only a negligible group of other agents.2
The above results have been achieved under the assumptions that agents' private types are purely informational (i.e., the utility functions and endowments are type-irrelevant) and the agents have contingent consumptions based on the type profiles for all the agents.3 A natural question then arises: what will occur if the agents' types are allowed to enter the utility functions and endowments, and each agent's consumption plan is contingent on her private type? In this paper, we study the relevant issues systematically. Besides the usual solution concepts of Walrasian equilibrium and core in such a setting, the informational negligibility also leads us to consider insurance equilibrium which allows complete insurance of individual-level risks.4 Under the assumption that the private types of a continuum of agents are independent, we establish the equivalence of the above three solution concepts in Proposition 1.5 It is clear that those three notions may not be equivalent to each other in the finite-agent setting.
Regarding the issue of incentive compatibility, a subtle difference between finite-agent and continuum-agent economies has been discussed in the first paragraph. In general, one would expect incentive compatibility to hold in a large economy even though it may fail in the corresponding finite-agent case. However, we will obtain a result contrary to such an intuition in the set-up as considered in this paper (namely, the agents' private types influence the utility functions, endowments and consumption plans). More specifically, in a finite-agent economy with private information measurable endowments, it is easy to see that the combination of exact feasibility and private measurability on allocations leads to type-independent net trades, which implies the incentive compatibility.6 It is surprising that there exists a large private information economy in which incentive compatibility fails completely in the sense that almost every agent in any Walrasian expectations equilibrium/private core/insurance equilibrium allocation has the incentive to misreport her type; see Proposition 2. The basic intuition behind this result is that unlike the finite-agent setting, the exact feasibility requirement no longer imposes major restrictions on the privately measurable allocations in large economies so that the private information of an individual agent has an effect on her net trade (via the type dependency of her endowment and utility function). Thus, (almost) any individual can misreport her private information and become better off.
This paper is organized as follows. Section 2 introduces a model of private information economies with a continuum of agents. Section 3 then states the definitions of Walrasian expectations equilibrium, private core, and insurance equilibrium. The main results are presented in Section 4. Section 5 concludes the paper while the proofs are given in Section 6.
Section snippets
Private information economy
Let an atomless probability space7 be the space of economic agents. Let a finite set be the space of all the possible signals/types for individual agents (its power set denoted by ), and a probability space that models
Basic definitions
In this section, we consider three solution concepts for private information economies: Walrasian expectations equilibrium, private core, and insurance equilibrium. Let be a private information economy.
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A consumption plan y for individual agents is an integrable mapping from to . For each state , is the consumption when t occurs. Let denote the set of consumption plans.
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If agent i's consumption plan is y, then her (ex ante) expected utility
The results
Before providing the main results, we first state a definition which formalizes the intuition of negligible private information.
Definition 4 Idiosyncratic signal process The signal process f is called an idiosyncratic signal process if it is essentially pairwise independent in the sense that for λ-almost all , the random variables and from to are independent for λ-almost all .14
Concluding remarks
Since reality suggests that the uncertainty in an economy with many agents comes from both macro and micro levels, one may work with the assumption that the private types of a continuum of agents are conditionally independent, given the macro level shocks.17 Our set-up and results (without macro states) can be generalized in a trivial way to the case with finitely many macro
Proofs
To prove Proposition 1, Proposition 2, we begin the analysis by stating the framework of Fubini extension and the exact law of large numbers in Subsection 6.1 and the induced large economies in Subsection 6.2.
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The authors are very grateful to the editor, an associate editor and an anonymous referee for their careful reading and helpful suggestions. They also thank Wei He, Qian Jiao, Bo Shen, Bin Wu, and Haomiao Yu for their comments. Part of this work was done when the authors visited the School of Management and Economics at CUHK Shenzhen and Department of Economics at CUHK in December 2016. Some of the results reported here were presented at the 2014 Asian Meeting of the Econometric Society in Taipei. This research was supported in part by the NUS grant (R-122-000-227-112), NSFC (No. 11401444) and SRF for ROCS, SEM.