http://cowles.econ.yale.edu / The Demand for Information: More Heat than Light ∗
Bibliographic record
Abstract
This paper produces a comprehensive theory of the value of Bayesian information and its static demand. Our key insight is to assume ‘natural units ’ corresponding to the sample size of conditionally i.i.d. signals — focusing on the smooth nearby model of the precision of an observation of a Brownian motion with uncertain drift. In a two state world, this produces the heat equation from physics, and leads to a tractable theory. We derive explicit formulas that harmonize the known small and large sample properties of information, and reveal some fundamental properties of demand: • Value ‘non-concavity’: The marginal value of information is initially zero. • The marginal value is convex/rising, concave/peaking, then convex/falling. • ‘Lumpiness’: As prices rise, demand suddenly suddenly chokes off (drops to 0) • The minimum information costs on average exceed 2.5 % of the payoff stakes • Information demand is hill-shaped in beliefs, highest when most uncertain • Information demand is initially elastic at interior beliefs • Demand elasticity is globally falling in price, and approaches 0 as prices vanish. • The marginal value vanishes exponentially fast in price, yielding log demand. Our results are exact for the Brownian case, and approximately true for weak discrete informative signals. We prove this with a new Bayesian approximation result. We acknowledge useful suggestions of Paavo Salminen and Xu Meng, and the comments from the theory seminar at the University of Toronto and Georgetown University. Lones thanks the NSF for financial support. 1 1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.005 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.324 | 0.077 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".