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Problem Set 6 Solution 1. Calculate Pratt’s risk aversion for the following functions: a. ()axu x e at 10x 2(10)axaxaeraae b. 2( ) lnu x x at 15x 221()21(15)15xrxxxr c. 5()u x x at 7x 4320 4()54(7 )7xrxxxr 2. A wealthy individual has hired you to analyze an investment project. The project consists on building a luxurious hotel for a total cost of $500 million. The project is going to be a total success with probability 30%, a mild success with probability 60% and a total failure with probability of 10%. The hotel will run for a hundred years. If a total success, the hotel is going to produce $60 million a year, if a mild success $50 million a year and only $30 million a year if it fails. The interest rate is 10%. Let x be the total wealth in present value of this individual (in millions). The total wealth of this individual (before hotel) is $600 million. Discount Factor 11 10% Present Value of Success: 100160 659.951 Present Value of Mild Success: 100150 549.961 Present Value of Failure: 100130 329.981 Net Present Value of Success: 600 – 500 + 659.95 = 759.95 Net Present Value of Mild Success: 600 – 500 + 549.96 = 649.96 Net Present Value of Failure: 600 – 500 + 329.98 = 429.98 a. If the utility function of this individual is ()u x x, should he build the hotel? Utility of NOT building the Hotel (600) 600u Problem Set 6 Solution Utility of building the Hotel: 30%759+60%649+10%429=661 He should build the Hotel b. If the utility function of this individual is ( ) lnu x x, should he build the hotel? Utility of NOT building the Hotel (600) ln600 6.39u Utility of building the Hotel: 30%ln(759)+60%ln(649)+10%ln(429)=6.48 He should build the Hotel 3. Consider the following game: U D L (1,1) (-1,-1) R (-1,-1) (1,1) a. Is there a dominant strategy? No, no player has a dominant strategy b. Is there a never best response strategy? No, there is no never-a-best-response strategy c. Find all pure strategy Nash Equilibria (L,U) and (R,D) d. Find all mixed strategy Nash Equilibria (1 ) (1 )(1 )12p p p pppp Player 1 plays L with 50% chance and R with 50% chance Player 2 plays U with 50% chance and D with 50%

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