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

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Economics 206 Spring 2007 (Prof. G. Loury) Assignment 8: Signalling Games (1) Solution for the Grade In f ation Problem : You were asked to show that there is an “equilibrium without grade in f ation” in the Grade In f ation Signalling Game if and only if: [ θ H θ L ] α β . You were also asked to show that, when this condition fails, then there is a separating equilibrium with grade in f ation, but only the professors of high ability students issue in f ated evaluations. An “equilibrium without grade in f ation” is de F nedasastrategyforeachtypeo fpro - fessor, r = r ( θ ) , and market beliefs, μ = μ ( r ) , such that θ { θ L , θ H } : (i) r ( θ )= θ ; such that r 0 : (ii) μ ( r ) [0 , 1] ,and μ ( θ L )=0=1 μ ( θ H ) ; such that r 0 : (iii) w ( r )= μ ( r ) θ H +(1 μ ( r )) θ L ; andsuchthat r 0 , θ { θ L , θ H } : (iv) αθ α w ( r ) β ( r θ ) 2 . Condition (i) above says that no professor in f ates his grade. Condition (ii) says that beliefs are consistent with Bayes’s Rule whenever possible. Condition (iii) says that, given beliefs, wages equal expected productivity. And, condition (iv) says that for both types of professor, given the market’s beliefs and the associated wages, the level of utility without grade in f ation [ u ( w ( θ ) , θ ; θ )= αθ ] is at least as great as the level of utility from reporting any other grade r [ u ( w ( r ) ,r ; θ )= α w ( r ) β ( r θ ) 2 ]. It follows immediately from the conditions above [taking θ = θ L and r = θ H in (iv)] that in any no-grade-in f ation equilibrium: (v) αθ L αθ H β ( θ H θ L ) 2
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answerseight - Economics 206 Spring 2007(Prof G Loury...

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