Incentivizing the acquisition and aggregation of information is a key task of the modern economy (e.g., financial markets). We study the design of optimal mechanisms for this task. A population of rationally inattentive agents can flexibly learn about a common state of nature. A principal, who aims to procure a given information structure from the agents at minimal cost, can design general dynamic mechanisms with report- and state-contingent payments. If the agents are risk-neutral, prediction markets implement the first-best. But if the agents are risk-averse, no mechanism can approximate the first-best--not even those that harness the "wisdom of the crowd" by employing a large number of "informationally small" agents--indicating a fundamental conflict between the dual goals of informational and allocative efficiency. Under broad conditions, we show that this inefficiency hinges on the combination of agents' moral hazard and adverse selection. Our characterization of incentive compatibility, which exploits an equivalence between proper scoring rules and "posterior separable" information costs, is tractable and portable to other design settings with rationally inattentive agents.