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The Demand for Information: More Heat than Light

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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: (a) Value 'non-concavity': The marginal value of information is initially zero; (b) The marginal value is convex/rising, concave/peaking, then convex/falling; (c) 'Lumpiness': As prices rise, demand suddenly chokes off (drops to 0); (d) The minimum information costs on average exceed 2.5% of the payoff stakes; (e) Information demand is hill-shaped in beliefs, highest when most uncertain; (f) Information demand is initially elastic at interior beliefs; (g) Demand elasticity is globally falling in price, and approaches 0 as prices vanish; and (h) 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.

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  • Jussi Keppo & Giuseppe Moscarini & Lones Smith, 2005. "The Demand for Information: More Heat than Light," Cowles Foundation Discussion Papers 1498, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1498
    Note: CFP 1258.
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    Cited by:

    1. Jeffrey L. Hoopes & Daniel H. Reck & Joel Slemrod, 2015. "Taxpayer Search for Information: Implications for Rational Attention," American Economic Journal: Economic Policy, American Economic Association, vol. 7(3), pages 177-208, August.
    2. Gryglewicz, Sebastian, 2011. "A theory of corporate financial decisions with liquidity and solvency concerns," Journal of Financial Economics, Elsevier, vol. 99(2), pages 365-384, February.
    3. Azevedo, Eduardo M. & Mao, David & Montiel Olea, José Luis & Velez, Amilcar, 2023. "The A/B testing problem with Gaussian priors," Journal of Economic Theory, Elsevier, vol. 210(C).
    4. Axel Anderson & Lones Smith, 2013. "Dynamic Deception," American Economic Review, American Economic Association, vol. 103(7), pages 2811-2847, December.
    5. Hoffmann, Florian & Inderst, Roman & Turlo, Sergey, 2018. "Regulating Cancellation Rights with Consumer Experimentation," EconStor Preprints 253659, ZBW - Leibniz Information Centre for Economics.
    6. Michael Jong Kim, 2020. "Variance Regularization in Sequential Bayesian Optimization," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 966-992, August.
    7. Maximilian Schaefer, 2025. "When Should we Expect Non-Decreasing Returns from Data in Prediction Tasks?," Papers 2503.03602, arXiv.org.
    8. Jiajia Nie & Qijun Wang & Gendao Li & Dun Liu, 2023. "To share or not to share? When information sharing meets remanufacturing," Annals of Operations Research, Springer, vol. 329(1), pages 815-846, October.
    9. Alessandra Fogli & Laura Veldkamp, 2007. "Nature or nurture? learning and female labor force dynamics," Staff Report 386, Federal Reserve Bank of Minneapolis.
    10. Jae Wook Kim & Jin Hwa Rhee & Chul Hung Park, 2024. "How Does Digital Transformation Improve Supply Chain Performance: A Manufacturer’s Perspective," Sustainability, MDPI, vol. 16(7), pages 1-16, April.
    11. Heyen, Daniel & Goeschl, Timo & Wiesenfarth , Boris, 2015. "Risk Assessment under Ambiguity: Precautionary Learning vs. Research Pessimism," Working Papers 0605, University of Heidelberg, Department of Economics.
    12. Dai, Min & Huang, Shan & Keppo, Jussi, 2019. "Opaque bank assets and optimal equity capital," Journal of Economic Dynamics and Control, Elsevier, vol. 100(C), pages 369-394.
    13. Kosenko, Andrew, 2024. "Beyond the Blackwell order in dichotomies," Economics Letters, Elsevier, vol. 245(C).

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    More about this item

    Keywords

    Value of information; Non-concavity; Heat equation; Demand; Bayesian analysis;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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