6. (15) A individual, with a utility function InW and an initial wealth level Wo, is faced with a fair gamble of winning or losing Sh (where Wo > h> 0) with 50-50 chance. (a) Is this individual risk averse? Explain. (b) Suppose that the individual is willing to pay up to an amount of f in order to avoid such a gamble. Give the equation that determines f, and solve the equation for f (i.e., express fin terms of Wo and h). (c) Show that fincreases as h increases.
a) An individual who always refuses fair bets is said to be risk averse. If individuals
exhibit a diminishing marginal utility of wealth, they will be risk averse. As a consequence, they will
be willing to pay something to avoid taking fair bets. To illustrate the connection between risk aversion and insurance, consider a person with a
current wealth of $100,000 who faces the prospect of a 25 percent chance of losing his or her
$20,000 automobile through theft during the next year. Suppose also that this person’s von
Neumann–Morgenstern utility function is logarithmic; that is, U(W ) ¼ ln (W ).
If this person faces next year without insurance, expected utility will be EU ( no insurance) = 0.75U (100, 000) + 0.25U (80, 000)
0.75ln 100,000 + 0.25ln 80,000= 11.45714
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