Noisy rational expectations equilibrium with optimal information acquisition
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Abstract
This dissertation studies a multi-asset, multi-agent continuous time economy with asymmetric information and noise traders with bounded rationality. There is a long strand of research studying the role that private information plays in markets, particularly how it is communicated in equilibrium, and how it affects prices and investors' welfare. However, the existing literature almost entirely considers the case where private information is communicated at the beginning of the trading period. Therefore, questions such as what induces investors to obtain private information, which conditions allow for the benefit of private information to outweigh its cost, and how much is private information worth, have not been fully addressed. This dissertation studies a model which allows us to provide answers to these questions in a continuous time economy where the underlying factor process driving the risky assets' payoffs is an Ornstein-Uhlenbeck process. There are three types of agents: informed, who have the opportunity to acquire (for a cost) a private information signal about the risky assets' terminal payoffs, uninformed, who have no access to private information, and bounded rational noise traders. The latter agents behave as if they were informed, in that both informed and noise traders use the same map to take private signals into optimal portfolio policies. However, in contrast to the informed agents (insiders), noise traders evaluate this map using the realization of an incorrect signal. Thus, on the one hand, for a given signal realization, noise traders behave optimally, choosing portfolios which maximize expected utility from terminal wealth. On the other hand, by trading on an incorrect signal rather than the true informational signal, they deviate from the informed agents. The novelty in the present work lies in the fact that we endogenize the information acquisition time by allowing the informed investor to optimally choose whether and when to acquire the private information signal. The private signal's precision fluctuates over time and is exogenous. The insider can only use publicly available information when deciding whether (and when) to acquire the signal, and, in particular she is not allowed to view the signal and then decline to obtain it. The insider does not trade prior to obtaining the signal. The cost of obtaining the signal can be thought of either in terms of money or in terms of effort, is both time and state dependent, and is exogenously specified. For general signal's precision and cost functions we prove that an equilibrium exists. However, in order to identify the optimal information acquisition time we connect the precision of the signal to its cost, and we formulate an optimal stopping problem in which the objective function is expressed via the insider's certainty equivalent from overall wealth and the cost of acquiring the signal. Under mild assumptions on the precision and cost functions we establish the existence of an optimal information acquisition time and the regularity of the associated value function. Under some further specifications we obtain an explicit characterization of the optimal information acquisition time by identifying an explicit condition on the signal's cost under which the insider obtains the signal and enters the market. Violation of this condition (particularly if the cost exceeds an explicitly identified threshold) leads to the situation where the insider never acquires the signal, and never enters the market. In this case, we have an equilibrium in which there is no private information and optimal policies are static. The analysis also establishes an upper bound on the benefit of obtaining private information in our setting.
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2025