\- Jamal has a utility function \(U=W^{1 / 2}\), where \(W\) is his wealth in millions of dollars and \(U\) is the utility he obtains from that wealth. In the final stage of a game show, the host offers Jamal a choice between A. 84 million for sure and B. a gamble that pays \(\$ 1\) million with probability 0.6 and $$ 9\( million with probability \)0.4 .\( a. Graph Jamal's utility function. Is he risk averse? Explain. b. Does \)A\( or \)B\( offer Jamal the higher expected prize? Explain your reasoning with appropriate calculations. (Hint: The expected value of a random variable is the weighted average of the possible outcomes, where the probabilities are the weights.) c. Does \)A\( or \)\underline{B}\( offer Jamal the higher expected utility? Again, show your calculations. d. Should Jamal pick \)\underline{\text { dr }} \underline{\mathrm{B}} ?$ Why?

Risk aversion is a key concept in economics and decision theory that reflects the extent to which a person prefers certainty over uncertainty when facing risky prospects. A person who is risk-averse has a strong preference for sure outcomes over gambles with potentially higher but uncertain payoffs. To understand risk aversion, we can take the case of Jamal from the exercise.

When faced with the decision between a sure sum of money and a gamble, Jamal's choice can reveal his risk preferences. His utility function, given by the equation \( U = W^{1/2} \), represents how he values wealth. Because the utility function has a concave shape, it indicates that Jamal derives less utility from additional wealth as his wealth increases. This is characteristic of a risk-averse individual, as they prefer to avoid risks that could result in lower levels of utility, even if there's a chance of achieving higher wealth.

The utility function is a mathematical representation that translates amounts of goods or wealth into levels of happiness or satisfaction, referred to as 'utility'. In the exercise, Jamal's utility function is defined as \( U = W^{1/2} \), where \( W \) is his wealth. This function illustrates how Jamal perceives the value of wealth for different amounts.

Utility functions differ from person to person, as they reflect individual preferences and tolerance for risk. The square root function in Jamal's case, with its concave shape, shows diminishing marginal utility – meaning as Jamal's wealth grows, each additional million provides him with less utility than the previous one.

Within the context of expected utility theory, wealth probability refers to the likelihood of various wealth levels occurring when faced with a risky decision. In Jamal's scenario, he had the option to take a guaranteed \(\$84\) million, or to take a gamble with a \(0.6\) probability of winning \(\$1\) million and a \(0.4\) probability of winning \(\$9\) million.

These probabilities are key to calculating the expected outcomes of risky decisions. They provide the weights in the calculation of expected value or expected utility, which helps individuals like Jamal evaluate the attractiveness of different options under uncertainty.

The expected value is a fundamental concept in probability and statistics used to calculate the average outcome of a random variable, taking into account all possible scenarios and their probabilities. It is the weighted sum of the outcomes, where each outcome is multiplied by its probability of occurrence.

For Jamal's decision in the exercise, the expected value for option A (the sure option) is simply \(\$84\) million. However, for option B (the gamble), the expected value is the sum of the products of each possible wealth amount and its associated probability: \(0.6\) times \(\$1\) million plus \(0.4\) times \(\$9\) million, giving us an expected value of \(\$4.2\) million. Despite this, the expected utility calculation, which incorporates Jamal's risk aversion, may lead to a different recommendation than the expected value alone would suggest.