assignmenteight

assignmenteight - Economics 206 Spring 2007 (Prof. G....

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Economics 206 Spring 2007 (Prof. G. Loury) Assignment 8: Signalling Games (1) Consider the following Grade InFation Problem : A professor must issue an evaluation of his student to “the market.” The professor perfectly observes his student’s productivity, but the market can only observe the professor’s evaluation. Let θ ∈ { θ L H } ⊂ [0 , ) be a student’s two possible levels of productivity – either “high” or “low,” and let r [0 , ) be a professor’s evaluation of student ability, which can be any non-negative number. Suppose that the market rewards a student according to how productive that student is believed to be, and that market beliefs about a student’s productivity are derived from the professor’s evaluation. SpeciFcally, let the market reward be denoted by w ( r ), and assume that: w ( r ) = E [ θ | r ] . In addition, assume that professors care both about the rewards received about their stu- dents, and about the accuracy of their evaluation. In particular, the professor derives utility from seeing a student get a higher market reward but experiences disutility from issuing an evaluation of a student which di±ers from the student’s true productivity. Denoting a professor’s utility by u ( w,r ; θ ), assume the particular funcitonal form: u ( w,r ; θ ) = αw - β ( r - θ ) 2 , where α > 0 and β > 0 are Fxed preference parameters. ²inally, suppose that Pr { θ = θ H } = λ (0 , 1) is the prior probability that a student’s productivity is high. We can think of this situation as a signalling game where a professor can be of either of two types (i.e., either with a high productivity or a low productivity student) and must issue an evalution. Let the professor follow a strategy in doing so, r = r * ( θ ), which indicates the evaluation that is issued for a student of each ability level. Moreover, let the market forms beliefs μ = μ * ( r ), taking these strategies into account and using Bayes’s Rule where possible, but otherwise in an unrestricted way. ²inally, let rewards to students be equal to the student’s expected productivity given these beliefs. Thus: w * ( r ) = μ * ( r ) θ H + [1 - μ * ( r )] θ L . Now, consider the possible equilibria in this signalling model. An equilibrium is characterized
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This note was uploaded on 07/06/2009 for the course ECON 206 taught by Professor G.loury during the Spring '07 term at Brown.

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assignmenteight - Economics 206 Spring 2007 (Prof. G....

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