Social learning and costly information acquisition

Short-lived agents want to predict a random variable $\theta $ and have to decide how much effort to devote to collect private information and consequently how much to rely on public information. The latter is just a noisy average of past predictions. It is shown that costly information acquisition prevents an unbounded accumulation of public information if (and only if) the marginal cost to acquire information is positive at zero $(C^\prime (0) > 0)$. When $C^\prime (0) = 0$ public precision at period n, $\tau_n$, tends to infinity with n but the rate of convergence of public information to $\theta $ is slowed down with respect to the exogenous information case. At the market outcome agents acquire too little private information. This happens either with respect to a (decentralized) first best benchmark or, for n large, with respect to a (decentralized) second best benchmark. For high discount factors the limit point of market public precision always falls short of the welfare benchmarks whenever $C^\prime (0) > 0$. In the extreme, as the discount factor tends to one public precision tends to infinity in the welfare-optimal programs while it remains bounded at the market solution. Otherwise, if $C^\prime (0) = 0$ public precision accumulates in an unbounded way both at the first and second best solutions. More public information may hurt at either the market or second best solutions.