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eco331h2solution - HOMEWORK 2 SOLUTIONS Northwestern...

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HOMEWORK 2 SOLUTIONS Northwestern University Marciano Siniscalchi Fall 2008 Econ 331-0 1. Forecasting and Random Variables The individual minimizes s P ( { s } )[ X ( s ) - ξ ] 2 with respect to ξ . Differentiating with respect to this variable yields s P ( { s } )2[ X ( s ) - ξ ]( - 1) = 0; dividing by two, recognizing that s P ( { s } ) ξ = ξ and rearranging terms yields the required result. 2. Forecasting and Expected Utility Suppose π is not admissible, so there is ρ such that L ρ ( s ) < L π ( s ) for all s S . Then W - L ρ ( s ) > W - L π ( s ) for all s , so u ( W - L ρ ( s )) > u ( W - L π ( s )) for all s because u is strictly increasing. But then E[ u ( W - L ρ )] > E[ u ( W - L π )], so the individual strictly prefers to provide the forecast system ρ rather than the forecast system π . It follows that, if a forecast system π maximizes the individual’s expected utility, it must be admissible. 3. Expected Utility Calculations (i) Write U ( Z ) for the expected utility of the generic lottery Z . Then we have U ( X ) = 1 3 100 + 1 3 50 + 1 3 0 5 . 69036; the certainty equivalet C[ X ] of X
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