Consider an expected utility maximiser with utility function u(w) = w^a, a > 0. Verify whether this individual always rejects the lottery {(w + 110), 0.5; (w −100), 0.5} for all wealth level. note - answer this question completely in detail, do not leave any calculations incomplete
Question
Consider an expected utility maximiser with utility function u(w) = w^a, a > 0. Verify whether this individual always rejects the lottery {(w + 110), 0.5; (w −100), 0.5} for all wealth level. note - answer this question completely in detail, do not leave any calculations incomplete
Solution
The question is asking whether an individual with a certain utility function would always reject a specific lottery, regardless of their wealth level. The utility function is u(w) = w^a, where a > 0, and the lottery is {(w + 110), 0.5; (w -100), 0.5}.
The expected utility of the lottery can be calculated as follows:
E[u(w)] = 0.5u(w + 110) + 0.5u(w - 100)
Substituting the given utility function into the equation, we get:
E[u(w)] = 0.5*(w + 110)^a + 0.5*(w - 100)^a
To determine whether the individual would always reject the lottery, we need to compare the expected utility of the lottery with the utility of the individual's current wealth level, u(w). If the expected utility of the lottery is less than the utility of the current wealth level, the individual would reject the lottery.
So, we need to solve the inequality:
u(w) > E[u(w)]
Substituting the given utility function and the calculated expected utility into the inequality, we get:
w^a > 0.5*(w + 110)^a + 0.5*(w - 100)^a
This inequality is not straightforward to solve analytically due to the non-linearity introduced by the exponent a. However, it can be solved numerically for different values of w and a.
If the inequality holds for all values of w and a > 0, then the individual would always reject the lottery. If not, there are some wealth levels at which the individual would accept the lottery.
Please note that this is a simplified analysis and actual decision-making may be influenced by other factors such as risk aversion, the individual's wealth level relative to the lottery amounts, and the individual's subjective valuation of the utility of wealth.
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