Economics+395+Fall+2008+problem+set+5

Economics+395+Fall+2008+problem+set+5 - Economics 395 Risk...

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Economics 395 Risk and Uncertainty Problem Set 5 Question Eighteen Adam owns an asset that has uncertain value. Its value, however, is bound to be $100, $200, $300 or $400. In the absence of any relevant information, each of these values is deemed equally likely. If we denote the asset’s value as V, then the prior probability distribution of V is P(V=$100) = P(V=$200) = P(V=$300) = P(V=$400) = ¼. The expected value of the asset is E(V) = $250. Both Adam and his friend Eve are able to gather private information about the value of the asset. But as they have different sources of information, they learn different things: Adam receives a signal that tells him if the asset will either have a “low” value (i.e. $100 or $200) or a “high” value (i.e. $300 or $400); Eve receives a signal that tells her if the asset will have a “square” value (i.e. $100 or $400) or a “non-square” value (i.e. $200 or $300). We wish to investigate the possibility that private information could motivate trade between Adam and Eve. (a) If Adam receives a “high” signal, what is his posterior distribution over possible values of the asset? What is his expected value of the asset, conditional on his private information? If he receives a “low” signal, what is Adam’s posterior distribution over possible values of the asset? What is his expected value of the asset, conditional on his private information? (b) If Eve receives a “square” signal, what is her posterior distribution over possible values of the asset? What is her expected value of the asset, conditional on her private information? If she receives a “non-square” signal, what is Eve’s posterior distribution over possible values of the asset? What is her expected value of the asset, conditional on his private information? (c) Suppose the price of asset was strictly less than $300 (P < $300). Based purely on his private information, under what circumstances would Adam refuse to sell the asset? If Adam finds trade acceptable at P < $300, what should Eve infer about Adam’s signal? (d) Suppose the price of the asset is strictly between $200 and $300 (i.e. $200 < P < $300). If Adam finds trade acceptable at this price, we already know that Eve infers something explicit about Adam’s signal, and therefore about the value of
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the asset (from part (c)). Would Eve ever wish to trade at such a price? Explain why. (e) Suppose the price of the asset is strictly between $100 and $200 (i.e. $100 < P < $200). Again, if Adam were to find the trade acceptable at this price, that would communicate information to Eve. In addition, Eve has her own private information. Under what circumstances would she immediately reject trade at a price such as this? If she were to find trade acceptable at this price, what should Adam infer about Eve’s private signal? (f) Given the inferences you made in part (e), would Adam be prepared to accept a
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Economics+395+Fall+2008+problem+set+5 - Economics 395 Risk...

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