Show that if an individual's utility-of-wealth function is convex (rather than concave, as shown in Figure 8.1 ), he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

A utility-of-wealth function represents the satisfaction derived from a given level of wealth. A convex utility function indicates that the individual's utility increases at an increasing rate. In contrast, a concave utility function, as shown in Figure 8.1, indicates that the individual's utility increases at a decreasing rate.

If the utility function is convex, the individual derives greater utility from wealth increases. This implies that the individual would prefer the potential for larger wealth increases in a gamble over a fixed, less-variable income. Let's use a simple example to illustrate this. Suppose an individual has the option between a certain income of \(X\), and a fair gamble where they have a 50% chance of winning \(2X\) and a 50% chance of losing \(X\). The expected value of the gamble is also \(X\). However, with a convex utility function, the utility derived from winning \(2X\) would be substantially more significant than the disutility from losing \(X\). The individual would, therefore, prefer the gamble over the guaranteed income of \(X\).

Similarly, an individual with a convex utility function would be more likely to accept unfair gambles because the potential utility derived from the larger wealth increases outweighs the disutility from losses. This risk-taking behavior will depend on the level of convexity of the individual's utility function, as well as their beliefs about the probability distribution of possible outcomes.

This sort of risk-taking behavior can be observed in certain circumstances, such as in entrepreneurship, investments, or gambling activities. However, not all individuals exhibit such risk-seeking behavior due to various factors. Factors that might limit risk-taking behavior include: 1. Personal preferences and risk tolerance: Some individuals have a more conservative attitude towards risk-taking and prefer stable and secure incomes to the possibility of large fluctuations in wealth. 2. Financial constraints and responsibilities: Individuals with financial obligations or limited resources may not be able to afford potential losses, thus limiting their exposure to risk. 3. Information and knowledge: A lack of understanding about potential returns and risks associated with certain activities may dissuade individuals from participating in them. 4. Market regulations and restrictions: Laws and policies aimed at reducing risk in financial markets can influence the decision-making process of individuals and limit the occurrence of risk-taking behavior. In conclusion, while the convex utility-of-wealth function may lead to risk-seeking behavior in some cases, personal preferences, financial constraints, knowledge, and market regulations can limit the occurrence of such behavior.