Jamal has a utility function U=W^1 / 2, where W is his...

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. $\$ 4$ 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 $\underline{\text { A }}$ or $\underline{\mathrm{B}}$ offer Jamal the higher expected utility? Again, show your calculations. d. Should Jamal pick A or B? Why?

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. $\$ 4$ 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 $\underline{\text { A }}$ or $\underline{\mathrm{B}}$ offer Jamal the higher expected utility? Again, show your calculations.
d. Should Jamal pick A or B? Why?

Key Concepts

Risk Aversion

Risk aversion is a property of an individual's preferences where they prefer a certain outcome over a risky option with the same expected monetary value. This behavior is typically represented by a concave utility function, indicating that additional units of wealth yield diminishing increases in utility (satisfaction).

Concave Utility Function

A concave utility function is characterized mathematically by a negative second derivative, which reflects diminishing marginal utility. This means that each additional increment of wealth provides less added utility than the previous one, a key indicator of risk-averse behavior.

Expected Value

The expected value is a fundamental statistical concept that computes the average outcome of a random process by weighting each possible outcome by its probability. It serves as a baseline measure to compare certain versus uncertain prospects.

Expected Utility

Expected utility theory extends the idea of expected value by incorporating an individual's utility function, allowing for the assessment of choices under risk not solely based on monetary outcomes but on the satisfaction derived from those outcomes. This framework is crucial for making decisions when facing uncertain outcomes, as it accounts for both the probabilities of outcomes and the decision-maker's risk preferences.

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