Good News and Bad News: Representation Theorems and ApplicationsPaul R. MilgromThe Bell Journal of Economics, Vol. 12, No. 2. (Autumn, 1981), pp. 380-391.Stable URL:http://links.jstor.org/sici?sici=0361-915X%28198123%2912%3A2%3C380%3AGNABNR%3E2.0.CO%3B2-QThe Bell Journal of Economics is currently published by The RAND Corporation.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/rand.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgMon Oct 15 10:53:57 2007Good news and bad news: representation theorems and applications Paul R. Milgrorn* This is an article about modeling methods in information economics. A notion of '~avorableness" of news is introduced, characterized, artd applied to four simple models. In the equilibria of these models, (I) the arrival of good news about afirm's prospects always causes its share price to rise, (2)more favorable evidence about an agent's effort leads the principal to pay a larger bonus, (3)buyers expect that uny product information withheld by a salesman is un- favorable to his product, and (4)biddersjgure that low bids by their competitors signal a low value for the object being sold. 1. Introduction IInformation economics is the study of situations in which different economic agents have access to different information. Many kinds of institutions and patterns of behavior have been treated as attempts to cope with such informa- tional asymmetries. For example, Spence (1973) has treated higher education as an attempt by talented workers to signal their talents to employers. Akerlof (1976) has offered a similar analysis of the "rat race," in which employees work faster than the socially optimal pace to distinguish themselves from less talented coworkers. Milgrom and Roberts (1979) offer a signaling analysis of the phenomenon of limit pricing, in which an established firm sets its price below the monopoly price in an attempt to discourage potential competitors. In each of these signaling models, the analysis is driven by a monotonicity property: more talented workers buy more education (Spence) or work faster (Akerlof) than their less talented counterparts, and lower cost firms set lower prices. Monotonicity also plays a key role in models of adverse selection. For example, in the insurance market models of Rothschild and Stiglitz (1976), C. Wilson (1977), and Pauly (1974) in which each individual knows his probabil- ity of suffering a loss but the insurers do not, the individuals with the greatest likelihood of loss buy the most comprehensive insurance coverage. Similarly, in Akerlof's (1970) famous "lemons" model, higher prices in the used car market result in a higher average quality of the cars available, since owners of good cars will simply keep them if the prevailing prices are too low. * Northwestern University. I am pleased to acknowledge the many useful suggestions of Sanford Grossman, Bengt Holmstrom, Alvin Klevorick, Roger Myerson, Mark Satterthwaite, Robert Weber, and Ward Whitt. This research was partially supported by the Center for Advanced Studies in Managerial Economics, by NSF grant SES-8001932, and by ONR grant N00014-79-C-0685.MILGROM 1 381 Additional examples of the role of monotonicity can be found in the litera- tures on search, advertising, and bidding. In bidding, for example, the typical analysis proceeds on the basis of the intuition that a buyer's bid should be an increasing function of his true reservation price. This price, of course, is known only to the buyer. For example, see Vickrey (1961, 1962) and Ortega-Reichert (1968). In view of the role of monotonicity in so much of information economics, it is surprising that studies of rational expectations equilibria and of the problem of moral hazard make no use of any such property. One might guess, for example, that in a rational expectations model the arrival of good news about a firm's prospects would cause the price of its stock to rise. Such results have, unfortunately, been out of reach because no device has been available for modeling "good news." The purpose of this article is to introduce such adevice. In the formal model treated in Section 2, there is a single, unknown, real- valued parameter 8 which is of interest to a decisionmaker. The variable 8 might represent "quality" or "intrinsic value" in a rational expectations or adverse selection model. The decisionmaker observes an informative signal s. Depending on the nature of 8, an appropriate signal might be an array of experi- mental data, a financial or geological report, a road map, a satellite photo- graph, or a television news show. In the absence of extra assumptions, the form that a signal takes is theoretically irrelevant to its ability to convey information. Thinking of 8 as "effort" or "ability" or "quality," I shall say that ob- servation x is more favorable than observation y if for every nondegenerate prior distribution on 8 the posterior corresponding to x dominates that corresponding to y in the sense of strict first-order stochastic dominance. In Section 2, I characterize the "more favorable than" relation and develop some related ideas. The usefulness of the ideas is illustrated by a series of four