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**Unformatted text preview: **Cowles Foundation for Research in Economics at Yale University Cowles Foundation Discussion Paper No. 1498 January 2005 THE DEMAND FOR INFORMATION: MORE HEAT THAN LIGHT Jussi Keppo, Giuseppe Moscarini, and Lones Smith This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection: http://ssrn.com/abstract= 671743 An index to the working papers in the Cowles Foundation Discussion Paper Series is located at: http://cowles.econ.yale.edu/P/au/DINDEX.htm The Demand for Information: More Heat than Light * Jussi Keppo University of Michigan IOE Department Giuseppe Moscarini Yale University Economics Department and Cowles Foundation Lones Smith University of Michigan Economics Department This version: January 24, 2005 JEL: D81, D83 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 Introduction Information acquisition is an irreversible process. One cannot return to the pristine state of ignorance once apprised any given fact. Heat dissipation also obeys the arrow of time: The heat equation in physics describing its transition is not time symmetric. This paper begins with an observation that this link is not merely philosophical. In static models of Bayesian information acquisition, the value function of beliefs and the quantity of information acquired obeys an inhomogeneous form of the heat equation. We show that a nonlinear transformation of the value function and beliefs exactly obeys the heat equation.nonlinear transformation of the value function and beliefs exactly obeys the heat equation....

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