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Economics 395 Topics in Risk and Uncertainty Practice Final Examination Question One Fred owns an asset of uncertain value. Ginger may be interested in buying the asset from Fred, if an appropriate price can be found. The possible values it can take are { } . 3 , 2 , 1 , 0 v Fred and Ginger have the same prior: that each of these values is equally likely. Before negotiating over the sale of the asset, Ginger and Fred each receive private information providing more accurate insight into the value of the asset. Fred receives one of two signals, f 1 or f 2 , where: Signal f 1 tells Fred that the asset has value v = 0 Signal f 2 tells Fred that the value of the asset is not 0 Similarly, Ginger receives one of two signals, g 1 or g 2 , where: Signal g 1 tells Ginger that the asset has value v = 3 Signal g 2 tells Ginger that the value of the asset is not 3 The signals that the agents receive are always true. This means that it is impossible for Ginger to receive signal g 1 at the same time that Fred receives the signal f 1 . Also assume that both agents are rational and risk-neutral and that each knows what private information the other might possibly obtain. The actual signal received by either agent will remain private information, not shared directly with the other. (a) If Fred receives signal f 1 , he knows that the value of the asset is 0. If he receives the signal f 2 , he knows that the value is not 0. What probability does Fred place on the value being v = 1, or v = 2, or v = 3 if he receives signal f 2 ? Fred’s prior suggested that v=1, 2 or 3 were equally likely events. After receiving the signal that v is not 0, he still believes that v=1, 2 or 3 are equally likely events. Therefore, P(v=1|f 2 ) = P(v=2|f 2 ) = P(v=3|f 2 ) = 1/3.

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(b) Suppose Fred receives signal f 2 , but also deduces that Ginger’s signal is g 1 . What probability will Fred now place on the asset’s value being v = 1, or v = 2, or v = 3? Calculate the same probabilities in the event that Fred receives signal f 2 and also deduces that Ginger received signal g 2 . If Fred knows his own signal (f 2 ) and Ginger’s signal, then he can make a finer distinction between possible states of the world. If Ginger’s signal is g 1 , then Fred will believe that P(v=3| f 2 , g 1 ) = 1. If Ginger’s signal is g 2 , then he will know that the value of the asset is not 0, nor is it 3. Since his prior told him that values of 1 and 2 were equally likely, he will believe that P(v=1|f 2 , g 2 ) = P(v=2|f 2 , g 2 ) =1/2. (c) Let p be the price at which Fred and Ginger agree to trade. Is it possible that 0 < p < 1? Is it possible that 2 < p < 3? Explain your answers. Suppose that 0 < p < 1. It is clear that Fred will not sell the asset for such a price if he received a signal of f 2 . Such a signal assures him that the value exceeds 1. Therefore, he would never accept a price below 1 for the asset if his signal was f 2 .
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