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eco331-10-ValueOfInformation - Northwestern University Fall...

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Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 THE VALUE OF INFORMATION 1. Introduction This lecture marks the beginning of our study of dynamic choice . This week we focus on a simple, but extremely important type of problems: prior to making a final decision, an individual can, typically for a price, receive information that may help her decide. These problems are dynamic, in that there are (at least) two points in time when the DM must decide: first, she must decide whether or not to acquire information; second, she must act upon the information she receives. We shall focus on how to best use information, and on its value to the DM. In order to study dynamic choice, the state-space framework turns out to be really useful. That is, we will specify a state space S and a set of prizes X (typically real numbers); the objects of choice are random variables, viewed as maps X : S → X . The collection of random variables of interest is denoted by F . 2. The Value of Information: An Example The basic take-home point of this lecture is that “information is always potentially beneficial to the decision-maker”. This emphasizes that: (1) more information is a good thing, but (2) information is not good ‘per se’: rather, it has the potential to be beneficial, provided the decision-maker makes good use of it . A simple example clarifies these points. You are the sales manager of a monopolistic firm who faces consumers with uncertain demand. More precisely, you know that, if you charge a price of p , demand will be equal to Q ( p, s ) = s - p , where s R is a parameter whose value you ignore. Your objective is to maximize the firm’s profits; suppose that your technology exhibits constant marginal cost 2 and no fixed costs. Let us map this problem to a state-space framework. We can think of s as being the state of the world—representing market conditions, if you wish. Prizes represent monetary outcomes, or profits. Utility is linear: u ( x ) = x . The acts we are interested in are those corresponding to price choices; thus, we can denote by X p the random variable corresponding to profits in case unit price equals p . Clearly, p, X p ( s ) = ( p - 2) · Q ( p, s ) . That is, in state s , fixing a price p results in sales equal to Q ( p, s ), and unit profits of ( p - 2). The collection F of random variables of interest is thus given by F = { X p : p R } . Turn now to the analysis of this model. Suppose first, as a preliminary step. that you actually know the value of s . Clearly, if s = 0, the optimal choice is to set p * = 0; if s > 0, we can solve for the optimal price p * via the first-order condition dX p ( s ) dp = s - p - ( p - 2) = s + 2 - 2 p = 0 . 1
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2 THE VALUE OF INFORMATION Note that the second derivative is negative, so the first-order condition identifies an interior maxi- mum: p * ( s ) = 1 2 s + 1 .
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eco331-10-ValueOfInformation - Northwestern University Fall...

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