quiz1solution - 1 Quiz 1 Solutions ECOS3012 Strategic...

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1 Quiz 1 Solutions ECOS3012 - Strategic Behavior, Semester 1, 2011 Note. Here is the sketch of solutions to Quiz 1. I will put up the explanation for Q7 a bit later – it is easiest to explain with a couple of pictures and I am not able to draw those on this computer. Begin Quiz Q1. Assume that there are two states of the world and a typical action is denoted by a vector ( x,y ) where x 0 and y 0 denotes monetary reward in the two states. Consider the utility functions U ( x,y ) = x α y 1 / 2 , V ( x,y ) = α log( x ) + β log( y ) where 0 < α < β < 1. A decision maker is necessarily consistent with the expected utility hypothesis if her preferences are give by the utility (a) Either U or V. (b) Neither U nor V. (c) Only U. (d) Only V. Solution : See solution to Q1, Problem Set 1 to note that the preferences of a DM that are described by U will be the as thsoe of a DM whose utility function is given by U 0 , where U 0 ( x,y ) = α α + 1 / 2 log( x ) + 1 / 2 α + 1 / 2 log( y ) Similarly both V and V 0 both describe the same preferences where V 0 ( x,y ) = α α + β log( x ) + β α + β log( y ) . Both U 0 and V 0 are utility functions that would conform to a DM who satisfies Expected Utiltiy Hypothesis. ± Q2. Suppose there are three states of the world and a typical action is given by a triple ( x,y,z ), denoting the monetary reward in the three states of the world. Consider two actions A = ( x,y,z ) and B = (ˆ x, ˆ y,z ). ( Both actions give the same reward in state 3). Now consider any other actions C = ( x,y,z * ) and D = (ˆ x, ˆ y,z * ) obtained from A and B by changing the reward in state 3. Suppose a Decision Maker (DM) is known to satisfy Expected Utility Hypothesis and it is known that she strictly prefers A to B. (a) DM must strictly prefer C to D. (b) DM may strictly prefer C to D but depends on the value of z * . (c) DM may prefer C to D but this depends on payoffs in the first two states (d) None of the above. Solution : This is straightforward. If the Expected utility hypothesis holds, then there must be a probability distribution ( p 1 ,p 2 ,p 3 , and a utility function u such that U ( A ) = p 1 u ( x ) + p 2 u ( y ) + p 3 u ( z ) (1) U ( B ) = p 1 u ( x 0 ) + p 2 u ( y 0 ) + p 3 u ( z ) (2) U ( C ) = p 1 u ( x
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This note was uploaded on 08/20/2011 for the course ECON 101 taught by Professor Etw during the Spring '11 term at Università di Bologna.

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quiz1solution - 1 Quiz 1 Solutions ECOS3012 Strategic...

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