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sol_ps4_101_2011

# sol_ps4_101_2011 - Economics 101 Solutions to Problem Set 4...

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Economics 101 Solutions to Problem Set 4 Due Tuesday, February 22, 2011 Soojin Kim 1. The certainty equivalent of a lottery (gamble) X is the amount of money for which the individual (with utility for money u and wealth w 0 ) is indifferent between the lottery X and the certain amount c ; that is, Eu ( w 0 + X ) = u ( w 0 + c ) . (The certainty equivalent may be negative.) Suppose Edward has utility function over money u ( w ) = w , and initial wealth w 0 = \$4. Consider the gamble X given by X = ( \$12 , with probabilty 1 3 , - \$4 , with probabilty 2 3 . (a) What is the expected value of the gamble X ? Solution The expected value of the gamble X is E ( X ) = 1 3 · 12 + 2 3 · ( - 4) = 4 3 . (b) What is Edward’s expected utility if he accepts the gamble X ? Solution Since Edward has the initial wealth of \$4, his ex- pected utility is Eu ( w 0 + X ) = 1 3 u (4 + 12) + 2 3 u (4 - 4) = 4 3 . (c) What is Edward’s certainty equivalent of the above gamble? Solution By definition, we want to find c such that Eu ( w 0 + X ) = 4 3 = 4 + c = u ( w 0 + c ) is satisfied. Thus, c = - 20 9 . (d) Discuss the relationship between the expected value of the gam- ble, Edward’s certainty equivalent of the gamble, and his atti- tudes towards risk. 1

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Solution Notice that expected value of the gamble is positive and Edward’s utility from getting expected value of the gamble is u (4 + 4 3 ) = q 16 3 2 . 03 which is greater than the expected utility from the gamble ( Eu ( w 0 + X ) = 4 3 ). This is due to the fact that Edward is risk averse (notice that his utility function is concave). Moreover, his certainty equivalent of the gamble is - 20 9 , which implies that Edward is indifferent between taking the gamble and being taken \$ 20 9 away from him. (Also refer to Figure
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