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Dracula - Econ 210A Problem Set 6 Answer Key Jehle and Reny...

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Econ 210A Problem Set 6 Answer Key Jehle and Reny Exercise 2.19 Using the de°nition of risk aversion given in the text, prove that an individual is risk averse over gambles involving nonnegative wealth levels if and only if her VNM utility funciton is strictly concave on R + . Suppose that an individual is risk averse, which implies that u ( E ( g )) > u ( g ) : u ( E ( g )) > u ( g ) ) u p i w i ) > ° p i u ( w i ) ° u ( p 1 w 1 + ::: + p n w n ) > p 1 u ( w 1 ) + ::: + p n u ( w n ) This holds for all p 0 s that are between 0 and 1 and sum to one and u ( p 1 w 1 + ::: + p n w n ) ° u ( w t ) and p 1 u ( w 1 ) + ::: + p n u ( w n ) ° T 1 u ( w 1 ) + ::: + T n u ( w n ) Hence, u ( w t ) > T 1 u ( w 1 ) + ::: + T n u ( w n ) which implies u is strictly concave. , Suppose that a consumer has a concave utility function over all wealth levels that are positive. This implies that u ( E ( g )) = u n i p i w i ) > ° n i p i w i = u ( g ) . Therefore the consumer is risk averse. 2.23 Consider the quadratic VNM utility function U ( w ) = a + bw + cw 2 . a) What restrictions if any must be placed on parameters a; b and c for this function to display risk aversion? Risk aversion is characterized by the utility function when U 0 ( w ) > 0 and U 00 ( w ) < 0 . U 0 ( w ) = b + 2 cw > 0 U 00 ( w ) = 2 c < 0 ) c < 0 b) Over what domain of wealth can a quadratic VNM utility function be de°ned? In order for the utility function to be de°ned, U 0 ( w ) > 0 ) b + 2 cw > 0 or w < ± b 2 c c) Given the gamble g = ((1 = 2) ( w + h ) ; (1 = 2) ( w ± h )) show that CE < E ( g ) and P > 0 . U ( CE ) = 1 2 U ( w + h ) + 1 2 U ( w ± h ) ) a + bCE + cCE 2 = 1 2 ° a + b ( w + h ) + c ( w + h ) 2 ± + 1 2 ° a + b ( w ± h ) + c ( w ± h ) 2 ± ) u ( CE ) = a + b ( CE ) + c ( CE ) 2 = a + bw + cw 2 + ch 2 > u ( w ) = u ( E ( g )) Because utility is strictly increasing, CE < E ( g ) . P = E ( g ) ± CE is then greater than 0. d) Show that this function, satisfying the restrictions in part (a), cannot represent preferences that display decreasing absolute risk aversion. Risk aversion R a ( w ) = ± u 00 ( w ) u 0 ( w ) = ± c b +2 cw dR a ( w ) dw = 4 c 2 ( b +2 cw ) 2 > 0 , which implies that R a ( w ) is increasing in wealth and cannot represent decreasing absolute risk aversion. 1
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2.24 Let u ( w ) = ± ( b ± w ) c . What restrictrions on w; b; and c are required to ensure that u ( w ) is strictly increasing and strictly concave? Show that under those restrictions, u ( w ) displays increasing absolute risk aversion.
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