Hugh's income is currently 5,000. His utility function is...

Hugh's income is currently $\$ 5,000$. His utility function is shown in the accompanying table. a. Calculate Hugh's marginal utility of income. What is his attitude toward risk? b. Hugh is thinking about gambling in a casino. With a probability of 0.5 he will lose $\$ 3,000,$ and with a probability of 0.5 he will win $\$ 5,000$. What is the expected value of Hugh's income? What is Hugh's expected utility? Will he decide to gamble? (Suppose that he gets no extra utility from going to the casino.) c. Suppose that the "spread" (how much he can win versus how much he can lose) of the gamble narrows, so that with a probability of 0.5 Hugh will lose $\$ 1,000,$ and with a probability of 0.5 he will win $\$ 3,000$. What is the expected value of Hugh's income? What is his expected utility? Is this gamble better for him than the gamble in part b? Will he decide to gamble?

Hugh's income is currently $\$ 5,000$. His utility function is shown in the accompanying table.
a. Calculate Hugh's marginal utility of income. What is his attitude toward risk?
b. Hugh is thinking about gambling in a casino. With a probability of 0.5 he will lose $\$ 3,000,$ and with a probability of 0.5 he will win $\$ 5,000$. What is the expected value of Hugh's income? What is Hugh's expected utility? Will he decide to gamble? (Suppose that he gets no extra utility from going to the casino.)
c. Suppose that the "spread" (how much he can win versus how much he can lose) of the gamble narrows, so that with a probability of 0.5 Hugh will lose $\$ 1,000,$ and with a probability of 0.5 he will win $\$ 3,000$. What is the expected value of Hugh's income? What is his expected utility? Is this gamble better for him than the gamble in part b? Will he decide to gamble?

Key Concepts

Expected Utility

Expected utility extends the idea of expected value by applying the utility function to each outcome before averaging. It represents the weighted average of the utilities of all possible outcomes, rather than the monetary amounts, and is used to capture the decision-maker's risk preferences in situations involving uncertainty.

Utility Function

A utility function represents an individual's preferences over different outcomes, assigning a numerical value (utility) to each outcome which reflects the satisfaction or benefit derived from that outcome. It is used to analyze decision-making under uncertainty by comparing the utility of various income levels or outcomes rather than their raw monetary values.

Marginal Utility

Marginal utility is the additional satisfaction or utility gained from an incremental increase in income or consumption. In the context of income, it measures how much utility increases when income is raised by a small amount, and its calculation can reveal the rate at which additional income contributes to overall well?being.

Risk Attitude

Risk attitude refers to an individual's predisposition towards risk when making decisions under uncertainty. It is determined by the shape of the utility function: a concave utility function indicates risk aversion (preferring a certain outcome over a risky prospect with the same expected value), a linear function indicates risk neutrality, and a convex function indicates risk seeking.

Expected Value

The expected value is a statistical concept calculated as the weighted average of all possible monetary outcomes, with weights corresponding to their probabilities. It provides a measure of the central tendency of the possible outcomes, serving as a benchmark in evaluating the financial desirability of uncertain prospects.

Decision Making under Risk

Decision making under risk involves evaluating uncertain prospects by comparing expected utilities rather than just expected monetary values. This approach accounts for both the probabilities of outcomes and the individual's risk attitude, ensuring that the decision reflects the trade-off between potential gains and losses, as well as the satisfaction derived from different outcomes.

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