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

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Economics 206 Spring 2007 (Prof. G. Loury) Assignment 8: Signalling Games (1) Consider the following Grade Inflation 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 , θ 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. Specifically, 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 differs 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 fixed preference parameters. Finally, 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. Finally, 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 .
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